Analysis and Management of Animal Populations
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Analysis and Management of Animal Populations Modeling, Estimation, and Decision Making
Byron K. Williams U.S. Geological Survey Cooperative Research Units Reston, Virginia
James D. Nichols
Michael J. Conroy
U.S. Geological Survey Patuxent Wildlife Research Center Laurel, Maryland
Cooperative Fish and Wildlife Research Unit DB Warnell School of Forest Resources University of Georgia Athens, Georgia
ACADEMIC PRESS An Imprint of Elsevier San Diego San Francisco New York Boston London Sydney Tokyo
Cover images: Top three images, @ 2001 PhotoDisc, Inc. Bottom image, @ 2001, Joe Lange This book is printed on acid-free paper. Copyright 9 2002 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Science and Technology Rights Department in Oxford, UK. Phone: (44) 1865 843830, Fax: (44) 1865 853333, e-mail:
[email protected]. You may also complete your request on-line via the Elsevier homepage: http://www.elsevier.com by selecting "Customer Support" and then "Obtaining Permissions". Academic Press An Imprint of Elsevier 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http: / / www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 ,'BY, UK http: / / www.academicpress.com Library of Congress Catalog Card Number: 2001094375 ISBN-13" 978-0-12-754406-9 ISBN-10:0-12-754406-2 PRINTED IN THE UNITED STATES OF AMERICA 06 07 EB 9 8 7 6 5 4
To my parents, Roger S. (deceased) and Mary F. Williams; my wife Genie; and my daughters ]aimin and Shannon. Byron K. Williams
To my parents, James E. and Barbara Irwin Nichols; and to Walt Conley, mentor and friend. James D. Nichols
To the memory of my parents, Edith M. and James R. Conroy. Michael J. Conroy
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Contents
Preface xiii Acknowledgments xvii
2.4.
2.5. 2.6. 2.7. 2.8.
PART
Hypothesis Confirmation 16 Inductive Logic in Scientific Method 17 Statistical Inference 18 Investigating Complementary Hypotheses Discussion 19
18
I FRAMEWORK FOR MODELING, ESTIMATION, A N D MANAGEMENT OF ANIMAL POPULATIONS
CHAPTER
3 Models and the Investigation of Populations
CHAPTER
3.1.
3.2. 3.3. 3.4. 3.5. 3.6.
1 Introduction to Population Ecology Some Definitions 3 1.2. Population Dynamics 4 1.3. Factors Affecting Populations 4 1.4. Management of Animal Populations 1.5. Individuals, Fitness, and Life History Characteristics 7 1.6. Community Dynamics 9 1.7. Discussion 9 1.1.
Types of Biological Models 22 Keys to Successful Model Use 22 Uses of Models in Population Biology 23 Determinants of Model Utility 28 Hypotheses, Models, and Science 30 Discussion 31
CHAPTER
4 Estimation and Hypothesis Testing in Animal Ecology
CHAPTER
2
4.1.
Scientific Process in Animal Ecology 2.1.
2.2. 2.3.
Causation in Animal Ecology 11 Approaches to the Investigation of Causes Scientific Methods 13
4.2. 4.3. 4.4. 4.5. 4.6.
12
vii
Statistical Distributions 34 Parameter Estimation 42 Hypothesis Testing 50 Information-Theoretic Approaches 55 Bayesian Extension of Likelihood Theory Discussion 58
57
viii
Contents CHAPTER
CHAPTER
5
8
Survey Sampling and the Estimation of Population Parameters
Traditional Models of Population Dynamics 8.1.
5.1.
5.2. 5.3. 5.4.
5.5. 5.6.
Sampling Issues 60 Features of a Sampling Design 61 Simple Random and Stratified Random Sampling 62 Other Sampling Approaches 67 Common Problems in Sampling Designs Discussion 76
74
CHAPTER
6
Density-Independent Growth--The Exponential Model 136 8.2. Density-Dependent GrowthmThe Logistic Model 139 8.3. Cohort Models 141 8.4. Models with Age Structure 143 8.5. Models with Size Structure 157 8.6. Models with Geographic Structure 159 8.7. Lotka-Volterra Predator-Prey Models 161 8.8. Models of Competing Populations 164 8.9. A General Model for Interacting Species 170 8.10. Discussion 171
Design of Experiments in Animal Ecology 6.1.
6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9.
Principles of Experimental Design 80 Completely Randomized Designs 83 Randomized Block Designs 89 Covariation and Analysis of Covariance 91 Hierarchical Designs 92 Random Effects and Nested Designs 97 Statistical Power and Experimental Design 100 Constrained Experimental Designs and Quasi-Experiments 102 Discussion 106
CHAPTER
9 Model Identification with Time Series Data 9.1. 9.2.
9.3. 9.4. 9.5. 9.6.
PART
II DYNAMIC MODELING OF ANIMAL POPULATIONS
9.7.
9.8. 9.9.
CHAPTER
7.1.
7.2. 7.3. 7.4. 7.5. 7.6. 7.7.
Model Identification Based on Ordinary Least Squares 174 Other Measures of Model Fit 176 Correlated Estimates of Population Size 178 Optimal Identification 178 Identifying Models with Population Size as a Function of Time 179 Identifying Models Using Lagrangian Multipliers 181 Stability of Parameter Estimates 181 Identifying System Properties in the Absence of a Specified Model 182 Discussion 184
CHAPTER
7
10
Principles of Model Development and Assessment
Stochastic Processes in Population Models
Modeling Goals 113 Attributes of Population Models 114 Describing Population Models 117 Constructing a Population Model 122 Model Assessment 126 A Systematic Approach to the Modeling of Animal Populations 131 Discussion 134
10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9.
Bernoulli Counting Processes 189 Poisson Counting Processes 192 Discrete Markov Processes 197 Continuous Markov Processes 202 Semi-Markov Processes 205 Markov Decision Processes 207 Brownian Motion 210 Other Stochastic Processes 213 Discussion 220
Contents
ix CHAPTER
CHAPTER
11
14
The Use of Models in Conservation and Management
Estimating Abundance for Closed Populations with Mark-Recapture Methods
11.1. 11.2. 11.3.
Dynamics of Harvested Populations Conservation and Extinction of Populations 231 Discussion 237
223
PART
III
Two-Sample Lincoln-Petersen Estimator 290 K-Sample Capture-Recapture Models 296 Density Estimation with Capture-Recapture 314 14.4. Removal Methods 320 14.5. Change-in-Ratio Methods 325 14.6. Discussion 331
14.1. 14.2. 14.3.
ESTIMATION M E T H O D S FOR A N I M A L POPULATIONS CHAPTER
15
CHAPTER
12
Estimation of Demographic Parameters
Estimating Abundance Based on Counts Detectability and Demographic Rate Parameters 334 15.2. Analysis of Age Frequencies 337 15.3. Analysis of Discrete Survival and Nest Success Data 343 15.4. Analysis of Failure Times 351 15.5. Random Effects and Known-Fate Data 361 15.6. Discussion 362 15.1. 12.1. Overview of Abundance Estimation 242 12.2. A Canonical Population Estimator 243 12.3. Population Censuses 245 12.4. Complete Detectability of Individuals on Sample Units of Equal Area 245 12.5. Complete Detectability of Individuals on Sample Units of Unequal Area 247 12.6. Partial Detectability of Individuals on Sample Units 250 12.7. Indices to Population Abundance or Density 257 12.8. Discussion 261
CHAPTER
16 Estimation of Survival Rates with Band Recoveries
CHAPTER
13 Estimating Abundance with Distance-Based Methods 13.1. 13.2. 13.3. 13.4.
Point-to-Object Methods 263 Line Transect Sampling 265 Point Sampling 278 Design of Line Transect and Point Sampling Studies 281 13.5. Other Issues 286 13.6. Discussion 287
Single-Age Models 366 Multiple-Age Models 383 Reward Studies for Estimating Reporting Rates 391 16.4. Analysis of Band Recoveries for Nonharvested Species 398 16.5. Poststratification of Recoveries and Analysis of Movements 402 16.6. Design of Banding Studies 406 16.7. Discussion 414
16.1. 16.2. 16.3.
x
Contents 20.3.
CHAPTER
17
20.4.
Estimating Survival, Movement, and Other State Transitions with Mark-Recapture Methods 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7.
Single-Age Models 418 Multiple-Age Models 438 Multistate Models 454 Reverse-Time Models 468 Mark-Recapture with Auxiliary Data Study Design 489 Discussion 492
Estimating Parameters of Community Dynamics 561 Discussion 572
PART
IV D E C I S I O N ANALYSIS FOR ANIMAL POPULATIONS 476 CHAPTER
21 Optimal Decision Making in Population Biology
CHAPTER
18 Estimating Abundance and Recruitment with Mark-Recapture Methods 18.1. 18.2. 18.3. 18.4. 18.5. 18.6. 18.7.
Data Structure 496 Jolly-Seber Approach 497 Superpopulation Approach 508 Pradel's Temporal Symmetry Approach Relationships among Approaches 518 Study Design 520 Discussion 522
511
21.1. Optimization and Population Dynamics 578 21.2. Objective Functions 579 21.3. Stationary Optimization under Equilibrium Conditions 579 21.4. Stationary Optimization under Nonequilibrium Conditions 580 21.5. Discussion 581
CHAPTER
22 CHAPTER
Traditional Approaches to Optimal Decision Analysis
19 Combining Closed and Open Mark-Recapture Models: The Robust Design 19.1. 19.2. 19.3. 19.4. 19.5. 19.6.
Data Structure
524 529 Likelihood-Based Approach 535 Special Estimation Problems 538 Study Design 552 Discussion 553
Ad Hoc Approach
22.1. 22.2. 22.3. 22.4. 22.5. 22.6.
The Geometry of Optimization 584 Unconstrained Optimization 585 Classical Programming 593 Nonlinear Programming 597 Linear Programming 601 Discussion 606
CHAPTER
23 CHAPTER
2O Estimation of Community Parameters 20.1. An Analogy between Populations and Communities 556 20.2. Estimation of Species Richness 557
Modem Approaches to Optimal Decision Analysis 23.1. Calculus of Variations 608 23.2. Pontryagin's Maximum Principle 23.3. Dynamic Programming 627 23.4. Heuristic Approaches 638 23.5. Discussion 639
618
Contents
Appendix C Differential Equations 693 C.1. First-Order Linear Homogeneous Equations C.2. Nonlinear Homogeneous Equations m Stability Analysis C.3. Graphical Methods
CHAPTER
24 Uncertainty, Learning, and Decision Analysis 24.1. Decision Analysis in Natural Resource Conservation 644 24.2. General Framework for Decision Analysis 649 24.3. Uncertainty and the Control of Dynamic Resources 650 24.4. Optimal Control with a Single Model 651 24.5. Optimal Control with Multiple Models 652 24.6. Adaptive Optimization and Learning 653 24.7. Expected Value of Perfect Information 654 24.8. Partial Observability 655 24.9. Generalizations of Adaptive Optimization 656 24.10. Accounting for All Sources of Uncertainty 658 24.11. "Passive" Adaptive Optimization 658 24.12. Discussion 660 CHAPTER
25 Case Study: Management of the Sport Harvest of North American Waterfowl 25.1. Background and History 664 25.2. Components of a Regulatory Process 667 25.3. Adaptive Harvest Management 671 25.4. Modeling Population Dynamics 672 25.5. Harvest Objectives 676 25.6. Regulatory Alternatives 677 25.7. Identifying Optimal Regulations 679 25.8. Some Ongoing Issues in Waterfowl Harvest Management 680 25.9. Discussion 684
Appendix A Conditional Probability and Bayes' Theorem
xi
685
Appendix B Matrix Algebra 687 B.1. Definitions B.2. Matrix Addition and Multiplication B.3. Matrix Determinants B.4. Inverse of a Matrix B.5. Orthogonal and Orthonormal Matrices B.6. Trace of a Matrix B.7. Eigenvectors and Eigenvalues B.8. Linear and Quadratic Forms B.9. Positive-Definite and Semidefinite Matrices B.10. Matrix Differentiation
Appendix D Difference Equations 709 D.1. First-Order Linear Homogeneous Equations D.2. Nonlinear Homogeneous EquationswStability Analysis Appendix E Some Probability Distributions and Their Properties 721 E.1. Discrete Distributions E.2. Continuous Distributions Appendix F Methods for Estimating Statistical Variation 733 Distribution-Based Variance Estimation El. Empirical Variance Estimation E2. Estimating Variances and Covariances with the E3. Information Matrix Approximating Variance with the Delta Method E4. Jackknife Estimators of Mean and Variance E5. Bootstrap Estimation E6. Appendix G Computer Software for Population and Community Estimation 739 G.1. Estimation of Abundance and Density for Closed Populations G.2. Estimation of Abundance and Demographic Parameters for Open Populations G.3. Estimation of Community Parameters G.4. Software Availability Appendix H The Mathematics of Optimization 745 Unconstrained Optimization H.1. H.2. Classical Programming H.3. Nonlinear Programming H.4. Linear Programming H.5. Calculus of Variations H.6. Pontryagin's Maximum Principle H.7. Dynamic Programming
References 767 Index 793
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Preface
This book deals with the assessment and management of animal populations. It is an attempt to pull together key elements of what has become a truly overwhelming body of theory and practice in population biology and to add by way of synthesis to our understanding of animal populations and their conservation. Such an effort requires perambulations through the sometimes strange worlds of mathematical modeling, probability theory, statistical estimation, dynamic optimization, and even logical inference. Happily, one need not establish residence in any of these places to absorb what is needed for the journey. On the other hand, one is well served by a visit and by spending at least some time exploring the terrain. The overarching theme of this book is that modeling, attribute estimation, and optimal decision making are linked together in the doing of science-based conservation. Models play key roles in both the science and management of biological systems, as expressions of biological understanding, as engines for deductive inference, and as articulations of biological response to management and environmental change. These roles are supported by the principles of sampling design and statistical inference, which focus on the use of field observations to identify and calibrate models according to their purposes and objectives. Both modeling and statistical assessment are key elements in formal decision analysis, which utilizes model-based predictions along with measures of sampling variation and other stochastic factors to support informed decision making. These thematic elements form the basis for Parts II-IV of the book. We are concerned here with animal populations, recognizing that population biology ultimately must be understood in a broader context of the habitats and communities of which populations are a part. We build on the notion of a population as a partially self-regulating ecological unit composed of potentially inter-
breeding individuals, with characteristics such as birth rate, death rate, age structure, and dispersion pattern through space and time. Our focus is on dynamic populations, in a context of interspecific interactions and environmental influences. It is through a complex network of biotic and abiotic influences that individuals choose and defend territories, select mates and engage in reproduction, compete for food resources, and avoid predators during the course of their life histories. In this book, we emphasize the processes of birth, growth, reproduction, maturation, and death, with the idea that these processes effectively integrate the influences of the biotic and physical environment and thus permit inferences about individual fitness and population status. The combined influences of structure and feedback among species and their habitats can lead to complicated patterns of change, and the attempt to represent these patterns precisely often results in complex models with large numbers of interactions and feedbacks. These in turn can give rise to certain analytic difficulties and an inability to recognize biological features that control population change. In the ensuing chapters, we address the modeling and assessment of animal populations in light of tradeoffs between understanding and complexity, accounting for model realism, precision, and generality. We acknowledge a bias for parsimony in the modeling of biological phenomena, in keeping with the principle that among acceptable alternatives the simpler explanation is preferred over its more complicated alternative. We believe that population modeling is especially useful when models are developed in a context of conservation and management, recognizing that the range of management practices is bewilderingly wide and often difficult to address systematically. Assessment of impacts can be a considerable challenge when the actions involved are as diverse as manipulation of habi-
xiii
xiv
Preface
tats, control of competition and predation, stocking of individuals, and, for those species subject to sport a n d / o r commercial harvest, the regulation of harvests. We provide the reader with examples of population assessment framed in a context of wildlife conservation and management. Animal ecologists often justify their work by claims that more information about animal populations ultimately will lead to better conservation decisions. Although we believe this claim to be true, we also believe that scientists can do much better than simply providing information. Indeed, a key message is that biological information is much more likely to be useful in solving conservation problems if it is collected in the context of a decision-theoretic approach to management. The book is organized into four thematically focused parts supported by a number of technical appendices. Part I sets out a framework for the role of modeling and the treatment of field observations. It begins with an exposition on scientific method in animal ecology, followed by a discussion on modeling in biological investigation and management. The remainder of Part I focuses on statistical estimation, sampling, and experimental design in ecological investigations. This information provides the reader with the conceptual tools needed for the chapters that follow. Part II focuses on modeling approaches for dynamic biological and natural resource systems, using as examples well-known ecological models. We review notation, objectives, and attributes of models of biological populations, discuss the modeling of stochastic influences such as environmental variation and other random factors, and describe some approaches to the identification of model structure based on time-series data. Part II concludes with applications of models in population management, especially harvest management, conservation biology, and experimentation. Part III builds on the statistical framework introduced earlier and treats more formally the problem of estimating population attributes with sampling data. In Chapters 12 through 14 we deal with estimation for "closed" populations, for which individuals neither enter the population through birth and immigration nor leave the population through mortality and emigration over the course of sampling. In Chapters 15 through 19 we discuss parameter estimation for "open" populations, for which population size a n d / o r composition can change during the course of sampling. Finally, in Chapter 20 we address the estimation of community parameters such as species richness, extinction rates, and species turnover rates. Part IV addresses management of biological populations in terms of optimal decision making through time, recognizing that management actions taken at any point in time can influence population dynamics
at subsequent times. We describe and illustrate a number of optimization techniques that originally were designed for nondynamic problems, and then introduce some modern techniques that make explicit the dynamic nature of animal populations. Part IV culminates in a unified framework for optimal management under uncertainty, recognizing multiple sources of uncertainty and accounting for the potential for learning through management. In particular, we describe adaptive optimization as a way to accommodate uncertainties about the structure of biological processes. The book concludes with a case study of modeling, estimation, and management of waterfowl populations. This science-based management system serves as a clear and successful example of how modeling, estimation, and decision analysis can be integrated into a biologically informed, adaptively managed program.
Book Objectives and Intended Audience A rationale for the selection of material covered in this book is given by way of an analogy between books and mathematical models. Many recognize that in a given biological situation, a model that is designed to be general can be less useful in meeting objectives than a model designed specifically for those objectives. The same holds true for a book: it is not possible in a single volume to treat subjects comprehensively and technically, while simultaneously making them accessible to those seeking a less rigorous treatment. Compromise between these competing objectives is always required, and indeed, the level of detail presented in this book is an example of such a compromise. Thus, we have attempted to explain concepts in a relatively straightforward manner, while still providing the background and detail required for more comprehensive understanding. Our primary purpose in doing so is to promote the integration of modeling, estimation, and decision analysis, which we regard as a unique feature of this book. The intended audience for the book consists of graduate students and advanced undergraduate students in animal ecology, biometrics, quantitative ecology, conservation ecology, and fish and wildlife biology; researchers in biology, biostatistics, and natural resource conservation; natural resource conservationists and managers; and libraries and natural resource reference collections. Readers of the book need a working knowledge of probability, statistics, and differential equations, though the subject matter in each chapter is organized so that key messages can be understood without the need for in-depth mathematical study.
Classroom Use The book is designed to be a single reference for modeling, estimation, and decision analysis, with frequent
Preface references in each chapter to supporting materials in other chapters. It would be an appropriate text for a two-semester course for graduate students and advanced undergraduates who have a background in population biology, probability, statistics, and differential calculus. However, the four thematic sections of the book (or combinations thereof) might be useful in a number of different courses. For example, Part I could be used in a course on quantitative methods in population modeling, with a strong emphasis on sampling design and the analysis of biological data. In particular, Chapters 4, 5, and 6 provide a good foundation for the use of statistical methods in the analysis of populations. The material in Part II may be useful in courses on modeling animal populations. For students with limited training in probability, the use of Chapter 10, which provides background material on stochastic processes in population models, can be restricted to the first few sections. We recommend using Chapters 1 and 2 as introductory materials for such a course. Part III could be used in courses on methods of estimating population parameters with count data. The chapters in Part III group naturally into methods for closed populations, which are covered in Chapters 12 through 14, and methods for open populations, which
xv
are covered in Chapters 15 through 20. A course on either topic can be taught with the corresponding chapters in Part III, along with Chapter 4. Of course, a more comprehensive course covering the whole subject matter of Part III could take advantage of materials on both open and closed populations in developing the robust design of Chapter 19. Finally, Part IV may be useful in courses on decision analysis in natural resources. Chapter 3 could help frame the role of models in decision analysis, and the first five sections of Chapter 7 could serve as an aid in describing and formulating dynamic models. The materials in the first section of Chapter 4 on statistical distributions would prove useful as background for decision making under uncertainty. We acknowledge throughout the book that modeling, estimation, and decision making are all very active areas of research in ecology. The necessary framework of theory, methods, and applications already is very broad and in many cases quite elegant, and over the course of writing this book we were both pleased and frustrated that the biological literature was expanding faster than our ability to absorb it. We look forward to continuing developments in these areas and hope that in some small way this book contributes to the effort.
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Acknowledgments
they realized it or not: Carl Bennett, George Burgoyne, Walt Conley, and Glenn Dudderar. Thanks also to an unknown teaching assistant and a pile of gassed Drosophila, in a long ago genetics lab at Michigan State, for convincingly demonstrating the advantages of sampling and estimation over enumeration. Finally, thanks to the many students who have taken my graduate course in population estimation over the years, for their feedback on what "worked" and what didn't, and for innumerable corrections to class notes, which formed the kernel for several sections of the book. I especially thank my current and former graduate students for critical input along the way and for keeping me honest. J.D.N. thanks Christy, Jonathan, and especially Lois for their support and tolerance and coauthors Ken and Mike for their patience, as I was the primary reason for "The Book That Never Ends" almost never ending. I thank my M.S. advisor Bob Chabreck and my Ph.D. advisor Walt Conley for teaching me about wildlife management and science, respectively. My first supervisor, Franklin Percival, provided lots of good advice that has served me well, and he has continued to provide friendship and support. I will follow the lead of Ken and Mike and forego the list of 40+ colleagues and collaborators who have been important influences, but I must acknowledge the special role of Ken Pollock as a friend and collaborator who has shared many ideas with me and patiently listened to mine. Finally, I thank my most frequent collaborator, Jim Hines, the best programmer I know, whose career has been so intertwined with mine that I simply cannot imagine working without him.
We wish to acknowledge the many colleagues, liberally cited in the references, with whom the authors have collaborated over the years on the ideas in the book. Our thanks also for reviews and constructive comments on various portions of the book by Chris Fonnesbeck, Bill Kendall, Clint Moore, Jonathon Nichols, Jim Peterson, Andy Royle, and Nathan Zimpfer. In addition, we thank Jim Hines for computing some of the capture-recapture examples and Shannon Williams for her help with word processing and copy editing. Special thanks from B.K.W. to my coauthors, whom I count myself most fortunate to have as colleagues and friends, and to Fred Johnson, also colleague and friend, who helped to shape many of the ideas expressed in this book about the interface of science and management, especially as concerns the adaptive management of migratory birds. Thanks also to an unnamed faculty member who, during my graduate days many years ago, opined that it was time for me to decide whether I should study math or biology and who thereby started me on a quest to do both, culminating years later in this book. Finally, endless thanks to Genie, who renamed the manuscript "The Book That Never Ends," but who stayed through to the end anyway. With her, the trip is never dull, and she proves daily that what really counts is the going, not the getting there. M.J.C. gratefully acknowledges the love and support of my family, Liz, Mary, and Laura, without whom none of this would have been worth it. Also thanked are key individuals who provided a spark, a kick, or some other form of inspiration at critical moments in the author's early professional development, whether
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PART
I FRAMEWORK FOR MODELING, ESTIMATION, AND M A N A G E M E N T OF ANIMAL POPULATIONS
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C H A P T E R
1 Introduction to Population Ecology
1.1. SOME DEFINITIONS 1.2. POPULATION DYNAMICS 1.3. FACTORS AFFECTING POPULATIONS 1.3.1. Population Regulation 1.3.2. Density Dependence and Density Independence 1.3.3. Population Limitation 1.4. MANAGEMENT OF ANIMAL POPULATIONS 1.5. INDIVIDUALS, FITNESS, AND LIFE HISTORY CHARACTERISTICS 1.6. COMMUNITY DYNAMICS 1.7. DISCUSSION
In this chapter we introduce the concept of a population that changes over time, in response to primary biological processes that influence population dynamics. We discuss the concepts of density dependence and density independence in these processes, and their roles in regulating and limiting population growth. We incorporate these concepts into a biological context of conservation and management of animal populations. The framework of population dynamics as influenced by primary biological processes and their vital rates will be seen to be useful across ecological scales, and in particular will be seen to contribute to a unified frame of reference for investigations at the scale of individuals (evolutionary ecology), populations, and communities.
1.1. SOME D E F I N I T I O N S
A population often is defined as a group of organisms of the same species occupying a particular space at a
particular time (e.g., Krebs, 1972), with the potential to breed with each other. Because they tend to prefer the same habitats and utilize the same resources, individuals in a population may interact with each other directly, for example, via territorial and reproductive behaviors, or indirectly through their use of common resources or occupation of common habitat. Spatial boundaries defining populations sometimes are easily identified (e.g., organisms inhabiting small islands or isolated habitat patches) but more typically are vague and difficult to determine. Spatial and temporal boundaries often are defined by an investigator; however, this arbitrariness does not detract from the utility of the population concept. A key quantity in population biology is population size, which refers to the number of individual organisms in a population at a particular time. In this book, the terms abundance and population size are used synonymously. We reserve the term density for the number of organisms relative to some critical resource. Typically the critical resource is space, so that density represents, e.g., the number of organisms per unit land area for terrestrial species, or the number of organisms per unit water volume for aquatic species. However, the concept of density is sufficiently general that it need not involve space. For example, a meaningful use of the term would be the number of organisms per unit food resource, or in the case of discrete habitat patches, the number of organisms per patch (e.g., the number of ducks per pond on prairie breeding areas). The structure of a population often can be described in terms of the number of individual organisms characterized by specific attributes of interest. For example, the age structure of a population refers to the respective
4
Chapter 1 Introduction to Population Ecology
proportions of individuals in discrete age classes. A population also may be described by its stage structure, with discrete stages defined by variables such as size (the proportions of animals in discrete size classes) (e.g., see Sauer and Slade, 1987a,b), reproductive behavior (e.g., breeders or nonbreeders), or physiological development. In fact, the structure of a population can be described in terms of any attribute thought to be relevant to population dynamics. A common example utilizes the sex ratio of a population, which expresses the proportionate sex composition of a population. 1.2. P O P U L A T I O N D Y N A M I C S Population ecology can be viewed as the study of the distribution of the individuals in a population over time and space. Population ecologists often focus on temporal change in abundance or population dynamics, asking how and why a population changes over time. Temporal population change can be expressed via a simple balance equation that incorporates gains and losses: N(t + 1 ) = N(t) + B(t) + I(t) -
D(t)-
(1.1)
E(t),
where N(t + 1), the population size at time t + 1, is written as a function of population size N(t) at time t, with increases to N(t) during the interval t to t + 1 as a result of reproduction B(t) and immigration I(t), and losses during the interval from mortality D(t) and emigration E(t). The four variables, B(t), I(t), D(t), and E(t), reflect the primary population processes responsible for changes in population size. If an environmental factor or a management action is to influence population size, its influence must be registered through one of these processes. The primary population processes in Eq. (1.1) describe gains and losses in terms of numbers of individual organisms. But births and deaths during the interval (t, t + 1) are likely to depend on the number N(t) of animals in the population at the beginning of the interval. For this reason, it often is useful to rewrite B(t) as B(t) = b(t)N(t), where b(t) is defined as a per capita reproductive rate, or the number of new individuals in the population at time t + 1 resulting from reproduction during (t, t + 1), per individual alive in the population at time t. Similarly, the number of deaths often is rewritten as D(t) = [1 - S(t)]N(t), where S(t) is an interval survival rate, reflecting the proportion of animals alive at time t that are still alive at time t + 1. For populations that are geographically closed (i.e., there are no gains or losses resulting from movement), Eq. (1.1) can be rewritten as N(t + 1) = N(t)[b(t) + S(t)].
(1.2)
For populations that are not geographically closed, it is tempting to write immigration and emigration as functions of N(t). This often is reasonable for emigration, and we can write E(t) as E(t) =e(t)N(t), where e(t) is the proportion of animals in a population at time t that emigrate out of the population by time t + 1. But it is less reasonable for immigration, given that the number of individuals immigrating into the population between t and t + 1 is more likely a function of abundance or density in the source population of immigrants, rather than the size of the recipient population. Immigration thus is treated differently than the other primary population processes, in that it usually is not modeled as a per capita rate based on the recipient population size. Equations (1.1) and (1.2) constitute simple mathematical models of population change, to be discussed in more detail in later chapters. For present purposes, models can be viewed generally as abstractions and simplifications of reality, and in particular, Eqs. (1.1) and (1.2) can be thought of as simple hypotheses about population change. In later chapters we expand and enhance these models, to incorporate a number of biologically relevant factors that influence population change. For example, single-species population models frequently incorporate information about the attributes of individuals in the population, with individuals grouped into classes as defined by variables such as age, size, and sex (e.g., Lefkovitch, 1965; Streifer, 1974; Caswell, 2001). The population then is characterized by a vector specifying the number of individuals in each class or stage. Model enhancements also can include spatial structure, as in Levins' (1970) description of a metapopulation as a "population of populations." Metapopulation models often include different habitat patches that may or may not contain individuals, with reproduction occurring among individuals within a patch and movement of individuals occurring between patches (Levins, 1969, 1970; Hanski and Gilpin, 1997; Hanski, 1999). Metapopulation dynamics are thus a function of both within-patch (reproduction, survival) and between-patch (emigration, immigration) processes. Finally, both single-location and multiplelocation models can be extended to include multiple species and their potential interactions.
1.3. F A C T O R S A F F E C T I N G POPULATIONS Equation (1.1) provides a framework for population change, but carries little information about why populations change. Many questions of ecological and man-
1.3. Factors Affecting Populations agement relevance involve factors that potentially influence the four primary processes driving population change. These can be categorized in many ways, but it often is convenient to think in terms of abiotic and biotic factors. Abiotic factors include physical and chemical characteristics of an organism's environment such as soil type, water availability, temperature, and fire frequency for terrestrial organisms, and water salinity, pH, currents, light penetration, and dissolved oxygen for aquatic organisms. Factors such as these commonly influence population dynamics via multiple rather than single population processes. For example, water and wetland availability on prairie breeding areas in North America can influence duck populations (Johnson et al., 1992) by affecting reproduction (lower probabilities of breeding and increased duckling mortality when conditions are dry), survival of adults (higher mortality of hens associated with predation when nesting during wet years), and movement (increased movement away from relatively dry areas and to relatively wet areas). On the other hand, biotic factors are understood in terms of interactions among members of the same species (intraspecific), or interactions involving species other than that of the population of interest (interspecific). Interspecific factors include vegetative components of the habitat as well as processes such as predation, interspecific competition, parasitism, and disease. Like abiotic influences, they also can affect more than one of the primary population processes. For example, predation clearly influences mortality, but may also influence movement (increased emigration from areas with large numbers of predators) and reproduction (decreased probability of reproducing in response to increased Predation risk). Intraspecific factors involve interactions among the individuals in a population, with potential influences on all of the primary population processes. They often involve direct behavioral interactions, in which some individuals in the population actively exclude other members of the population from habitat patches or deny access to food resources or even to members of the opposite sex. But they also can involve indirect interactions, through the possible depletion of common resources and the occupation of common habitat. Indirect interactions such as these almost always involve other biotic and abiotic factors.
1.3.1. Population Regulation Because population processes are influenced simultaneously by abiotic and biotic factors, there may be only limited value in trying to ascertain which class of factors is most relevant to population change. Never-
5
theless, the history of population ecology has been characterized by repeated arguments about the relative importance of abiotic vs. biotic factors in controlling population dynamics, and the importance of interspecific vs. intraspecific factors (e.g., see Nicholson, 1933; Andrewartha and Birch, 1954; Lack, 1954; Slobodkin, 1961; Reddingius, 1971; Murdoch, 1994). Much of this debate has focused on explanations for the simple observation that populations do not increase indefinitely (Malthus, 1798). The terms population regulation and population limitation refer to concepts that emerge from the impossibility of indefinite population increase. Population regulation refers to the process by which a population returns to an equilibrium size (e.g., Sinclair, 1989). A glance at Eq. (1.1) indicates that in order for a population to grow [i.e., N(t + 1) > N(t)], gains must exceed losses, or B(t) + I(t) > M(t) + E(t). On the other hand, the equilibrium condition N(t + 1) = N(t) is attained when additions to the population equal losses, that is, when B(t) + I(t) = M(t) + E(t). A growing population eventually must reach a state in which the primary population processes change in the direction of equilibrium, that is, births and immigration decrease a n d / o r deaths and emigration increase until gains equal losses. Population ecologists have expended considerable effort in attempting to identify factors that can influence the primary processes of growing populations and thereby produce equilibrium. In reality, such an equilibrium is not likely to be a single fixed population size. Instead, regulation can be viewed as producing a "long-term stationary probability distribution of population densities" (Dennis and Taper, 1994; Turchin, 1995). Murdoch (1994) identified regulation with "boundedness," noting that some cyclic and chaotic populations can also be viewed as regulated.
1.3.2. Density Dependence and Density Independence The debate about population regulation often is framed in terms of density dependence and density independence. Sometimes these concepts are defined in terms of the rate of population change ~'t = N(t + 1) / N(t), although such definitions can become relatively complicated (Royama, 1977, 1981, 1992). Our preference is to define density dependence and density independence in terms of the vital rates associated with the primary population processes. For example, the vital rates associated with a geographically closed population are the survival rate S(t) and reproductive rate b(t) in Eq. (1.2). Though the absolute numbers of births b(t)N(t) and deaths [1 - S(t)]N(t) occurring during the interval (t, t + 1) obviously depend on the population
Chapter 1 Introduction to Population Ecology size at the beginning of the interval [see Eq. (1.2)], density dependence is defined by the functional dependence of a vital rate on abundance or density {i.e., S(t) = fiN(t)] a n d / o r b(t) = g[N(t)]}. Density independence refers to the absence of such a functional dependence. Examples of density dependence might include survival and reproductive rates, which typically decrease as abundance or density increases. The relevance of this concept to population regulation is that regulation requires negative feedback between ~'t (and thus the vital rates that produce kt) and population size at t or some previous period. Finally, we note the possibility of Allee effects, in which survival and reproductive rates may decrease in populations at very low density (e.g., Allee et al., 1949; Courchamp et al., 1999; Stephens and Sutherland, 1999). The concepts of density dependence and density independence provide another means of classifying factors affecting animal populations. Some factors operate as functions of density or abundance (i.e., in a density-dependent manner) and represent dynamic feedbacks. For example, in some rodent populations, intraspecific aggressive behavior among individuals appears to increase as density increases, leading to decreased rates of survival and reproduction (Christian 1950, 1961). Interspecific factors also can act in a density-dependent manner, as when rates of predation or parasitism depend on the abundance of the prey or host population (e.g., Holling, 1959, 1965). On the other hand, some factors act in a densityindependent manner, absent dynamic feedback. When flooding reduces alligator reproductive rates by destroying nests, the magnitude of the reduction in reproductive rate depends on the proportion of nests that are constructed in susceptible locations (e.g., Hines et al., 1968), but not on alligator density. Similarly, severe grassland fires may cause direct mortality of insect and small mammal inhabitants, but the increase in mortality associated with fire events typically is independent of the density of the affected population. In some situations, factors acting in density-dependent and density-independent manners interact, as when density-dependent decreases in reproductive rate occur because of increases in numbers of cavity-nesting birds using a fixed supply of cavities (Haramis and Thompson, 1985).
1.3.3. Population Limitation Every population is restricted in its growth potential, with a range of conditions beyond which the population tends to decrease because of reductions in survival rates, reproduction rates, or both. Consider a population at equilibrium, such that gains equal losses
over time and population size does not deviate greatly from some average or expected value. Limitation refers to "the process which sets the equilibrium point" (Sinclair, 1989) or, more generally, that determines the stationary probability distribution of population densities. Limitation can involve factors that act in a densitydependent manner as well as factors that are density independent. A limiting factor can be defined as one in which changes in the factor result in a new equilibrium level (Fretwell, 1972) or, more generally, a new stationary distribution of population densities. For example, if predation is a limiting factor for a prey population, then a sustained decrease in predation should bring about an increase in equilibrium abundance of the prey. This new equilibrium level would itself be determined by the action of other factors on the primary population processes. Consistent with this definition of a limiting factor is the recognition that populations potentially have multiple equilibria, and a given population may move among equilibria as conditions and limiting factors change (e.g., Hestbeck, 1986).
1.4. M A N A G E M E N T OF ANIMAL POPULATIONS Interest in certain animal populations has led to management efforts to try to achieve population goals. These goals frequently involve a desired abundance and, for harvested species, a desired level of harvest. Some animal species exist at abundances thought to be too great, and management efforts are directed at reducing abundance. These include pest species associated with human health problems [e.g., Norway rats (Rattus norvegicus); see Davis, 1953] and economic problems such as crop depredation [e.g., the use of cereal crops by the red-billed quelea (Quelea quelea) in Africa; see Feare, 1991]. Other species are viewed as desirable, yet are declining in number or persist at low abundance. Relevant management goals for the latter typically involve increases in abundance, in an effort to reduce the probability of extinction in the near future. Such a goal is appropriate for most threatened and endangered species, and methods for its achievement dominate the field of conservation biology (e.g., Caughley, 1994; Caughley and Gunn, 1996). Still other species are judged to be at desirable abundances, and management efforts involve maintenance of population size. Finally, for harvested species, an abundanceoriented goal must be considered in the context of maintaining harvest yield that is consistent with recreational a n d / o r commercial interests (e.g., Hilborn and Walters, 1992; Nichols et al., 1995a).
1.5. Individuals, Fitness, and Life History Characteristics If management is to influence animal abundance, then it must do so by influencing at least one of the four primary population processes in Eq. (1.1). For example, white-tailed deer are judged to be overabundant in portions of eastern North America, and management efforts to reduce abundance have been directed at both increasing mortality (via hunting and culling operations) and decreasing reproduction (via sterilization and chemical contraception) (McShea et al., 1997; Warren, 1997). Management efforts directed at endangered species frequently involve attempts to decrease mortality via predator control, or attempts to influence reproduction, emigration, and mortality by setting aside or maintaining good habitat. For harvested species, the regulation of harvests focuses on both harvest yield (harvest regulations should influence yield directly) and abundance (harvest regulations influence abundance by changing rates of mortality and, sometimes, movement). The concepts of population limitation and regulation underlie population management, especially as they factor into the roles of density dependence and independence. For example, the manager of a threatened or endangered species can utilize an understanding of limiting factors to effect management actions to improve the species status. Many endangered species are habitat specialists that are thought to be limited by the amount of suitable habitat available to them. Thus, the purchase or creation of additional habitat represents an effort to remove a limiting factor and to permit the population to increase to a new equilibrium level commensurate with the expanded habitat. Of course, a population increase occurs because of changes in the primary population processes corresponding to the increase in habitat, and it often is useful to focus on the processes as well as the limiting factors. The concept of density dependence is especially important in management of harvested populations. As a direct mortality source, harvest acts to reduce abundance. However, reduced abundance may lead to increases in reproductive rate or to decreases in nonharvest mortality or emigration, depending on which vital rates behave in a density-dependent manner. For example, much fisheries management is based on stock-recruitment models that incorporate densitydependent reproductive rates (e.g., Beverton and Holt, 1957; Ricker, 1975; Hilborn and Walters, 1992). Management of North American mallard (Anas platyrhynchos) populations is based on competing models that represent different sets of assumptions about the density dependence of survival and reproductive rates (Johnson et al., 1997). Because our definitions of density dependence and independence involve the populationlevel vital rates of survival, reproduction, and move-
7
ment, density dependence again directs the manager's attention to the primary population processes.
1.5. I N D I V I D U A L S , F I T N E S S , A N D LIFE HISTORY CHARACTERISTICS The comments above, and indeed most chapters in this book, focus on the population level of biological organization. However, it is important to remember that the constituents of populations are individual organisms, and the characteristics of these organisms are shaped by natural selection. Characteristics associated with relatively high survival or reproductive rates are favored by natural selection, in that organisms possessing them tend to be represented by more descendants in future generations than do other organisms. Individuals with greater potential for genetic representation in future generations are said to have relatively high fitness. Though they typically are thought to deal with different levels of biological organization, fitness and population growth are closely related. Thus, the growth rate of a geographically closed population is determined by survival rate and reproductive rate, whereas the fitness of an individual organism is determined by its underlying probabilities of surviving from year to year and of producing 0, 1, 2 , . . . offspring each reproductive season. Indeed, fitness associated with a particular genotype can be defined operationally as the growth rate of a population of organisms of that genotype (see Fisher, 1930; Stearns, 1976, 1992; Charlesworth, 1980). An important consequence of the close relationship between population growth and individual fitness is that evolutionary ecologists, population ecologists, and population managers are often interested in the same population processes and their vital rates. Nevertheless, a subtle difference can exist between definitions of survival and reproductive rates at the population and individual levels of organization. We defined the interval survival rate S(t) as the proportion of animals in the population at time t that survives until time t + 1. This quantity is not so useful at the level of the individual organism, because an organism either survives or it does not; however, it can be thought of as having some underlying probability of surviving the interval between times t and t + 1. These two distinct quantities, the probability that an individual survives and the proportion of animals in a population that survive, are closely related. Consider a population of individuals with identical underlying survival probabilities for some interval of interest. The
Chapter I Introduction to Population Ecology proportion of individuals that survives the interval likely is not identical with the underlying individual survival probability. On the other hand, the proportion that survives is expected to deviate little from the individual survival probability. More precisely, multiple realizations of population dynamics over comparable time intervals would produce an average proportion of survivors approaching the individual survival probability. In Chapter 8 we define the terms needed to specify the relationship between population-level survival rate and individual probability of survival. The important point for now is that these quantities are closely related. Throughout most of this book, we will use the terms survival rate and survival probability interchangeably to refer to the underlying individual survival probability. When discussing survival at the population level we will use the term survival rate to denote the surviving proportion of a population or group. Of course, the latter quantity is of interest regardless of whether all individuals in the population have the same survival probability. A similar situation exists for reproductive rate. An individual can produce some integer number of offspring {0, 1, 2 , . . . } during a single reproductive season, but a reproductive rate refers to the number of offspring produced per adult in the population. In essence, this offspring/adult ratio is a population-level attribute. The term reproductive rate could refer in concept to (1) the average number of young produced if we could observe an individual over many replicate time intervals or (2) the average number of young produced per adult in the population if we could observe the population over many replicate time intervals. Our intention here is not to dwell on subtle differences in the terms used for individuals and populations, but instead to emphasize the role of vital rates in determining both fitness and population growth. In the discussion above we suggested that the concepts of population limitation and regulation follow naturally from the simple observation that populations do not increase indefinitely. Similarly, evolutionary ecology is based on the observation that neither species nor populations of genotypes can increase indefinitely, though temporary increases are possible. Species and populations of genotypes must eventually reach a state in which temporary increases and declines in numbers of individuals fluctuate about some equilibrium over time. The necessary balance between average survival and reproductive rates has led to various classification schemes [e.g., r- and K-selected species, "fast" versus "slow" species (Cody, 1966; MacArthur and Wilson, 1967; Boyce, 1984; Stearns, 1992)] for species based on these average values. A basic idea underlying all of
these schemes is that species with high reproductive rates must also be characterized by high mortality rates, whereas species with low reproductive rates must also have low mortality rates. The underlying survival and reproductive rates that apply at each age throughout an organism's lifetime are frequently referred to as life history characteristics (Cole, 1954; Stearns, 1976, 1992). Most discussions of life history characteristics also include features such as age at first reproduction, individual growth rate, body size, and age at which individuals can no longer reproduce (see Chapter 8). However, the relevance of these features to life history evolution involves their relationship to the age-specific schedule of survival and reproductive rates. The magnitudes of survival and reproductive rates throughout the organism's lifetime often are viewed as species-specific characteristics, allowing for variation in survival and reproduction rates among individuals. The expectation is that variation among individuals within a species typically is much smaller than variation among individuals of different species. The suite of life history characteristics is important not only for understanding and predicting population dynamics, but also for managing populations. Consider, for example, the management of two harvested species, one with high mortality and reproductive rates (e.g., several commercially harvested fish species) and one with low reproductive and mortality rates (e.g., harvested whales). Imposition of a fixed harvest rate (proportion of animals in the population harvested) typically has a larger influence on the population dynamics of the species with the otherwise low mortality and the low reproductive rate. In addition to low per capita reproductive rates, such species tend to exhibit delayed sexual maturity, with the consequence that they take longer to recover from decreases in abundance. In summary, there is a close relationship between fitness and population change, despite the fact that these quantities apply to different levels of biological organization. One consequence of this relationship is that even. though population ecologists, population managers, and evolutionary ecologists address different kinds of questions and have different objectives, they are all concerned with population vital rates. Thus, the methods presented in this book for estimating vital rates should be relevant to scientists in these different disciplines. Another consequence is that life history characteristics molded by natural selection are relevant to population dynamics and population management. Knowledge of a species' life history characteristics is of key importance in predicting population
1.7. Discussion responses to management, and thus should play an important role in management decisions.
1.6. C O M M U N I T Y D Y N A M I C S In this book, our focus occasionally shifts to the community level of biological organization, where the term community refers to a group of populations of different species occupying a particular space at a particular time. A community may include all the different plant and animal species represented in the space, or, more commonly, may refer to a subset of species defined by taxonomy (e.g., the bird community of an area), functional relationships (e.g., vegetative or herbivore community), or other criteria that are relevant to a question of interest. One way to model community-level dynamics is to model the population for each species, perhaps linking the models via the sharing of resources to induce interactions. For example, consider a simple model of a single predator species and a single prey species. The survival and reproductive rates of the predator species might be modeled as functions of prey species abundance, such that larger numbers of prey lead to higher survival and reproductive rates of the predator species. In the same model, the survival rate for the prey species could be written as a function of predator abundance, with more predators leading to reduced survival for the prey species. A similar approach frequently is taken for the modeling of interspecific competition. The importance of population-level vital rates is again emphasized in this modeling approach, as the interactions between populations are specified as functional relationships involving the vital rates (or composite quantities that combine vital rates). A less mechanistic and more descriptive approach for community-level modeling does not focus on interspecific interactions. This modeling approach has been used by community ecologists (e.g., MacArthur and Wilson, 1967; Simberloff, 1969,1972) and by paleobiologists (Raup et al., 1973; Raup, 1977) and simply involves models such as those of Eqs. (1.1) and (1.2) shifted to the community level. Thus, instead of projecting changes in numbers of individual organisms within a population, the models specify change in the numbers of different species in the community. The primary population processes and their corresponding vital rates are replaced by analogous processes and vital rates at the community level.
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To see how, let N(t) denote the number of species in the community at time t, with S(t) the species-level survival rate (the complement of local extinction rate) for the interval t to t + 1, and I(t) the number of colonists during the interval (species absent from the community at t, but present at t + 1). Using notation similar to that of Eqs. (1.1) and (1.2), the natural expression for change in the number of species in the community is
N(t + 1 ) = N(t)S(t) + I(t). Consideration of the processes determining S(t) and I(t) again leads back to the primary population processes and associated vital rates. Local extinction rate for a species-population is a function of populationlevel rates of survival, reproduction, immigration, and emigration, and the number of colonizing species is a function of immigration at the population level. The approach of representing a "population" of species via a model for which local extinction plays the role of mortality, and immigration/colonization plays the role of reproduction, is a natural extension of the biological framework portrayed in Eq. (1.1). This analogy has been used in biogeography for many years (MacArthur and Wilson, 1967) and is used frequently in other fields such as conservation biology (e.g., Rosenzweig and Clark, 1994; Russell et al., 1995; Boulinier et al., 1998, 2001; Cam et al., 2000). 1.7. D I S C U S S I O N In this chapter we have introduced the biology of animal populations in terms of the fundamental processes of survival, reproduction, and migration, along with their associated vital rates. These quantities define the balance equation [Eq. (1.1)] by which population dynamics can be investigated, and they also provide a basis for understanding the factors that influence population dynamics. In the chapters to follow we make liberal use of this framework, as we focus on the modeling of populations and the estimation of population attributes. We will see that quantities such as population size, harvest numbers and rates, recruitment levels, and migration patterns are key to an understanding of population dynamics. We focus much of what follows on the use of field data to estimate these and other population parameters. A careful accounting of the statistical properties of these estimates will be seen to be an essential component in the informed conservation of animal populations.
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C H A P T E R
2 Scientific Process in Animal Ecology
2.1. CAUSATION IN ANIMAL ECOLOGY 2.1.1. Necessary Causation 2.1.2. Sufficient Causation 2.2. APPROACHES TO THE INVESTIGATION OF CAUSES 2.3. SCIENTIFIC METHODS 2.3.1. Theory 2.3.2. Hypotheses 2.3.3. Predictions 2.3.4. Observations 2.3.5. Comparison of Predictions against Data 2.4. HYPOTHESIS CONFIRMATION 2.5. INDUCTIVE LOGIC IN SCIENTIFIC METHOD 2.6. STATISTICAL INFERENCE 2.7. INVESTIGATING COMPLEMENTARY HYPOTHESES 2.8. DISCUSSION
ology. Thus, the objective of this chapter is to provide a biological context for scientific methodology, and in so doing to clarify the respective roles of theory development, statistical inference, and the structures of formal logic in animal ecology.
2.1. C A U S A T I O N IN ANIMAL ECOLOGY Science is about the identification and confirmation of causes for observed phenomena, whereby "cause" is meant as a generic explanation of patterns observed for a class of phenomena. The explanatory power of a cause results from the ability to entail many, often apparently disparate, phenomena under its rubric. Causes are recognized as "explanatory" in the context of a scientific theory of which they are components, the theory itself consisting of relatively few causal factors entailing a wide range of phenomena. More formally, causation can be described in terms of antecedent conditions, consequent effects, and a rule of correspondence for their conjoint occurrence. In population biology the "effect" of a cause typically is a biological event (e.g., mortality, growth, population change) that occurs subsequent to the occurrence of some prior condition. Provided the joint occurrence of the prior condition and the subsequent event meet certain theoretical and logical requirements, the prior condition is held to be the cause of the event. The causal linkage between a prior condition and a subsequent effect can be described in terms of the logic of material implication (Copi, 1982). The expression A --~ B describing material implication is taken to mean
However varied the practice of animal ecology, a common feature is the comparison of predictions, deduced from biological hypotheses, with data collected pursuant to the comparison. Much has been written about the testing of biological/ecological hypotheses (Romesburg, 1981; Hurlbert, 1984; Peters, 1991), and specifically about sampling designs and statistical inferences for hypothesis testing (Green, 1979; Hairston, 1989; Skalski and Robson, 1992). However, much of this documentation has focused on the characterizing of biological hypotheses in terms of statistical distributions, and on the investigation of distribution attributes with sample data (Brownie et al., 1985; Burnham et al., 1987; Lebreton et al., 1992). It is useful to consider how these activities fit into a broader context of theory, logic, and data analysis that is definitive of scientific method-
11
12
Chapter 2 Scientific Process in Animal Ecology
that affirmation of the premise A implies affirmation of the conclusion B. However, material implication is silent about tile affirmation of A given that B is affirmed. More formally, material implication establishes the equivalence of A ~ B with the assertion that either A is false or B is true. Thus, one can look to the premise of A ~ B to confirm its conclusion, but one cannot look to the conclusion of A ~ B to confirm its premise. The concept of causation in scientific inquiry is informed by the logic of material implication, by identifying cause (C) and effect (E) as either premise or conclusion. Two distinct definitions of causation can be identified.
2.1.1. Necessary Causation In this case an effect E points to a presumptive cause C, in that the occurrence of the effect guarantees the occurrence of condition C. A logically equivalent argument is ---C --~ ---E, i.e., the nonconcurrence of C guarantees the nonconcurrence of the effect (the symbol ~-in this expression is used to indicate logical negation, so that ---C, which is read "not C," means that the truth of C is negated). Thus, necessary causation asserts that the absence of an effect follows from the absence of the cause. However, it is silent about effect E in the presence of C. Examples of necessary causation might include light as a cause of photosynthesis, Salmonella bacteria as a cause of typhoid fever, and fuel loads as a cause of forest fires. In each example the effect may or may not be present when the presumptive cause occurs; however, the effect is held to be absent when the cause is missing.
2.1.2. Sufficient Causation In this case the presumptive cause C points to the effect E, in that the occurrence of condition C guarantees the occurrence of the effect. Thus, sufficient causation asserts that the occurrence of an effect follows from the presence of condition C. However, it is silent about the effect in the absence of C. Sufficient causation might underlie an argument that heat causes fluid dynamics; that a low level of ambient oxygen during respiration causes the production of lactic acid; that oxygenation of pig iron under high pressure causes the production of steel; that drought causes physiological stress in nonsucculent plants. In these examples the presence of the cause is held to ensure the presence of the effect; however, the effect may or may not be present in the absence of the cause. Sufficient causation is a logically stronger definition than necessary causation, in that necessary causation specifies C as one condition (possibly among many) that must be present to ensure the occurrence of an
effect, whereas sufficient causation specifies that C alone ensures its occurrence. An otherwise necessary cause can be recognized as sufficient by restricting the range of conditions in which it is operative. Thus, a concentrated source of heat (e.g., a lighted match) is a necessary cause of combustion, but a heat source in the presence of combustible material in a cool, dry, oxygenated environment becomes a sufficient cause (under these conditions). The importance of maintaining a clear distinction between necessary and sufficient causation can be illustrated by the controversy about smoking as a potential cause of lung cancer. Advocates for restricting the advertisement and sale of tobacco products base their arguments on the strong statistical association between tobacco use and the occurrence of lung cancer, wherein the great majority of lung cancer victims in the United States also have a history of smoking. On the other hand, opponents of tobacco restrictions have argued repeatedly that the association between smoking and lung cancer is not causal, and cite as evidence the fact that a majority of smokers in the United States do not have lung cancer. Clearly, these conflicting positions (and different assessments of evidence) point to inconsistent uses of the concept of causation. Apparently advocates of tobacco restrictions assume necessary causation, such that a history of tobacco use is inferred from the occurrence of lung cancer. Evidence for smoking as a necessary cause of lung cancer focuses on the fact that lung cancer victims overwhelmingly have a history of smoking, and a key implication is that the avoidance of smoking implies the near absence of lung cancer. On the other hand, opponents of tobacco restrictions appear to use sufficient causation, wherein smoking should lead to the occurrence of cancer. By implication, the absence of cancer therefore should imply the absence of smoking, which is inconsistent with the fact that the overwhelming proportion of smokers have no record of lung cancer. Hence, tobacco is held not to be a cause of lung cancer by opponents of tobacco restrictions. Given the inconsistent uses of causation, it is not surprising that the controversy between advocates and opponents of tobacco restrictions has not been amenable to data-based resolution. Indeed, the evidence likely will continue to indicate that tobacco use is simultaneously a cause of lung cancer (in the necessary sense) and not a cause of lung cancer (in the sufficient sense).
2.2. APPROACHES TO THE O F CAUSES
INVESTIGATION
A definition of cause as necessary often applies to the control of unwanted effects, whereby the elimina-
2.3. Scientific Methods tion of an effect (e.g., typhoid fever) is assured by the elimination of the cause (e.g., destruction of Salmonella bacilli through sterilization). Scientific investigation thus involves a search for conditions that are predictive of the nonconcurrence of an effect of concern. Necessary causation often is implied in population biology when biological effects in the presence of a particular condition are attenuated by the restriction or removal of the condition. A particular example is duck nest predation as a presumptive (necessary) cause of reproductive failure in cultivated prairie lands under nondrought conditions. The implication is that reducing predation will reduce reproductive failure. On the other hand, a definition of cause as sufficient applies to causes (e.g., drought) that guarantee an effect (e.g., physiological stress in plants). Scientific investigation in this case involves the search for conditions that are predictive of the occurrence of an effect. Sufficient causation is implied in population biology when the influence of a prior condition is both direct and adequate to produce an effect of concern. A relevant example is the investigation of sport hunting as a potential cause of declining waterfowl population trends. Thus, heavy hunting pressure is hypothesized to reduce survival and depress population levels, recognizing that population declines can occur even in the absence of hunting. Necessary and sufficient forms of causation share a natural linkage with the experimental elements of treatment and control. A typical experiment investigates the association between some putative causal factor C and an effect E, with the idea that the cooccurrence (along with joint nonoccurrence) of C and E provides evidence for causation. The experiment has treatment C imposed to determine whether effect E occurs in its presence, i.e., to investigate whether C is a sufficient cause of E in the sense of C ~ E. By the rules of logical inference, the occurrence of E is insufficient by itself to support a claim of causation. However, one can infer from an absence of E that the treatment cannot be a (sufficient) cause of E. On the other hand, experimental control allows one to investigate whether the absence of E follows from the absence of C, i.e., whether ---C ~---E. But this is logically equivalent to the assertion of necessary causation, that is, E--~ C. Under experimental control, the absence of effect E is inadequate to support a claim of causation. However, one can infer from the occurrence of E that the treatment cannot be a (necessary) cause of E. It is the coupling of inferences from both treatment and control in an experiment that confers logical rigor to designed experiments. Experimental results in which E occurs in the presence of C but not in its absence provide the evidence for necessary and suffi-
13
cient causation. Under these conditions no other factor than C can cause the effect, for otherwise E presumably would be recorded in the absence of C, in violation of the requirement for co-occurrence. C is therefore recognized as the cause, and the only cause, of E under the experimental conditions. This very high standard illustrates the value (and rigor) in establishing causation through experiment and helps to explain why experimental design is a near-imperative in much of biological science.
2.3. S C I E N T I F I C M E T H O D S A useful context for scientific method involves scientific investigation both before and during a period when it is guided by a recognized theoretical framework. Thus, in its early stages, scientific activity consists of observation guided primarily by intuition, tradition, guesswork, and perceived pattern. Its function initially is to organize observations into coherent categories, to explore these observations for patterns, and to describe the patterns clearly. The process of recognizing the underlying causes of patterns comes as the scientific discipline matures, and a set of relationships, which are accepted as "explanatory," is formulated. These relationships are sometimes called a theoretical paradigm or, more briefly, a theory (Kuhn, 1970). A standard for the operation of science, including biological science, involves a comparison of theoretically based predictions against data, recognizing that a match between data and prediction provides evidence of hypothesis confirmation, and the lack of such a match disconfirms a hypothesis (Hempel, 1965). A somewhat more detailed treatment includes five elements: theory, hypotheses, predictions, observations, and comparisons of prediction against data. 2.3.1.
Theory
First, an explicit statement of a relevant theory is necessary, or at least the reference to it is necessary. The theory is expressed in terms of the axioms, postulates, theoretical constructs, and causal relationships among constructs that constitute the corpus of the theory. This corpus, involving biological elements such as genetics, taxonomy, evolutionary principles, and ecological relationships (Hull, 1974), is operationally accepted as verified and true. A theory is noted in what follows by {T}. Every scientific discipline is founded on an operational theory, which provides a conceptual framework through which the world is observed and facts about the world are discerned. Broadly recognized examples
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might include the theory of relativity, electromagnetic field theory, the theory of plate tectonics in geology, thermodynamic theory, and the theory of evolution by natural selection. An operational theory allows one to recognize patterns among apparently disparate phenomena and to explain relationships among them. It also is the foundation for hypothesis formulation, prediction, and testing. In short, a theory is essential to the conduct of scientific investigation.
amended theory and observed reality. The derivation of predictions is designated by {T} + H --~ P, where P represents a prediction and the arrow indicates logical inference. The notion here is that the addition of H to {T} allows for inferences that otherwise would not follow from {T} in the absence of H. At least some of these inferences are testable, in that they predict observable phenomena that potentially are verifiable with field or experimental data. The key here is that P consists of potentially observable predictions.
2.3.2. Hypotheses Second, a hypothesis that is relevant to the theory is identified, often through field or laboratory observations that appear to be anomalies to the theory, i.e., that appear not to be explained adequately by the theory as it currently is understood. A hypothesis, denoted here by H, asserts a claim about relationships among components of the theory, or about relationships of these components to observed reality, or about relationships among entities in the observed world that are presumed to follow from the hypothesis. An example of the first kind of claim might be the recognition that one component of the theory entails another; an example of the second kind is the predicted existence of heretofore unrecognized sociobiological patterns; and an example of the third kind is the dynamics of dispersal following certain kinds of environmental disruption. We emphasize in what follows the investigation of causal hypotheses, involving antecedent conditions and consequent effects that are identified in a theoretical context. A hypothesis is recognized as potentially true or false. When added to a theory, it renders the theory potentially inconsistent, or potentially false. In what follows, an amended theory is designated by {T} + H, to indicate that H is included as one of the elements defining the amended theory. This notation suggests an attendant increase in theory complexity. Alternatively, H can replace a particular hypothesis H 0 within the body of the theory. This is designated by {T0} + H, where {T0} + H 0 represents the theory before amendment. Scientific investigation then becomes a comparison of the relative explanatory power of the two theoretical constructs {T0} + H 0 and {T0} + H. To simplify notation we use {T} + H to represent both the appending of H to {T} and the replacement of H 0 in {T} b y H .
2.3.3. Predictions Third, potentially observable conclusions are deduced from the amended theory. These follow from logical relationships inherent in the amended theory, or they are derived from relationships between the
2.3.4. Observations Fourth, field or experimental data are collected that are pertinent to the predictions. The investigator's attention is directed to these data by the amended theory, which is used as above to derive predictions for which the data are relevant. Field a n d / o r experimental data, designated by observation O, are essential components by which the amended theory is to be evaluated. Key to successful data collection are statistically sound surveys, experiments, and other data collection instruments.
2.3.5. Comparison of Predictions against Data Fifth, predictions from the amended theory are compared to observations O from the field or laboratory. This comparison is used to determine the acceptability of the amended theory and hence the acceptability of the hypothesis H. If O conforms to P, i.e., if the predicted results of {T} + H are in fact observed, then the investigation provides evidence to confirm H. If O does not conform to P, then the evidence disconfirms H. Statistical testing procedures play a crucial role in the process of hypothesis confirmation. An ideal approach to scientific investigation consists of repeated applications of this sequence across all levels of investigation. Thus, alternative hypotheses often are part of a study design, wherein two or more hypotheses may be considered as alternatives for theory amendment. For a given hypothesis H, numerous predictions may be identified, each worthy of field investigation. For each prediction P, data from several different field and laboratory studies may be appropriate. In addition, studies involving the same hypothesis, the same prediction, and the same kind of data collection often are repeated numerous times, to add to the strength of evidence for confirmation or disconfirmation. We note that in concept, one could identify hypotheses without theoretical justification or guidance, and provided that predictions of the hypotheses are directly
2.3. Scientific Methods measurable, one could collect data that are relevant for testing. However, there are two serious problems with hypothesis testing that is not informed by theory: (1) it is much less likely that one can identify potentially useful and informative hypotheses for investigation, and (2) it is more difficult to determine the appropriate data to collect in support of confirmation or disconfirmation. Theory plays a key role in resolving both these problems, by directing the investigator's attention to theoretically interesting questions, testable predictions, and useful data for comparison against those predictions. Absent a theoretical context for the play of logic in recognizing testable predictions, it becomes much less likely that scientifically meaningful hypotheses can be identified, or that relevant data can be targeted for their testing (e.g., Johnson, 1999).
Example Consider a wildlife species that is exposed annually to sport hunting. A traditional concern in game management is the effect of harvest on future population status, and in particular the effect of harvest on annual survival. Two competing hypotheses have been identified: 1. The hypothesis of additive mortality asserts that harvest is additive to other forms of mortality such as disease and predation. Under this hypothesis the annual mortality rate increases approximately linearly in response to increases in harvest rate. 2. The hypothesis of compensatory mortality asserts that harvest mortality may be compensated by corresponding changes in other sources of mortality. Thus, increases in harvest rate have no effect (up to some critical level c of harvest) on the annual mortality rate. In the standard formulation of the compensatory hypothesis, harvest rates beyond c result in an approximately linear increase in annual mortality. We refer the reader to Anderson and Burnham (1976) and U.S. Department of the Interior (1988) for a more complete development of these relationships. The compensatory and additive hypotheses provide a convenient point of reference for the process of scientific investigation. Research on the effect of hunting is conducted in the context of a theory of population dynamics recognizing structural, functional, and dynamic characteristics of wildlife populations in an ecosystem of interrelated organisms and abiotic processes. Elements of the theory involve reproduction, survival, and migration as influenced by factors such as interspecific interactions, physiological condition, behavioral adaptations, and seasonal habitat conditions. The edifice of concepts, relationships, axioms, and terms relating to populations constitutes the scientific para-
15
digm of population ecology [see Baldasarre and Bolen (1994) for a review of theory and management as concerns waterfowl populations]. It is in the context of this paradigm that the relation between mortality and harvest rate can be investigated. The investigation proceeds with deduction of testable predictions, following from the paradigm along with the compensatory and additive hypotheses. Three general predictions can be recognized for waterfowl populations (Nichols et al., 1984a): 1. The compensatory mortality hypothesis leads to a prediction that there is no relationship between annual survival rate and hunting mortality, so long as harvest rate is less than the critical value defined in the hypothesis. On the other hand, the additive mortality hypothesis suggests that there is negative relationship between annual survival rate and hunting mortality over the whole range of potential harvest rates. 2. Under reasonable conditions, the compensatory mortality hypothesis leads to a prediction that there is a negative relation between hunting mortality rates (during the hunting season) and nonhunting mortality rates (during and after the hunting season). The additive mortality hypothesis leads to a prediction that there is no such relationship. 3. The compensatory mortality hypothesis leads to a prediction that there is a positive relation between nonhunting mortality rate and population size or density at some time in the year. In many circumstances nonhunting mortality rate after the hunting season should be positively related to population size at the end of the hunting season. The additive mortality hypothesis leads to a prediction that there is no relationship between nonhunting mortality rate and population size. These predictions differ considerably in the degree to which they represent explanatory causes of population dynamics, and the difficulty with which data can be collected and used informatively for testing (Conroy and Krementz, 1990). Indeed, it always is an outstanding challenge in scientific investigation to devise ways of collecting data that are pertinent to testable predictions. In this particular case, population surveys (Thompson, 1992), radiotelemetry (White and Garrott, 1990), mark-recapture procedures (Nichols, 1992), banding studies (Brownie et al., 1985), and other field procedures can provide valuable data by which to test the predictions. Such studies can be replicated at different times and different locations, under a variety of different field conditions and different harvest strategies, with a focus on one or any combination of the predictions listed above. Each study adds evidence by which investigators can confirm or disconfirm the
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hypotheses. Replication and redundancy of this kind play an important role in preventing unwarranted generalizations of study results.
In contrast, argument (2) above has a very different logical content. Here the assertion is of the form A--4 B B .'. a
2.4. HYPOTHESIS CONFIRMATION The logic of hypothesis confirmation can be expressed in general terms by means of material implication. The process is denoted by
(1)
{T} + H--~P O---~ --,P {T} O .'. ---H
or
(2)
{T} + H - - 4 P O--4 P {T} O .'. H
In these formulations the first premise asserts that prediction P is a consequent of an amended theory, as described above. The second essentially asserts that P is disconfirmed by observation (argument 1) or that P is confirmed by observation (argument 2). The third premise asserts the truth of theory {T}, and the fourth represents the observed data O. A horizontal line separates the argument's premises and evidence from its conclusion, which is stated on the last line. Again, the symbol --- in argument (1) is used to indicate logical negation, so that the expression O--4---P means "the truth of O implies that P is false" (i.e., the observation indicates that the prediction is incorrect). Though the two arguments above appear to be analogous in their forms, there is a crucial asymmetry in their logical content. Argument (1) is an example of the syllogistic form modus tollens (Copi, 1982), wherein rejection of the conclusion in an argument of material implication implies rejection of the premise: A --4 B ---B .'. ---A
Applying modus tollens to the scientific argument above, the observations O do not correspond to what was predicted; thus, O --4 ---P in the second line of the argument. But ---P implies ---{T} + H from the first line of the argument, which in turn implies either {T} or H (or both) is untrue. Because {T} is assumed in the third line of the argument to be a confirmed and operational theory, this leaves the falsity of H as a conclusion of the argument. Hence the conclusion ---H. Simply put, this argument states that evidence contrary to a hypothesis is logically sufficient to disconfirm the hypothesis.
Thus, the evidence O in argument (2) confirms prediction P, the consequent of {T} + H --4 P. The confirmation of P in turn is held to confirm the amended theory {T} + H. Because {T} + H is held to be true, the component H in particular is presumed to be confirmed. This argument is common in scientific investigation, including research in population biology. Unfortunately, it is logically invalid. Thus, the confirmation of P and the truth of {T} + H--4 P cannot be used to assert the truth of H. Simply put, evidence supporting a hypothesis is logically insufficient to confirm that hypothesis: factors other than H might well lead to confirmation of the prediction P, independent of the truth or falsity of H. The fallacy of affirming the premise of an implication based on its conclusion is an example of the fallacy of false cause, known as affirming the consequent (Copi, 1982). Scientific investigation thus faces an asymmetry in the confirmatory role of experimental or field evidence. On the one hand, a hypothesis can be disconfirmed by evidence contrary to prediction; on the other, a hypothesis cannot be (logically) confirmed by evidence supporting prediction. It is in the context of this asymmetry that scientific hypotheses are held by some to be meaningful only if they are theoretically amenable to disconfirmation (Popper, 1968). The fallacy of false cause can be avoided in argument (2) only if the prediction P can arise in no other way than by the truth of H, i.e., only when P and H have the same truth content (if H is true, P is also; if H is false, P is also). Under this much more restrictive condition the proposition {T} + H ~ P is replaced by {T } + H ~-4 P, whereby the arrow pointing in both directions means that P can serve either as premise or conclusion in material implication. Thus, to avoid the fallacy of false cause all alternative hypotheses must be eliminated through experimental design or otherwise must be identified, investigated, and rejected, so that by process of elimination only the hypothesis H remains as an explanation of a confirmed prediction P. Hypothesis confirmation through the elimination of alternatives was termed "strong inference" in an important paper by Platt (1964). Although relatively simple in concept, such an approach obviously requires thorough field observations as well as careful analysis to identify and properly examine all reasonable alternative hypotheses.
2.5. Inductive Logic in Scientific Method We note that this approach to science includes the essential features of Popper's hypothetico-deductive method of scientific inquiry (Popper, 1963, 1968). However, it differs from Popper's in at least one important feature, namely, the procedure for comparing hypotheses against data. The Popperian model describes a process in which a hypothesis H is tested by experiment to determine its acceptance or rejection, with hypothesis rejection in the event of nonconformance to the evidence, and provisional acceptance otherwise, pending further evidence. The process then is repeated with another hypothesis H', with evidence from another critical experiment leading to acceptance or rejection of H' depending on conformance with experimental data. In this scenario hypotheses are subjected to testing one at a time, with decisions about hypothesis acceptance or rejection made sequentially. Our approach to hypothesis investigation also could be applied one hypothesis a time, as per the Popperian model. However, sequential investigation of hypotheses is only one available option, and not a requirement of the approach. It is possible, and intuitively preferable, for alternative hypotheses to be compared simultaneously against evidence, so as to measure their relative conformance one against the other. Two important benefits accrue from this more comprehensive approach to hypothesis testing. First, it allows every feasible hypothesis to compete in an arena of evidence against all other feasible hypotheses. This is as opposed to the Popperian model, in which previously rejected hypotheses are no longer candidates for comparison against alternatives considered later in the testing process. Second, simultaneous testing allows the process to carry a "memory" of previous test results, via the measures of conformance between individual hypotheses and the evidence. The conformance measures provide a natural mechanism, through the use of updating procedures such as Bayes' Theorem (see Sections 3.3.2 and 4.5, and Appendix A.3), for confirmatory evidence to accumulate as scientific investigation proceeds. The use of weights to express hypothesis likelihoods will be explored in considerable detail in Part IV.
2.5. I N D U C T I V E L O G I C IN SCIENTIFIC METHOD Inductive as well as deductive logic is required for hypothesis confirmation in biological science. That inductive logic is an essential feature of scientific enquiry is seen in the identification of hypothesized biological mechanisms, as well as the testing of these hypotheses with data. Indeed, a key activity in scientific enquiry is to identify, from a limited set of observations,
17
hypotheses that explain more than the particular observations giving rise to them. Thus, a limited body of data generates possible explanations for their occurrence, and these are folded as hypotheses into an extant body of theory for elaboration and testing with additional observations. Because any particular set of data constitutes only a subset of all possible observations that could be used, testing procedures are designed to be robust to inherent variation in the evidence. The formulation and testing of scientific hypotheses, based on only a partial record of potentially relevant observations, render the practice of science inductive. Simply put, causal mechanisms are asserted to hold for a general class of phenomena, based on examination of only limited observations from that class. The inductive nature of the process inevitably gives rise to the possibility of incorrect inference and necessitates the conservative rules of scientific and statistical inference that have been developed to accommodate, and protect against, such a possibility. As described above, the logic of hypothesis confirmation suggests that evidence contrary to a hypothesis is sufficient for its rejection, whereas evidence conforming to the hypothesis is insufficient for its confirmation. In practical applications, however, the situation is less clear-cut. Biological systems are replete with uncertainties, and hypothesized explanations are never wholly sufficient to explain behaviors. Thus, natural variation (and sampling error) can lead to the rejection of a hypothesis that otherwise would be seen as appropriate, just as it can support acceptance of an inappropriate hypothesis. Because biological inferences must be confirmed via inductive logic from particular instances, some of which can be misleading, these inferences lack the logical certainty of deductive arguments such as modus tollens and modus ponens. Biological investigation is by its very nature open to the risk of incorrect inference, which can decline as evidence accumulates but never vanishes. It is the role of probability and statistics in biological science to characterize and account for this risk. Just because the rules of inductive logic are not as prescriptive as in deductive logic, one should not conclude that induction is somehow inferior to deduction. As in all scientific disciplines, both inductive and deductive inference are required in biology. The "truth" of a biological hypothesis can only be confirmed inductively, through an ever-growing body of evidence that lends it credence. But the derivation of observable predictions from hypotheses must be facilitated by deductive argument, building on an extant theory and the evidence supporting it. Indeed, derivation of predictions, rather than the logical confirmation (or disconfirmation) of hypotheses, constitutes the principal role of
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deduction in science. In any observation-based discipline it is the clever interplay of inductive and deductive reasoning that is a hallmark feature of robust scientific investigation.
2.6. STATISTICAL I N F E R E N C E Statistical procedures are involved in hypothesis testing at the point at which data are collected and subsequently used for comparison against predictions. The principles of survey and experimental design serve to improve the efficiency of data collection, and to ensure that the data are relevant and useful in the investigation of predicted responses. Subsequent to data collection, procedures for statistical inference play a key role in determining whether the predicted responses are supported by the data. A correspondence between data and predictions provides evidence for hypothesis confirmation, and the lack of a correspondence leads to hypothesis rejection. Statistical testing procedures often are framed in terms of mutually exclusive and exhaustive "null" and "alternate" hypotheses (Mood et al., 1974). By null hypothesis usually is meant (1) an assertion of extant theory that includes an accepted, sometimes simplified, form of some relevant biological relationship, or (2) a biological relationship per se, to be considered for replacement by an alternate hypothesis. By alternate hypothesis is usually meant a logically distinct, sometimes more complex, and often more appealing biological relationship that potentially can replace a particular null hypothesis. The mechanics of statistical testing involve the matching of observed evidence against predictions based on the null and alternate hypotheses, with the idea that both hypotheses cannot be true, but one must be. Thus, rejection of the alternate hypothesis leads automatically to acceptance of the null hypothesis as its only alternative. Several benefits accrue to the framing of test procedures in this manner. First, one retains the logical consistency afforded by modus tollens, whereby hypothesis rejection is inferred logically from the disconfirmation of a predicted response. For example, a lack of supporting evidence for predictions based on the additive mortality hypothesis leads to its rejection. Second, disconfirmation of predicted responses based on the alternate hypothesis leads automatically to acceptance of the null hypothesis. Thus, the rejection of the additive mortality hypothesis leads to the acceptance of the compensatory mortality hypothesis. Third, confirmation of predictions based on an alternate hypothesis leads automatically to rejection of the null hypothesis, and therefore to acceptance of
the alternate hypothesis. Thus, confirmatory evidence for the additive mortality hypothesis leads to rejection of the compensatory mortality hypothesis and to acceptance of the additive hypothesis. In this case the test discriminates cleanly between hypotheses irrespective of test results, and thereby avoids the fallacy of false cause. Though they appear to be analogous, acceptance/ rejection of the null and alternate hypotheses suffer disproportionate burdens of evidence. Indeed, the use of statistical procedures in hypothesis testing expresses an asymmetry that parallels that of syllogistic logic, based on a requirement that evidence must be quite strong to reject a null hypothesis in favor of the alternate. Thus, testing procedures express a scientific conservatism in which amendment of an extant theory, or acceptance of a favored alternate hypothesis, is to be discouraged without strong evidence that it is warranted. In this sense the asymmetry in statistical testing is analogous to that of logical inference, whereby hypothesis confirmation accrues only through a preponderance of evidence, in striking contrast with the relatively modest evidentiary requirements for hypothesis disconfirmation.
2.7. I N V E S T I G A T I N G COMPLEMENTARY HYPOTHESES Scientific methodology is framed above in terms of hypotheses about alternative mechanisms for an effect of interest, with the idea that only a single hypothesis is operative. Thus, one or more hypothesized mechanisms are considered as potentially explanatory, with repeated use of scientific methodology ultimately identifying the appropriate hypothesis. An underlying assumption is that there is only a single "appropriate" hypothesis, and that other hypothesized mechanisms under consideration will be found to be inadequate through proper use of scientific methodology. Although this scenario no doubt applies to causal mechanisms in many disciplines, it fails to apply to many interesting problems in population biology and ecology. In fact, biological science is replete with examples of complementary factors that interact in complex ways to produce observed effects. For example, it often is less a question of whether interspecific competition, predation, or habitat degradation is the cause of declines in a population, but rather the contribution each factor makes in the declines. In this case all factors may be operating simultaneously, playing important but unequal roles in influencing population dynamics. That issues involving simultaneous complementary
2.8. Discussion factors arise frequently in population biology is indicative of the complexity of the biological systems under investigation. Physical, ecological, and thermodynamic processes simultaneously influence these systems in a complicated network of interactions between populations and the communities and environments of which they are a part. A natural outgrowth of such complexity is the framing of many scientifically interesting issues about cause and effect in terms of the relative contribution of multiple causal factors (Quinn and Dunham, 1984). A useful approach then may involve the estimation of parameters measuring the level of factor influence, based on statistical estimation procedures (see Chapter 4).
2.8. D I S C U S S I O N Some researchers believe too much emphasis is placed on hypothesis testing as a signature feature of scientific methodology (Quinn and Dunham, 1984; Loehle, 1987). This concern is especially prevalent in the use of standard hypothesis testing procedures in statistics (Yoccoz, 1991; Johnson, 1999; Franklin et al., 2001). Quite often much of the information residing in sample data is overlooked in the process of hypothesis testing, because statistical tests address sometimes irrelevant questions about "significant" differences between treatments and controls. The lack of relevance is in large part a consequence of the fact that hypothesis tests often compare hypotheses, one of which (the null hypothesis) is unacceptable by design. Thus, the testing procedure is uninformative, in that it is designed at the outset to confirm what one already knows. The more biologically important information concerning the magnitudes of differences, or the parametric values defining the differences, or the biological structures underlying those differences, remains inadequately treated by statistical testing. The bottom line is that many, arguably most, scientifically interesting questions in biology are addressable by way of the estimation of parameters such as abundance, location, and proportionate influence, or by the selection of alternative models in which these parameters are imbedded. Both parameter estimation and model selection often are handled more effectively outside the context of hypothesis testing. We note, however, that irrespective of statistical method, biological investigation still depends on identification and/or parameterization of theoretically based relationships. It is unclear how such relationships can be recognized, or how assessed, separate from a foundation of theory. It is important to recognize that however hypotheses are investigated, investigation is actually an examina-
19
tion of both the hypothesis and the background theory. This can be seen in the arguments for hypothesis confirmation presented above. Thus, the rejection of predicted response P leads to rejection of the theory {T} as amended by hypothesis H. The argument above concluded that because the theory was assumed to be true, the hypothesis was necessarily false. Of course, it is always possible that the theory itself is false and the hypothesis is true (or both are false). Indeed, the history of science contains many examples of accepted theories that were shown eventually to be false (Kuhn, 1996). This ambiguity likely is an inevitable consequence of scientific methodology, whereby theories are constantly subjected to amendment and revision through the examination of hypotheses. Scientific methodology as described above involves theory amendment either by the addition of hypotheses to a theory, or by the replacement of one hypothesis by another. Standard practices of statistical testing fit well with the latter description, primarily because they are framed in terms of the comparison of null and alternate hypotheses. Two exceptions to this framework should be mentioned. First, it sometimes is the case that of two hypotheses under consideration, neither is easily recognizable as established, and there is a question about which hypothesis is to be identified as the null hypothesis and which as alternate. The decision is obviously of some operational consequence, because of the differential burden of evidence for null and alternate hypotheses. Under such circumstances nonscientific criteria, involving potential costs and benefits of hypothesis acceptance, often influence the decision. When this occurs it is important to recognize, and acknowledge, that the investigation is guided by objectives that go beyond the objective pursuit of understanding. Testing of the compensatory and additive mortality hypotheses provides a good example, with hypothesis acceptance/rejection strongly influenced according to which hypothesis is identified as null, although neither hypothesis is unambiguously recognizable as null. A second exception involves multiple comparisons of more than two hypotheses. Standard statistical procedures such as likelihood ratio testing do not lend themselves to the testing of multiple hypotheses, except with omnibus test procedures such as analysis of variance (Graybill, 1976) or by the comparison of hypotheses taken two at a time (Mood et al., 1974). However, some promising approaches have been identified that allow for the comparison and selection of hypotheses from among multiple candidates. For example, model selection criteria proposed by Akaike (1973,1974) have been used by Burnham and Anderson (1992, 1998) and others in the selection of biological
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Chapter 2 Scientific Process in Animal Ecology
relationships, and adaptive resource management (Waiters, 1986) provides a promising approach to the identification of population models from among multiple alternatives (Williams, 1996a). It is worth emphasizing that scientific methodology as described above is fully complementary to the traditional goals and objectives of population management. Indeed, many of the presumptive causes of biological patterns are recognized from observations made during the course of resource management, and in some instances management has been included in designs for their scientific investigation. The linkage between population management and scientific assessment, in which management both supports and is supported by research, is definitive of an adaptive approach to resource management (Waiters, 1986). Adaptive resource management, in concert with the use of sound scientific methodology, holds great promise for accelerating our understanding of biological processes, while simultaneously improving resource manage-
ment based on that understanding. In the long term, the melding of research and management may offer the only feasible approach to resolving many longstanding problems that confront wildlife and fisheries managers. We deal in considerable detail with adaptive management, and in particular with optimal adaptive decision making, in Chapter 24. We note in closing that population models represent hypotheses to be investigated, with components ranging from those known with great certainty to those derived only from guesses. The challenge is to analyze a model in such a way that the hypotheses strongly influencing model performance can be recognized and scientifically investigated. This task is almost never easy, and becomes increasingly difficult with increasing model size, complexity, scope, and amount of uncertainty as to model components. In Chapter 3 we turn to the relationship between hypotheses and models, and the use of both constructs in the conduct of science.
C H A P T E R
3 Models and the Investigation of Populations
3.1. TYPES OF BIOLOGICAL MODELS 3.2. KEYS TO SUCCESSFUL MODEL USE 3.3. USES OF MODELS IN POPULATION BIOLOGY 3.3.1. Theoretical Uses 3.3.2. Empirical Uses 3.3.3. Decision-Theoretic Uses 3.4. DETERMINANTS OF MODEL UTILITY 3.4.1. Simple versus Complex Models 3.4.2. Mechanistic versus Descriptive/ Phenomenological Models 3.4.3. More Integrated versus Less Integrated Model Parameters 3.5. HYPOTHESES, MODELS, AND SCIENCE 3.6. DISCUSSION
parameterization, and subsequent use of models provide one conceptual thread linking the themes of this book. In this chapter we are concerned with the relationship between theory (and associated hypotheses), as discussed in Chapter 2, and modeling, defined here as the abstraction and simplification of a real-world system (see Chapter 7). Our focus is on scientific models, which are used in the evaluation of hypotheses, and management models, which are used in making management decisions. We limit our discussion to models in population ecology and management, with a focus on model utility and the factors that make some biological models more useful than others. A key point in the chapter is that model utility is strongly influenced by the degree of correspondence between model structure and intended model use. The linkages between structure and function highlight some useful dichotomies in model development, and suggest a classification of models based on their utilization in science and management. The scientific and management literature includes many definitions of theories, hypotheses, and models. Some authors recognize little distinction among these concepts. For example, Neyman (1957) stated that "scientific theories are no more than models of natural phenomena." Hawking (1988) asserted that "a theory is just a model of the universe, or a restricted part of it, and a set of rules that relate quantities in the model to observations that we make." He also wrote that "any physical theory is provisional, in the sense that it is only a hypothesis" (Hawking, 1988). Other authors view the concepts hierarchically. For example, Pease and Bull (1992)stated that "hypotheses
As argued in Chapter 2, models are closely related to hypotheses, and as such are important components of both science and management. Indeed, progress in both science and management depends to a substantial degree on the recognition of a priori hypotheses, along with their articulation and assessment via biological models. The role models play in biological thinking is prominent throughout this book, so much so that the book might well be viewed as an exposition on population models. From this perspective, Part I provides background and a context for models with respect to science and management; Part II concerns the development of population models, with examples of model structures arising in population ecology and management; Part III deals with the estimation of attributes that parameterize population models; and Part IV describes the use of models in making decisions about the management of animal populations. The development,
21
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Chapter 3 Models and the Investigation of Populations
address much narrower dimensions of nature than the models from which they are derived." Hilborn and Mangel (1997) stated that "one can think of hypotheses and models in a hierarchic fashion with models simply being a more specific version of a hypothesis," thereby reversing the hierarchical positions of the terms. We note that the variety of definitions and uses of "theory," "hypothesis," and "model" need not be of great concern, provided the terms are at least operationally defined when they are used. On the other hand, needless confusion and miscommunication can arise in the absence of agreement as to their meanings. In what follows we utilize the conceptual framework developed in Chapter 2, which recognizes hypotheses as identifiable (and testable) elements of a broader scientific or management paradigm.
3.1. TYPES OF BIOLOGICAL MODELS It is a commonplace to identify different kinds of models depending on their uses. For example, a conceptual model refers to a set of ideas about how a particular system works. By translating these ideas into words, we create a verbal model. Similarly, translation of ideas or words into a set of mathematical equations yields a mathematical model. These different model types all correspond to our operational definition of the term model, in that they reflect an abstraction of key features of a system into a simple set of ideas, words, or equations that represents the system. One typically thinks of abstraction in terms of mathematical rather than physical models. However, just as physical phenomena can be modeled by mathematical constructs, so mathematical schemes can be modeled by physical constructs. Skellam (1972) stated that this "reverse modeling" includes "the most powerful instrument known for advancing empirical knowle d g e - t h e designed experiment." Experiments can be viewed as models based on our definition of the term, because they abstract from a real-world situation only a limited number of features to be investigated. In fact, the term empirical model often is used to mean a biological system that is amenable to experimentation. Perhaps the most famous empirical model in animal population ecology is the Tribolium model, a laboratory experimental system developed in the mid-1920s by R. N. Chapman for studying population growth and regulation using flour beetles (Chapman, 1928), and most commonly associated with the later work of T. Park and his students at the University of Chicago (e.g., Park, 1948; Neyman et al., 1956; Mertz, 1972; Wade and Goodnight, 1991). Over the years, work with the
Tribolium model has been characterized by close interaction between empirical and mathematical modeling and has led to strong inferences about many important aspects of population dynamics. A recent example involves prediction of points of transition in parameter values of a nonlinear mathematical model of animal population dynamics (Constantino et al., 1995). The testing of these predictions by altering adult mortality in experimental Tribolium populations produced shifts from point equilibria to stable periodic oscillations to aperiodic oscillations (Constantino et al., 1995). The impressive success of investigations using the Tribolium model highlights the value of investigation that involves the interactive use of mathematical and empirical models. Yet another model type is the physical model (e.g., a scale replica of an individual organism), examples of which have been used to good effect in physiological ecology to estimate heat exchange between organisms and their environments (e.g., Porter et al., 1973; Tracy, 1976). The mechanical model of Pearson (1960) can be viewed as one kind of physical model and is certainly one of the most interesting models ever developed for use in animal population ecology. Pearson's model had the appearance of a large pinball machine, with steel balls (representing individual animals) released onto an incline board. Holes were drilled into the board, and balls falling into these represented deaths. When a ball rolled over pairs of bronze strips, an electric circuit was completed and new balls (reproduction) were released from the top of the board. Pearson (1960) developed an algebraic model to describe the functioning of the mechanical model and utilized both models in producing counterintuitive results that eventually led to an improved understanding not achievable without both approaches.
3.2. KEYS TO S U C C E S S F U L M O D E L USE Just as there are many kinds of models, there also are many ways in which models can be used in the conduct of science and management (see Section 3.3). The large variety of applications precludes specific, detailed instructions about how to build and use a model; however, the following guidelines are offered as keys to successful model use (e.g., see Conley and Nichols, 1978; Nichols, 1987): 1. Clearly define the objectives of the modeling effort; i.e., provide an unambiguous statement of the way the model is to be used in the conduct of science a n d / o r management.
3.3. Uses of Models in Population Biology 2. Include in the model only those system features that are critically relevant to the objectives. Using these guidelines, we discuss below some classes of modeling objectives and provide suggestions for selecting biological features that enhance model utility with respect to objectives. We defer to Chapter 7 a discussion of specifics in developing and assessing a model.
3.3. USES OF MODELS IN POPULATION B I O L O G Y In a restrictive sense, the primary use of mathematical models in population ecology and management is to project the consequences of hypotheses. As noted in Chapter 2 (also see Johnson, 1999), it is possible to distinguish between scientific and statistical hypotheses, and useful to distinguish between their corresponding models. Scientific hypotheses represent stories about how a system works or responds to management actions, and scientific models are used to project the consequences of such hypotheses. For example, we may be able to use a simple conceptual model to anticipate population growth in a stable environment, or track population responses to harvest regulations, or predict species distributions in altered habitats. Because most models are too complicated to project system responses in our heads, scientific models can serve as "calculating engines" (Lewontin, 1963) to project the consequences of scientific hypotheses. Statistical hypotheses are derived from scientific hypotheses and represent stories not just about the system of interest, but also about observable quantities that are relevant to system behavior. A statistical model projects the behavior and values of these observable quantities (data) that are expected if the system is operating in a manner consistent with the statistical, and hence scientific, hypothesis. The construction of a statistical model is based not only on the underlying scientific hypothesis but also on aspects of sampling design and data collection. Thus, scientific models are used to project system dynamics, whereas statistical models are used to project the dynamics of observable quantities under one or more scientific hypotheses. Projections based on statistical models are used to estimate quantities of interest, including parameters of scientific models, and to discriminate among competing hypotheses by addressing the question: "Which statistical, and hence scientific, hypothesis corresponds most closely to the data?" Note that this distinction between scientific and statistical models corresponds closely to the distinction
23
between uses of deductive and inductive logic in the conduct of science (see Section 2.5). Thus, scientific models are used to deduce the consequences of their corresponding scientific hypotheses. On the other hand, statistical models are used to draw inferences about a particular hypothesis and to discriminate among competing hypotheses, through an inductive process based on limited observations. Finally, note that the organization of this book largely follows this distinction between model types, with Part II devoted to scientific models, Part III focusing on statistical models, and Part IV elaborating the interplay between these classes of model for the purpose of managing and understanding system dynamics. In addition to the broad classification of models as scientific and statistical, it is possible to categorize models based on the different classes of problems to which they are applied. In animal population ecology and management it is useful to identify theoretical, empirical, and decision-theoretic uses of models. Empirical and decision-theoretic uses typically require both scientific and statistical models, whereas theoretical uses are largely restricted to scientific models.
3.3.1. Theoretical Uses Here we define "theoretical" model use as the investigation of system responses that are possible under specific hypotheses. Model uses in this context do not entail a comparison of model predictions with data or observations, and indeed, the lack of a confrontation between predictions and data is the distinguishing feature of theoretical model use. The term theoretical as used here is consistent with Lewontin's (1968) view of theoretical population biology as "the science of the possible," and the views expressed in Caswell's (1988) essay on theory and models in ecology. For example, one might investigate with a model whether densitydependent migration can stabilize a particular metapopulation, or whether populations governed by a certain class of nonlinear equations exhibit chaotic behaviors, or whether populations subjected to certain harvest strategies exhibit thresholds in their responses. If such questions are tied to particular a priori hypotheses, then the use of models incorporating these hypotheses constitutes a form of hypothesis assessment and testing. Note that this use of the term theoretical has nothing to do with whether the model is used to address management-oriented questions. In fact, theoretical uses of models can be very important in the management of animal populations. For example, models that exhibit substantial mechanistic differences may lead to very similar management policies. From the perspective of
24
Chapter 3 Models and the Investigation of Populations
the manager, it thus would be unwise to devote resources to learning which model corresponds most closely to reality, because biological distinctions among the models would not be relevant to management decisions (e.g., Johnson et al., 1993; Williams and Johnson, 1995). For this reason one should investigate management implications prior to any effort to distinguish among management-oriented hypotheses and their corresponding models. Even when different models do lead to distinct management actions, it is useful to assess the management value of discriminating among them. It may be that a particular suboptimal policy performs adequately in terms of model objectives (e.g., number of animals harvested), regardless of variation among model-specific optimal policies (Hilborn and Walters, 1992). Modeling can be used to estimate the "expected value of perfect information" (see Section 24.7) as an aid in deciding whether it is worthwhile to expend effort discriminating among competing hypotheses (e.g., Hilborn and Walters, 1992; Johnson et al., 1993). Though modeling exercises of this kind are "theoretical" in the sense that they do not involve a confrontation with data, they nonetheless can be extremely useful from a pragmatic, management perspective.
3.3.2. Empirical Uses By "empirical" uses of models, we refer to predictions of population behaviors for the purpose of comparison with realized population behaviors. The confrontation of model predictions against data in an effort to discriminate among hypotheses (see Section 2.3) is a definitive feature of science-based investigation. Although various authors have identified a number of approaches to science (e.g., Hilborn and Mangel, 1997), here we focus on two generic alternatives and discuss the role of models in each.
Scientific models are used in step 2 to deduce predictions from the scientific hypothesis, whereas statistical models are used in the comparison of test results with these predictions (step 4). Advocates of this approach emphasize that the use of a critical experiment in step 3 is most likely to yield strong inferences (Platt, 1964). However, it is the single a priori hypothesis, rather than the nature of the test, that is the defining feature of this investigative approach. In the situation in which a hypothesis is rejected, there are two options (Fig. 3.1). One is to develop a completely new hypothesis and proceed as above with its investigation. The other is to revise the original hypothesis in a manner that renders it consistent with test results that led to rejection and then proceed with investigation of the revised hypothesis as above. In the event of a failure to reject the tested hypothesis, we again are left with two options (Fig. 3.1). One is to subject the hypothesis to still another test, using either the same or different predictions as those tested initially, recognizing that a hypothesis can be corroborated but can never be "proved" to be true (see Section 2.5) (Popper 1959, 1963). Alternatively, the hypothesis can be extended or otherwise modified, and a test can be formulated that focuses on the extension or modification. Iterative hypothesis testing and refinement as above eventually may identify a hypothesis that survives repeated efforts at falsification and consistently predicts system behaviors. Under these conditions the hypothesis then is accepted as provisionally true, in that we view it as our best approximation of reality (subject, of course, to subsequent investigation and possible refinement).
3.3.2.2. Multiple-Hypothesis Approach This approach usually is traced to a paper by Chainberlin (1897) on multiple working hypotheses (Platt,
3.3.2.1. Single-Hypothesis Approach This approach frequently is associated with the writing of Popper (1959, 1963, 1972) and the influential paper by Platt (1964) on strong inference. The approach is outlined in the following steps, using the elements of the scientific approach identified in Section 2.3: 1. Develop or identify a hypothesis (typically from existing theory). 2. With the help of the associated model, deduce testable predictions. 3. Carry out a suitable test. 4. Compare test results with predictions. 5. Reject or retain the hypothesis.
H
/ H 1
Accept
1
Reject
,/',,, H/1
H//1
H 2
FIGURE 3.1 Schematic representation of the single-hypothesis approach to scientific inquiry. H1 denotes the original hypothesis tested, H{ denotes an extension or elaboration of H1, and H~ denotes a revision of H1 designed to account for those aspects of test results that deviate from predictions of H1. H2 is a new hypothesis.
3.3. Uses of Models in Population Biology 1964). Multiple hypotheses are also an important part of the scientific research programs described by Lakatos (1970). The application described here for biological investigation is adapted primarily from adaptive resource management (Chapter 24; also see Walters, 1986; Johnson et al., 1993; Williams, 1996a), but the joint use of multiple hypotheses is relevant regardless of the motivation for learning. It is outlined in the following steps: 1. Develop a set of competing hypotheses. 2. Derive a set of probabilities associated with these hypotheses. 3. Use associated models to deduce testable predictions. 4. Carry out a suitable test. 5. Compare test results with predictions. 6. Based on this comparison, compute new probabilities for the hypotheses. Mathematical models again are prominent in this approach. Thus, scientific models corresponding to the different hypotheses are used to deduce competing predictions (step 3), and statistical models provide a framework for comparison of test results against these predictions (step 5), leading to new probabilities for the competing hypotheses (step 6). The probabilities in step 2 can be viewed as measures of our relative faith in the different hypotheses. Let P ( H i) denote the probability associated with hypothesis H i, with ~i P(Hi) = 1. Then the comparison of test results (step 5) with predictions of the different hypotheses/models leads to an updating of these probabilities (step 6). We note that this approach is not as widely utilized as the single-hypothesis approach to science, in part because of the need to identify and update hypothesis probabilities or "likelihoods." One approach to probability updating is based on likelihood functions (see Chapter 4) in conjunction with Bayes' Theorem (e.g., see Hilborn and Mangel, 1997). The likelihood function ~s for hypothesis H i describes the "likelihood" (for discrete random variables these are probabilities) of collecting the test data for parameters 0 i of the statistical model corresponding to H i (see Section 4.2.2). The likelihoods corresponding to different hypotheses in the set can be computed directly, using the observations in conjunction with the statistical models associated with H i. Given the set {P[Hi]} of prior probabilities, the test data, and the likelihoods ~(_0i]data) for the different hypotheses, we can compute updated probabilities P' (H i) for the different hypotheses by ~(Oi[data)P(Hi) P' (H i) = ~, ~(Oi]data)P(Hi )
i
(3.1)
25
P(H1)
L1
P(H3) FIGURE 3.2 Schematicrepresentation of the multiple-hypothesis approach to scientific inquiry. H i denotes each of several (e.g., three) alternative hypotheses, with associated probabilities P(Hi). Following an experiment or management intervention, predictions from each hypothesis are compared to observations, to form likelihoods (Li). Bayes' Theorem [Eq. (3.1)] then is used to provide updated values for P(Hi), and the process repeats, now using the updated values P(Hi).
[e.g., see Section 4.5 and Hilborn and Mangel (1997)]. The updated probabilities P'(Hi) then become prior probabilities for subsequent updates with additional data (Fig. 3.2). Likelihood functions, maximum likelihood estimation, and Bayes' Theorem are discussed in detail in Chapter 4, and their application in probability updating is described in Chapter 24. Learning can be thought of as a change over time in the probabilities associated with the different hypotheses (Fig. 3.3). These hypotheses are viewed as competing for our confidence, and each comparison of field data against model-based predictions leads to a change in their probabilities. We expect the probability to increase for the most appropriate hypothesis, and to decrease for the other hypotheses. For example, the accumulation of probability for model 3 in Fig. 3.3 reflects increasing faith in hypothesis 3 as an approximation of reality.
1
-
0.8 A
0.6-
~"
0.4-
--~- M1 1
-__ M 2 ' .
M3
0.2 0
.
0
.
.
.
.
2
.
.
.
4
.
6
8
FIGURE 3.3 Hypothetical changes in probabilities associated with three hypotheses under the multiple-hypothesis approach to scientific inquiry. P denotes probability, and M i denotes the model associated with hypothesis Hi. An investigation (e.g., an experiment) occurs between each pair of steps, and comparison of model-based predictions with test results leads to changes in the probabilities associated with the different models [e.g., using an approach such as Eq. (3.1)].All three hypotheses begin with equal probabilities (e.g., assuming the absence of prior knowledge by which to discriminate among them) at step 1, and investigation leads to high probabilities associated with M 3 and its corresponding hypothesis, H3.
26
Chapter 3 Models and the Investigation of Populations
3.3.2.3. Popper's Natural Selection of Hypotheses In discussing the role of theory in scientific investigation, Popper (1959) wrote that "We choose the theory which best holds its own in competition with other theories; the one which, by natural selection, proves itself the fittest to survive." He later expanded on this analogy (Popper, 1972), noting "the growth of our knowledge is the result of a process closely resembling what Darwin called 'natural selection'; that is, the natural selection of hypotheses: our knowledge consists, at every moment, of those hypotheses which have shown their (comparative) fitness by surviving so far in their struggle for existence; a competitive struggle which eliminates those hypotheses which are unfit." Thus, candidate hypotheses are subjected to falsification tests, and some survive the testing whereas others do not. Popper's analogy between hypothesis testing and natural selection extends easily to the multiple-hypothesis approach to science. Instead of focusing attention on a single hypothesis (analogous to an individual or a genotype) and its survival in the various confrontations with data, our attention is on the hypothesis probabilities P(Hi), which can be viewed as analogous to gene frequencies. Just as selective events bring about adaptive changes in gene frequencies within the population, so do our experiments and tests bring about changes in the probabilities associated with the hypotheses under consideration. Changes in gene frequencies over time reflect the action of natural selection, and changes in hypothesis probabilities reflect the relative predictive abilities of the different hypotheses and their models. The focus is on natural selection and learning, respectively, as the prime determinants of change, recognizing that other sources of variation influence changes in gene frequencies (e.g., environmental variation; "drift" associated with the stochastic nature of fitness components) as well as hypothesis probabilities (e.g., environmental variation; uncertainty about population size).
3.3.2.4. Recommendations Based on the Multiple-Hypothesis Approach The multiple-hypothesis approach to science is not as widely used as the single-hypothesis approach, and as a result, not as much thought has been devoted to it by those interested in scientific methodology. We offer two methodological recommendations. The first is simply to reiterate and reinforce the view long held by scientists, that science is a progressive endeavor. For example, in 1637 Descartes wrote "I hoped that each one would publish whatever he had learned, so that later investigations could begin where the earlier
had left off" [Descartes (translation), 1960]. Modern ecologists often pay only limited attention to the previous work of others, as evidenced by the perfunctory paragraph or so found in introductory sections of most scientific papers (though authors of review papers frequently do attempt to generalize the results of previous work). Our recommendation is to take full advantage of knowledge gained from past work, by accounting when practicable for previous investigation via assignment of probabilities to hypotheses based on past research. A key to this approach is the development of explicit models associated with members of a hypothesis set, which can be used to identify hypothesisspecific predictions from past investigations. Comparison of these predictions with the test results then permits one to update the hypothesis probabilities [e.g., as in Eq. (3.1)]. This approach of course depends on the amount of detail provided in the reporting of past work; but even in cases in which the level of detail is less than optimal, it still may be possible to design and revise hypothesis probabilities, though perhaps less formally. We believe the multiple-hypothesis approach provides a means of better utilizing results from previous investigation, via the updating of prior probabilities. A second recommendation involves study design and statistical methodology, and it emerges from optimal management designs (Part IV) under the rubric of adaptive management (also see Walters, 1986; Johnson et al., 1993; Nichols et al., 1995a; Williams, 1996a). Hilborn and Mangel (1997) note that the historical development of the single-hypothesis approach to science was accompanied by a corresponding development of associated statistical methods. A great deal of thought and effort have been devoted to the design of experiments, with the intent of rejecting or tentatively accepting a priori null hypotheses (Chapters 4 and 6; also see Fisher, 1947, 1958; Cox, 1958). After incorporating the critical design elements (e.g., randomization and replication) for reliable inference, investigators frequently turn their attention to test power, i.e., the probability of rejecting a null hypothesis when it is false (see Sections 4.3 and 6.7). Power frequently is viewed as an optimization criterion in experimental design (e.g., Skalski and Robson, 1992), and efforts are made to maximize power for fixed values for other test characteristics. Under a multiple-hypothesis approach, design criteria based on the rejection of a single hypothesis are no longer relevant. Instead of maximizing test power, the multiple-hypothesis approach seeks to maximize discrimination among models, via sampling and experimental designs for that purpose. Formal, actively
3.3. Uses of Models in Population Biology adaptive management can utilize optimal control methods to identify management policies supporting this objective (Part IV) (see Walters, 1986; Johnson et al., 1993; Nichols et al., 1995a; Williams, 1996a; Conroy and Moore, 2001). Thus, we should be able to use optimization (Chapters 21-23) (see Bellman, 1957; Williams, 1982, 1989, 1996a,b; Lubow, 1995; Conroy and Moore, 2001), in conjunction with objective functions that focus on discrimination among hypotheses, to develop optimal designs. For example, one might use as an objective function a diversity index such as the Shannon-Wiener H' (e.g., Krebs, 1972), computed with the prior probabilities. Diversity indices such as H' are minimized when one of the P(H i) approaches one and the remaining P(H i) approach zero (i.e., when we are confident that one of the hypotheses approximates reality better than all of the others). Regardless of specifics in the investigation, we recommend the use of optimization methods to assist in discriminating among multiple hypotheses.
3.3.3. Decision-Theoretic Uses An important application of models involves projecting the consequences of hypotheses about how a system behaves, for the purpose of identifying appropriate management actions. Just as two approaches to science were discussed under empirical model uses, two approaches to decision-making can be identified here. The following ideas are developed more fully in Part IV.
3.3.3.1. Single "Best Model" Approach This approach to decision-making is common in natural resource management. It relies on a single model that is judged to be the best available for predicting system responses to management actions, and it utilizes (1) an objective function (a formal statement of management objectives), (2) a favorite hypothesis (and corresponding scientific model) for the managed system, (3) a set of available management actions that can be taken to achieve management objectives, and (4) a monitoring program that provides time-specific information about system status and other variables relevant to the objective function. Based on these prerequisites, implementation of the single "best model" approach to management involves the following iterative steps: 1. Observe the current state of the system. 2. Update model parameter estimates, if appropriate, based on current information. 3. Identify an appropriate (or optimal) management action.
27
4. Implement management action and return to step 1. Step 2 usually is based on a statistical model, whereas step 3 typically uses a scientific model of the system. Given the objective function and information on the current state of the system, the scientific model is used in step 3 to identify the management action most likely to meet management objectives. In some cases, the model may be used to project the consequences of a suite of management actions, and the optimal decision is chosen based on the results. Alternatively, optimization algorithms (e.g., Williams, 1982, 1989, 1996a,b; Lubow, 1995) can be used to identify optimal management actions with respect to objectives. In either case, implementation of the management action (step 4) drives the system to a new state, and the process is repeated.
3.3.3.2. Multiple-Model Approach This approach to making management decisions is most commonly associated with adaptive management (Waiters, 1986; Johnson et al., 1993, 1997; Williams, 1996a; Conroy and Moore, 2001). Prerequisites for the approach include the following: (1) an objective function, (2) a model set consisting of the scientific models associated with competing hypotheses about how the managed system responds to management, (3) prior probabilities associated with the different hypotheses (and thus their models) in the model set, (4) a set of available management options, and (5) a monitoring program providing time-specific information about system status and other variables relevant to the objective function. Implementation of the multiple-model approach to management then involves the following iterative steps: 1. Observe the current state of the system. 2. Update model probabilities based on current information. 3. Derive the optimal management action. 4. Implement management action, and return to step 1. The information in step 1 about the current state of the system is provided by the monitoring program, and the estimated state of the system at time t + 1 is compared with predictions made at time t by each of the models as a basis for revising model probabilities (step 2). The updating of model probabilities is accomplished using statistical models with an algorithm [e.g., Eq. (3.1)], whereby probabilities increase for models that effectively track the observations and decrease for models that do not. Derivation of optimal management
28
Chapter 3 Models and the Investigation of Populations
actions is based on the competing scientific models and utilizes optimal control methods (e.g., Bellman, 1957; Anderson, 1975b; Williams, 1982, 1989, 1996a,b; Lubow, 1995; Conroy and Moore, 2001) that account for future effects of present actions. Implementation of the optimal management action (step 4) then drives the system to a new state, and the process is repeated.
benefits to the fisheries of such progress can hardly be exaggerated.
Beverton and Holt clearly recognized fishery biology to be a dual-control problem and recommended an essentially adaptive approach to the management of fishery resources.
3.4. DETERMINANTS OF MODEL UTILITY
3.3.3.3. Learning through Management The growth of knowledge in the field of wildlife management has not been as rapid as many would like (Romesburg, 1981, 1991). One path to faster learning is to make more intelligent use of management for learning (e.g., Holling, 1978; Walters and Hilborn, 1978; MacNab, 1983; Walters, 1986; Murphy and Noon, 1991; Sinclair, 1991; Johnson et al., 1993, 1997; Lancia et al., 1996; Williams, 1997; Conroy and Moore, 2001). Learning through management can occur with either a single-model or a multiple-model approach. Under the single-model approach, predicted system responses to management are compared with the observed (estimated) response. Based on this comparison, the model and its associated hypothesis are either retained for future use, or are rejected and replaced by a new hypothesis and model. Learning thus occurs in the same manner as under the single-hypothesis approach to science, except that here it is an unintended by-product of efforts to meet direct management objectives (see Chapter 24). The multiple-model approach to management also involves a comparison of model predictions with observed system responses, except that the comparison leads to changes in model probabilities (i.e., to learning). For example, "active adaptive management" (Walters and Hilborn, 1978; Walters, 1986; Hilborn and Walters, 1992; Williams, 1996a) uses multiple models to identify optimal management decisions as solutions to the so-called "dual-control problem" (e.g., Walters and Hilborn, 1978) of trying simultaneously to learn (because learning increases our ability to manage in the future) while achieving management objectives. The idea of using management adaptively to discriminate among competing models was articulated in the 1970s by Holling (1978) and Walters and Hilborn (1978). However, their work was presaged as early as 1957 in the pioneering book by Beverton and Holt (1957): It is the changes produced in the fisheries by the regulations themselves...that provide the opportunity of obtaining, by research, just the information that we may have been lacking previously. Thus the approach towards optimum fishing, and the increase in knowledge of where the optimum lies, can be two simultaneous and complementary advances; the
The successful use of models requires clear, unambiguous objectives of the modeling effort, and a focus on biological features of the modeled system that are critically relevant to the objectives. In the previous section we discussed model objectives in population ecology and management in terms of theoretical, empirical, or decision-theoretic uses. Here we focus on the selection of critical system features for a model, with the recognition that this selection ultimately determines model utility. We emphasize three gradients that are especially relevant to model development, which provide a convenient format to illustrate some issues for consideration when one develops a model.
3.4.1. Simple versus Complex Models By definition, the process of modeling involves abstraction and simplification (see Chapter 7), and thus entails a loss of information in the modeling of any real biological system. For that reason every modeler must face a question about model complexity (e.g., Levins, 1966; Walters and Hilborn, 1978). We believe that the modeling process can be usefully viewed as a filter, in which the full complement of information of a real system is passed through the filter and only the system attributes that are essential to the modeling objectives are retained. When the filter is informed by an intended application, the modeling process becomes an effort to match model complexity with model use.
Biologists often overlook the importance of matching complexity to intended use, and indeed, many have a natural tendency to create models that are more complex than necessary. For example, Nichols et al. (1976b) used a detailed simulation model of an alligator population to draw general inferences about the relative effects of size- and age-specific harvest on alligator population dynamics. The model included various components of reproductive and survival rates, but many of the general objectives of the modeling effort could have been met using a much simpler population projection matrix approach (Nichols, 1987). It is not difficult to carry the tendency for complex
3.4. Determinants of Model Utility explanation beyond the point of usefulness. Referring to the science of geographical ecology, MacArthur (1972) wrote that "The best person to do this is the naturalist...But not all naturalists want to do science; many take refuge in nature's complexity as a justification to oppose any search for patterns." Biologists have a natural tendency to focus on complexity. Indeed, the central guiding paradigm of all the biological sciences is Darwinian evolution by natural selection, and the raw material for this process is natural (and heritable) variation. Ecologists thus are taught to focus on differences between individual organisms, between organisms and their behaviors in different habitats, between species, etc., and to build selective stories to explain these perceived differences [see discussion in Gould and Lewontin (1979)]. In an extreme view of variation and complexity, the behavior or fate of an individual organism at a particular point in space and time is a unique event, one which is often of little use in predicting fate or behavior of another (or even the same) individual at a different point in space and time. Under this view, biologists are involved in descriptive work, and perhaps in a posteriori story telling, but not in science. On the other hand, a scientific view searches for generalizations among individual events, in the expectation that at some scale biological phenomena are at least stochastically predictable. This view leads back to modeling and to the recommendation that we incorporate in a model only those aspects of system complexity that are essential for meeting the objectives of the modeling effort.
3.4.2. Mechanistic versus Descriptive/ Phenomenological Models By mechanistic models we mean those that depict causal relationships between variables, in the sense that changes in one variable are directly responsible for changes in another. On the other hand, descriptive/ phenomenological models define statistical relationships between variables, without incorporating underlying mechanisms that are responsible for the relationships. We note that to a certain extent this distinction is in the eye of the beholder, because all models can be viewed as descriptive and phenomenological at some level, and most express at least some degree of biological mechanism. To illustrate the dichotomy, consider the relationship between hunting and population survival rates (Johnson et al., 1993). Different hypotheses about the effects of hunting mortality on annual survival rates of mallard ducks can be incorporated into the equation
29 S i --
0(1 - ~Ki),
(3.2)
where $i is the probability that a bird alive at the beginning of the hunting season in year i is still alive at the beginning of the hunting season the next year (Anderson and Burnham, 1976; Burnham et al., 1984; Nichols et al., 1984d). The parameter 0 usually is viewed as the probability of annual survival in the absence of hunting, and K i is the probability that a bird alive at the beginning of the hunting season in year i dies as a result of hunting during the subsequent season. The parameter ~ denotes the slope of the relationship between annual survival and hunting mortality rate. If = 1, then Eq. (3.2) corresponds to the completely additive mortality hypothesis, under which hunting and nonhunting mortality sources act as independent competing risks. If ~ = 0, at least for some values of K i (e.g., for K i < c, where c reflects a threshold value), then Eq. (3.2) corresponds to the completely compensatory mortality hypothesis under which variation in hunting mortality (below c) brings about no corresponding variation in annual survival (Anderson and Burnham, 1976; Burnham et al., 1984; Nichols et al., 1984d). Chapters 8 and 10 provide more details on the compensatory and additive mortality hypotheses. Recent analyses of band recovery data for North American mallard ducks have produced very different estimates of the slope parameter 13when based on data from different decades. A proposed explanation for this difference identifies density-dependent nonhunting mortality as the most likely mechanism underlying compensatory mortality (e.g., Anderson and Burnham, 1976; Nichols et al., 1984d). Thus, density-dependent responses to changes in hunting mortality would be expected to differ in years of high and low mallard abundance. Johnson et al. (1993) recommended the survival model S i --
with
0i
0i(1 - Ki),
(3.3)
given by ea+bNi(1 -Ki) Oi--
1
+ e a+bNi(1-Ki)"
(3.4)
where N i is the number of mallards alive at the beginning of the hunting season in year i, and a and b are parameters to be estimated. The finding that different estimates of 13 are necessary for different time periods is indicative of the inadequacy of Eq. (3.2) to account for the essential features of the modeled system. Indeed, the density dependence expressed in Eqs. (3.3) and (3.4) guarantees that no single value of ~ in Eq. (3.2) will perform well for populations with widely varying abundances. Given density dependence, Eq. (3.2) might represent survival
30
Chapter 3 Models and the Investigation of Populations
reasonably well over the range of mallard abundance used in estimating its parameters, but it would not be expected to perform well beyond that range (Johnson et al., 1993). Of course, it may be that the true relationship between annual survival and hunting mortality is not well represented by Eqs. (3.3) and (3.4) either, in that they omit some other essential feature of system response to hunting. The main point here is that if density dependence really does underlie the compensatory mortality phenomenon, then Eq. (3.2) inadequately represents the system, and Eqs. (3.3) and (3.4) provide a somewhat more mechanistic, and possibly more useful, model. Our general recommendation regarding this dichotomy is to tend toward more mechanistic models, because they are more likely to provide useful predictions when state a n d / o r environmental variables assume values outside the range used in estimating model parameters. The notion of a mechanistic model is closely related to the idea of extracting essential features of the modeled system, recognizing that mechanism often begets model complexity. Certainly, models that are mechanistic in ways not essential to the purpose of the model (e.g., a model of the physiological death process as steel pellets enter the body cavity of a duck in our example) should be avoided.
rate, K. Some predictions of the hypothesis can be tested using annual survival rate Si, but other predictions can be tested only with estimates of seasonal survival 0 i and hunting mortality K i. Levins (1966, 1968) introduced the term sufficient parameter, as "a many-to-one transformation of lower level phenomena" (Levins, 1966, p. 429), emphasizing its role in integration and aggregation in his discussion of the term. By analogy with "sufficient statistic" from mathematical statistics (e.g., Mood et al., 1974), it seems reasonable to think of a sufficient parameter as one that contains all of the information needed to accomplish the function for which the model is intended. Thus, we return to the second general determinant of model utility and note that the degree to which model parameters are aggregated should reflect the intended model use. For example, if we develop models with the intent of comparing predictions of competing hypotheses under some sort of treatment or manipulation, then the models should include a parameter structure that accommodates the treatment or manipulation, and they must yield predictions that are useful in discriminating among the competing hypotheses.
3.5. HYPOTHESES, MODELS, A N D SCIENCE
3.4.3. More Integrated versus Less Integrated Model Parameters Integration of model parameters reflects the degree to which process components that could be modeled with separate parameters are aggregated into a single parameter. The concept is easily illustrated via an example involving animal population ecology. The finite rate of increase ~ of a population (defined here as the ratio of population sizes in two successive years) is sometimes used to model population change (see Chapters 7-9). We view K as an integrated parameter in the sense that it aggregates effects of survival, reproduction, and movement on the population. In the case of a population closed to movement (e.g., an island population), it sometimes is useful to decompose into two parameters, an annual survival rate and a reproductive rate. For still other modeling purposes, it is better to decompose annual survival rate into component survival probabilities corresponding to different seasons of the year, and to decompose reproductive rate into functional components such as breeding probability, clutch size, hatching success, and brood survival. Assume, for example, that our a priori hypothesis about effects of hunting mortality on mallard survival is given byEqs. (3.3) and (3.4). This hypothesis would be very difficult to test using only population growth
Regardless of the scientific approach, effective learning is conditional on, and accomplished relative to, a priori hypotheses and their associated models. In our view this point is not adequately appreciated by some practitioners of biological science. We emphasize that under a single-hypothesis approach to science (or management), inferences are tied to an a priori hypothesis and its associated model-based predictions, with investigation leading to a decision to reject or provisionally retain the hypothesis. Similarly, a multiple-hypothesis approach to science and management is conditional on a set of a priori hypotheses and their corresponding models. The associated prior probabilities are standardized in the sense that they sum to one over the hypothesis set, and changes in hypothesis probabilities (learning) are entirely conditional on the members of that set. Indeed, the conditional nature of learning holds even if none of the hypotheses under consideration provides a reasonable approximation to reality. These considerations lead to the suggestion that more thought and effort should be devoted to the development of a priori hypotheses and their associated models. It is common for research papers to begin with the statement of a statistical null hypothesis to be tested and thus to give the appearance of scientific rigor. Though the expression of a null hypothesis is not neces-
3.6. Discussion sarily a bad thing (but see Burnham and Anderson, 1998; Johnson, 1999; Anderson et al., 2000), its value to science depends heavily on the nature of the alternative hypothesis. For example, the testing of a null hypothesis of "no difference" or "no variation" against an omnibus alternative of "some difference" or "some variation" is not likely to be useful. At a minimum, testing should be based on numerical, or at least directional, predictions from a priori biological hypotheses and their associated models. As indicated in Chapter 2, competing hypotheses often can be investigated through estimation of parameters reflecting ratios, absolute differences, or other measures of variation in system variables under different treatments. The utility of experimentation is strongly emphasized in ecology, so much so that ecologists tend to view manipulations and perturbations as inherently good and useful to the scientific endeavor. It is true that experimentation can be an extremely powerful means of learning about natural systems (see Chapter 6). However, a priori biological hypotheses are key to informative experimentation, just as they are to more descriptive studies. Manipulations and perturbations conducted simply to "see what happens" are not likely to be nearly as useful to science as those conducted to see what happens relative to model-based predictions. The conditional nature of scientific learning argues that we devote substantial effort in identifying useful
31
hypotheses and developing models corresponding to them. When experience indicates that the hypotheses under consideration are inadequate predictors, it is important to devote additional effort in developing new hypotheses. We believe that the disciplines of animal population ecology and management would be well served by renewed emphasis on the articulation of meaningful hypotheses and their associated models.
3.6. D I S C U S S I O N In this chapter we have focused on the role of models in the conduct of science and management on animal populations. Models are useful, and often essential, in the conduct of science and management, and this theme will be continued in subsequent parts of this book. In terms of model development, we emphasize the importance of specifying objectives of a modeling effort and then tailoring the model to those objectives. This involves an effort to include in the model only those features of the system that are critically relevant to the modeling objectives. In the chapters to follow we describe various model structures for animal populations (Part II), methods for estimating parameters and relationships required for model development (Part III), and applications of models to management decisions (Part IV).
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C H A P T E R
4 Estimation and Hypothesis Testing in Animal Ecology
4.1. STATISTICAL DISTRIBUTIONS 4.1.1. Some Discrete Distributions for Animal Populations 4.1.2. The Normal Distribution for Continuous Attributes 4.1.3. Distribution Parameters 4.1.4. Replication and Statistical Independence 4.1.5. Marginal and Joint Distributions 4.1.6. Conditional Distributions 4.1.7. Covariance and Correlation 4.2. PARAMETER ESTIMATION 4.2.1. Bias, Precision, and Accuracy 4.2.2. Estimation Procedures 4.2.3. Confidence Intervals and Interval Estimation 4.3. HYPOTHESIS TESTING 4.3.1. Type I and Type II Errors 4.3.2. Statistical Power 4.3.3. Goodness-of-Fit Tests 4.3.4. Likelihood Ratio Tests for Model Comparisons 4.4. INFORMATION-THEORETIC APPROACHES 4.5. BAYESIAN EXTENSION OF LIKELIHOOD THEORY 4.6. DISCUSSION
and the scientific process. In both chapters we mentioned uncertainties that result from only partially observing a biological system. However, we are yet to account in a systematic way for this "partial observability," or to describe procedures for using field and experimental observations to parameterize and test models. Ecological systems are variable over space and time, and among individuals, and typically are observed only incompletely, that is, by means of samples. In this chapter we focus on stochastic variation that arises in parameter estimation with sample data, leaving until later a discussion of stochastic effects that arise through random environmental influences. Stochastic influences of the latter kind are distinct from sampling variation, in part because of the manner in which they propagate through time (in that the influence of a stochastic input at time t carries forward to time t + 1 and possibly beyond). The statistical modeling of such phenomena constitutes the discipline of stochastic processes and is discussed in Chapter 10. In this chapter our concern is to investigate the stochastic nature of sample-based parameter estimates, preparatory to an assessment of variation in model predictions. Population parameters typically are estimated by means of a representative sample of field data that are appropriate for the parameter of interest. These data are collected according to some scientifically supportable protocol (see Chapters 5 and 6), by means of which statistical properties of the sample can be ascertained. Mathematical formulas are used to combine the data into parameter estimates, with follow-up analyses of their statistical properties. Because only a sample of potentially available data is used in the process of esti-
In Chapters 2 and 3 we discussed the role of models in evaluating hypotheses about natural processes, and the use of models in making management decisions. Chapter 2 articulated a framework for examining theoretically based hypotheses and highlighted the comparison of predictions from theoretical models against field or experimental data. In Chapter 3 we discussed theoretical, empirical, and decision-theoretic uses of models in natural resources and further elaborated some operational linkages among models, hypotheses,
33
34
Chapter 4 Estimation and Hypothesis Testing
mation, the estimates are subject to sampling variability. The actual or "true" values of the parameters are not known with certainty, and assertions about them must be framed in terms of the statistical likelihood of their values. In this chapter we explore the estimation of population parameters based on maximum likelihood and other estimation procedures. The organizing concepts in the following discussion are (1) a statistical population, from which samples of individuals are to be drawn; (2) a distribution of values associated with individuals in the population; and (3) a formula, or estimator, for combining sample values into a numerical value or estimate of a population parameter. For purposes of this development a statistical population is defined as a collection of individuals that potentially can be sampled in an investigation. A population becomes statistical when (1) the sampling procedure is according to some sampling design whereby probabilities can be ascribed to samples, and (2) specific characteristics of interest (such as sample size, measurements of individuals in a sample, etc.) are recorded for the sample. Statistical populations might include the following examples: 9 White-tailed deer (Odocoileus virginianus) in Georgia. Measurements: length, weight. Sampling design: stratified random sample, with stratification based on sex and age. 9 American black ducks (Anas rupribes) in their breeding grounds. Measurements: age, sex. Sampling design: stratified random sample, with stratification based on geographic location. 9 Wood duck (Aix sponsa) nest boxes in the Missisquoi National Wildlife Refuge, Vermont. Measurement: use/nonuse. Sampling design: simple random sample, with stratification based on geographic location. 9 Striped bass (Morone saxatilis) in Lake Hartwell, Georgia. Measurement: age. Sampling design: stratified random sample. 9 Students at the University of Georgia. Measurement: eye color. Sampling design: simple random sample. 9 Hunting licenses purchased by Vermont residents. Measurement: county of residence. Sampling design: simple random sample. 9 Fish in the Connecticut River. Measurement: species. Sampling design: stratified random sample with stratification based on latitude. 9 Duck nests in North Dakota. Measurement: success/ failure. Sampling design: random sample of nests, followed by repeated visits to individual nests until either fledging or failure of the nest occurs.
9 Harvested waterfowl. Measurement: species, age, and sex. Sampling design: stratified cluster sample, with stratification by state of harvest and clustering by county within state. 9 Trees above 300 m elevation on Camel's Hump Mountain, Vermont. Measurements: tree size, tree height. Sampling design: Cluster sample, with clustering defined by grid points.
4.1. STATISTICAL
DISTRIBUTIONS It is useful to think of natural variation in organisms and their environments in terms of some underlying pattern or distribution of occurrence. To illustrate, consider some observable characteristic of individuals in a population--for example, the size, weight, or age of each organism in a biological population. Ideally the value of the characteristic for any individual can be determined unambiguously, simply by observing (and perhaps measuring) the individual. The relative frequencies with which different values occur in the population constitute a frequency distribution. The frequency distribution reflects a probability distribution for these values when individuals in the population are sampled randomly. In that case, the attribute values for randomly selected individuals occur with probabilities equal to the frequencies of occurrence in the population, so that the frequency and probability distributions are identical. It is useful to distinguish discrete distributions, for which the attribute of interest can assume only a countable number of values, from continuous distributions, for which attribute measures can range over a continuous set of values. Examples of discrete distributions include the following entities: 9 Survival or mortality (two classes). 9 Nest box s t a t u s w u s e d / u n u s e d (two classes). 9 Capture status--captured/not captured (two classes). 9 Taxonomic class of harvested waterfowl (e.g., dabbling ducks, diving ducks, geese). 9 Age/sex class of migrating black ducks (four combinations of age and sex). 9 Atlantic salmon ages (e.g., five age classes). 9 Duck eggs per nest (positive integers). Note that in each of these examples there are only countably many values for the attribute of interest. For continuous distributions the attribute can assume any value over a continuous range. Examples of continuous distributions could include bear weights (positive real numbers), tree heights (positive real numbers), and
35
4.1. Statistical Distributions deviations from average heart rate (positive and negative real numbers).
4.1.1. Some Discrete Distributions for Animal Populations The distribution of a population attribute often can be described with a mathematical function f(x), which allows one to specify with a single formula the frequency of occurrence of any attribute value x. The function f(x) also specifies the probability of occurrence of x for r a n d o m l y sampled individuals, and in that case is called the probability density function. Some important examples follow. 4.1.1.1. Binomial Distribution R a n d o m events in which one of two outcomes can occur are k n o w n as Bernoulli trials. For example, suppose 1 and 0 designate head and tail, respectively, for outcomes of a coin toss. Then the distribution of outcomes can be described by
f(x) = (0.5)x(0.5)1-x. Thus, the frequency of occurrence of heads (x = 1) is f(1) = 1A, and the frequency of occurrence of tails (x = 0) is f(0) = 1A. N o w assume that the head side of the coin is m a d e of lead and the tail side is m a d e of copper. Then the relative frequency of a head is no longer 1/2, but is some general value p. In essence, p is a p a r a m e t e r for the distribution of outcomes (i.e., a constant that provides information about distribution structure). The frequency distribution for this more general case
is
f(xlp) = pX(1
_
p)l-x,
and is k n o w n as a Bernoulli distribution. Note that w h e n p = 1A the general distribution reduces to the distribution for a fair coin. In fact, each value of p defines a different Bernoulli distribution. Instead of considering the outcome of a single coin toss, we can consider the total n u m b e r of heads resulting from, say, two coin tosses. The distribution of outcomes then is given by
f(x)
x!(2
2~ -
X)! (0"5)x(O'5)2-x"
where p = 1A is the probability of getting a head on any single coin toss. Thus the frequency of occurrence of two heads (x = 2) is f(2) = 1/4, the frequency of occurrence of one head (x = 1) is f(1) = 1A, and the frequency of occurrence of no heads (x = 0) is f(0) = 1/~. In contrast to the first example, this distribution has two parameters: the probability of getting a head on
a single toss (p = 1/2), and the total n u m b e r of tosses (n = 2). In general the distribution of the n u m b e r of heads in n tosses, with probability p of a head on any one toss, is described by the probability density function
tnt,x l
x
Each combination of parameters n and p defines a specific binomial distribution, which is designated by B(n, p) to emphasize the role of n and p. The binomial distribution plays a crucial role in the investigation of biological populations. The distribution can be derived as a realization of n i n d e p e n d e n t Bernoulli trials, via the product of separate Bernoulli distributions with c o m m o n parameter p (see A p p e n d i x E). The assumption of independence, a key feature that determines the probability distribution for aggregate data, will be invoked repeatedly in the d e v e l o p m e n t of statistical estimation models. 4.1.1.2. Multinomial Distribution Instead of sampling with dichotomous outcomes (e.g., head or tail), we can consider sampling with several possible outcomes. For instance, a forward pass in football can result in a completion, an incompletion, or an interception. A s s u m e that over the course of last year a quarterback's percentages are Pl = 0.6, P2 = 0.38, and P3 = 0.02, respectively, for these outcomes. N o w consider the distribution of outcomes for any five r a n d o m l y chosen forward passes, based on these frequencies. If x I is the n u m b e r of complete passes, x 2 is the n u m b e r of incomplete passes, and 5 - x I - x 2 is the n u m b e r of interceptions, the distribution of outcomes for the quarterback's passes is given by the probability density function f(x1, X2) =
( 5X2) (0.6)xl(o.38)x2(o.02)
Note that the possible results of five passes include anything from 0 to 5 completed (the latter an unlikely but possible event), from 0 to 5 not completed (extremely unlikely but possible), from 0 to 5 intercepted (just about impossible, but not quite). Note also that two variables (x 1 and x 2) rather than one are required to specify the range of possible outcomes. Finally, note that three parameters are involved in the specification: the n u m b e r of passes (n = 5) and probabilities of two of the possible outcomes (Pl = 0.6, P2 = 0.38). In general the distribution of outcomes for n passes, with probabilities Pl and P2 of a complete and incomplete pass, respectively, is
36
Chapter 4 Estimation and Hypothesis Testing
f(xl, x2ln, pl, P2) =
F/ ~.Xl...x2t 1 ~x1, x 2 / P 1 F 2 ~'
--pl--p2)n-xl-x2.
Each combination of parameters defines a different distribution. Both this distribution and the binomial distribution from the previous example are members of the general class of multinomial distributions, which are appropriate for certain kinds of count data (see Appendix E). Multinomial distributions are especially useful for estimation of biological parameters such as population size, survival rate, and harvest rate. Because these parameters are key to the management of animal populations, multinomial distributions incorporating them are used frequently in the material to follow.
g=-I
0.4
~=0
g=l
0.3
0.2
\
0.1
\ 0.0
0.4
4.1.2. The Normal Distribution for Continuous Attributes 0.3
Many data in biological samples are continuous (e.g., weights, sizes, durations) rather than discrete (e.g., counts, qualities, category memberships). The probability distributions for continuous data are represented by smooth distribution functions over a range of values for the data. By far the most intensively studied and most frequently used continuous distribution is the univariate normal distribution with probability density function
0.2
0.1
~2= 1 0.0 -2
0
2
4
X
f(xllx, r =
1
2x/GGr
exp
-
2\
/
As indicated in this formula, the univariate normal is a two-parameter distribution, parameterized by the distribution mean IXand the variance 0.2 (or the associated standard deviation 0.) (see Section 4.1.3). As shown in Fig. 4.1, it is bell-shaped, symmetric about the mean, and more or less peaked, depending on the variance. The mean IXis a location parameter, in that it specifies the location (but not the shape) of the distribution. Figure 4.1a illustrates the effect of changing the mean, while holding the variance constant. Thus, increasing the mean effectively shifts the distribution to the right along the x axis. On the other hand, the variance is a shape parameter, in that it specifies the shape (but not the location) of the distribution. Figure 4.1b shows the effect of changing the variance, while holding the mean constant. Thus, an increase in the variance leads to a distribution that is less peaked and more evenly spread over the range of x values. Typically the normal distribution is denoted by N(I,, 0.2) or N(xll~, r to emphasize the role of the mean and variance in specifying shape and location of the distribution. As with all continuous probability density
FIGURE 4.1 Normalprobability distribution. (a) Effect of changing the mean (IX)of the distribution. (b) Effect of changing the variance (or2) of the distribution.
functions, the area under the curve is 1 (i.e., f~_oof(x) dx = 1). Furthermore, the area under the curve to the left of any specific ordinate value, say, v, is the probability that a randomly chosen value x from this distribution will be less than or equal to v. The probability is expressed mathematically as
f(u) = P(x 1 (because the trajectory of deviations exhibits exponential growth a n d / o r increasing oscillations). Appendix D provides a more comprehensive treatment of stability for discrete systems.
7.5.5. The Influence of Initial C o n d i t i o n s ~ M o d e l Ergodicity From Eqs. (7.3) and (7.4) it should be clear that the behavior of a model is influenced, at least in part, by its initial conditions. Two interesting possibilities can be identified as to long-term system dynamics: (1) the asymptotic behavior of x(t) is influenced by x(t0), or (2) the asymptotic behavior of x(t) is independent of x(t0). The latter case describes model ergodicity (a literal meaning is "recurring states," in this case recurring with respect to different initial conditions; see Section 10.3.2). System ergodicity anticipates biological processes that are asymptotically insensitive to accidents of environment and other factors affecting the starting point for system dynamics. Time-varying attributes of such systems typically converge to a single stationary value, irrespective of where the system trajectory originates. The Leslie matrix model, referred to in Section 7.5.3 and described in more detail in Section 8.4, is a case in point, in that the trajectory of a Leslie matrix
7.6. A Systematic Approach to the Modeling of Animal Populations model converges asymptotically to a stable age distribution irrespective of the initial age distribution. For ergodic systems, any variation among the long-term behaviors necessarily represents differences in system processes, rather than differences in system initial conditions. The other possibility for system behavior is that the system starting value x(t0) does influence long-term system dynamics. This influence defines nonergodic systems, for which variation in asymptotic behaviors can be induced by simply changing the system initial conditions. A nonergodic system is intrinsically nonlinear, typically possesses multiple equilibria, and often exhibits patterns of local stability whereby perturbations in local zones are followed by a return to local equilibria. The assessment of nonergodic systems is complicated by the fact that observed variation in longterm system behaviors may be induced by differences among systems in their initial conditions or by differences in their system processes. A trivial example is a logistic model of population dynamics for each of two geographic areas in the absence of migration between them. In this case the asymptotic behavior of the aggregate system varies depending on whether initial population size on each of the areas is nonzero. Computer simulation offers one approach to the investigation of initial conditions. The following steps are a possible sequence: 9 Identify the system attribute(s) to be investigated (e.g., population rate of growth, sex ratio, age distribution). 9 Select an appropriate range of initial conditions {x(t0)} to be investigated. 9 For each particular set of initial conditions x(t0), simulate system dynamics over some extended time frame and record the resulting values of the attributes of interest. 9 Assess the variation among recorded values of the attributes. An absence of variation in attribute values is indicative of ergodicity. On the other hand, a clustering of attribute values into discrete groupings may suggest further inquiry into the structure of the system processes inducing nonergodicity.
7.6. A S Y S T E M A T I C A P P R O A C H T O THE M O D E L I N G OF ANIMAL POPULATIONS Having introduced the major components in modeling and some tools for model assessment, we now can describe systematically the process of development
131
and application of models of animal populations. As mentioned earlier, the process begins with identification of goals and objectives for the effort and leads systematically to model identification, computerization, and testing. The modeling process includes the following key steps:
1". Establishment of model goals and objectives. Because the purposes for which a model is to be developed determine in large measure its structure (and thus its dynamic behaviors), this crucial step should begin the process. As discussed earlier, goals and objectives often are associated with properties such as model generality, model realism, model accuracy, model identifiability, and potential uses of the model for management or other purposes. Depending on its goals and objectives, a population model can have very different structural features and can exhibit substantially different behaviors. Given the importance of establishing model goals and objectives, it is surprising how frequently this step is overlooked (or inadequately reported). 2. Identification of system features and system boundaries. Early on it is necessary to determine what is to be included in a model and what is not. This decision establishes which population features to characterize, which environmental and management variables to include, and what biological mechanisms to model. Identification of model components includes (1) state, control, and exogenous variables, (2) flows among state variables, (3) intermediate variables and parameters, (4) information connectors among state variables, intermediate variables, flows, and model parameters, and (5) mechanisms by which flows are regulated and intermediate variables are updated. These features are captured in a diagram of the system that characterizes system components by means of, e.g., stocks, flows, and information connectors, and recognizes system boundaries by means of sources and sinks that are associated with elements outside (but connected with) the system. An intuitive order in identification of model features starts with the state variables, then includes the flows among state variables, followed by intermediate variables, parameters, and connectors among the information components. Exogenous influences are added as flows across system boundaries (e.g., precipitation events) or as converter components (e.g., temperature regimes). Control variables also are included as flows (e.g., stocking or harvest) or as converter components (e.g., alteration of mortality rates). Those model parameters that are likely to be altered as the model is investigated should be represented with converters. Through the process of identifying system features and boundaries, model goals and objectives often are
132
Chapter 7 Principles of Model Development and Assessment
clarified. System identification forces the modeler to consider the feasibility of developing a population model in terms of, e.g., spatial and temporal comprehensiveness, level of biological aggregation, and the degree of biological mechanism to be included. These considerations often lead in turn to revision and refinement of the model goals and objectives. Indeed, model identification and the establishment of goals and objectives can be seen as an iterative process, with goals and objectives shaping the model structure, with model identification in turn helping to refine model goals and objectives, with the refined goals and objectives further influencing the model structure, and so on. In practice, iterations of this process can continue for the duration of the modeling exercise.
3. Development of the mathematical~simulation model. It is useful to think of the mathematical formulation and coding of a model as occurring subsequent to identification of its features. However, the process of mathematical formulation actually begins as its features are being identified. Indeed, feasibility issues in model identification often relate to mathematical feasibility, in that mathematical relationships among system components must be known (or at least be estimable), to be included in the model. On assumption that the identified model features can be characterized mathematically, a natural order for model development is (1) mathematical description of functional forms for the flows among state variables, (2) mathematical description of functional forms for the intermediate variables, (3) incorporation of values for the parameters identified as converters, and (4) identification of initial values for the stock variables. If the model requires simulation, a computer program corresponding to the mathematical model also must be developed. The simulation model includes the following important features: (1) Time specifications for the simulation. These include the length of the time frame, the time step, and the output interval. (2) Specification of the format of the desired output (e.g., data protocols, graphical a n d / or tabular formats). This includes identification of system variables to be displayed, scaling of graphical displays, and the layout and labeling of output. (3) Documentation of the computer code. The latter step, which often is overlooked, helps to ensure that the model can be understood by others not involved in program development. Documentation of computer code involves both a review and description of the computing logic and thus is an integral part of the verification process. It is important to recognize that mathematical and computer formulations of a biological model can be mutually informative. On the one hand, simulations
with a computer model can be used to good mathematical effect, e.g., to highlight inconsistencies in the mathematical formulation of the model, to focus on important structural features that control model behaviors, or to suggest interesting mathematical features worthy of further exploration. On the other hand, the mathematical formulation of a model can focus one's attention on model parameters or other features that can be explored usefully by means of computer simulation. Indeed, the interaction between these two approaches (mathematical analysis and computer simulation) really constitutes an iterative process, in which analysis is used to direct one's efforts in simulation, the follow-up simulations suggesting relationships among system features to be explored analytically, the follow-up analyses in turn suggesting further simulations, and so on. The interplay of simulation and analysis thus provides insights about system structure and function that extend beyond either approach considered alone. 4. Model sensitivity analysis. As described above, sensitivity analysis involves an assessment of variation in model behavior, with the idea of varying some component of the model and evaluating the impact on model performance. For all but the simplest models it is not possible to conduct a comprehensive sensitivity analysis. For example, if we consider only two levels for each model parameter in a deterministic model, the number of simulations required to comprehensively examine a model with k parameters is 2k. Thus, a simple Lotka-Volterra model for three competing species (see Section 8.8) involves 2 1 2 - - 4096 simulations. Clearly, it is necessary to devise strategies for sensitivity analysis that avoid most parameter combinations and yet focus on combinations of importance. Often one can adapt certain experimental designs from statistics, e.g., stratified or fractional factorial designs (see Section 6.2) to aid in this effort. Nevertheless, the choice of parameters remains largely a matter of "probing" over the set of potential parameters, aided by intuition, previous investigation, and luck. We note that sensitivity analysis, like verification and validation, is neither a one-time exercise nor an ending point in model assessment. Indeed, sensitivity analysis is perhaps most useful in highlighting model features that should be identified with a high degree of accuracy. Thus, it serves as a guide for the allocation of effort in model development, as well as model revision and refinement. 5. Model verification. As mentioned earlier, verification consists of a review of the model structure and computer code, as well as an evaluation of model performance with data used in model development. The purpose of the review is to ensure the model "looks"
7.6. A Systematic Approach to the Modeling of Animal Populations the way it is intended to look, in the sense that the mathematical forms of the relationships are as intended, the parameter values are correctly specified, the sequence of logic in the computer code is as intended, and so on. The evaluation of model performance ensures that the model adequately represents important patterns in the data used to create it. 6. Model validation. Validation extends the assessment of model behavior to include evaluation of model performance based on data not used in its development. The idea is to test whether the model remains "valid" for representing independent, representative data. In this sense the model acts as a complex hypothesis, to be evaluated by comparison with observations of the system. A correspondence of model predictions and independent data over the intended range of operation of the model supports the validity of the model for its intended purposes. If the model fails to correspond with independent data, further model refinement is necessary. Typically this involves retention of the independent data in the data base, refinement of the model based on this extended data base, verification of the refined model, and comparison of predictions from the refined model against additional independent data. Thus, validation and verification are not one-time activities, but instead are part of an iterative process by which a model evolves in its structure and function. The process is driven by the comparison of field data against model predictions, and it is a part of model assessment and evaluation. 7. Stability analysis. In addition to sensitivity analysis and verification/validation, it is useful to determine the equilibria of a system and to explore equilibrium stability. Stability properties for both discrete and continuous systems can be investigated by means of a first-order Taylor series expansion of the transition equations, with stability determined by the eigenvalues of a matrix of transfer function derivatives. Similarly, an investigation of initial conditions can prove helpful in anticipating their influence on both transient and asymptotic patterns of population change. Computer simulation is one way to explore the effect of initial conditions on population dynamics.
8. Application to management of animal populations. Population models developed as management tools ultimately are used to provide information to managers about the population consequences of management. Pursuant to this goal, models are used in essentially two ways: (1) to play "what-if" games, wherein potential management policies are imposed on the model (through identification of a control variable trajectory) and the model is used to simulate population dynamics under the policy, and (2) to
133
identify optimal management policies, based on some well-defined measure of model performance. For example, models of a harvested population sometimes can be used to identify optimal harvests through time, based on an objective of maximizing total harvest over an extended time frame. The use of models in dynamic optimization is discussed in some detail in Part IV. A point worthy of strong emphasis is that the modeling process does not end with validation and application to population management/assessment. Models represent biological systems that change through time in response to ecological, environmental, and management factors. As these systems evolve, the models representing them should incorporate new information about the structure and function of the system. The information on which a model is based is sometimes in the form of data and sometimes in the form of intuition, theory, or anecdotal evidence. In all cases, the information base grows as system changes are observed, and the system model can be updated as additional information becomes available. The need for adaptive updating is especially important for management-oriented models. Even if they are carefully constructed and properly verified and validated, such models nevertheless are useful only over a limited range of values for the biological system under investigation. This range often is defined by "normal system conditions" or by behaviors in an unperturbed state. On the other hand, the management of biological systems almost always involves considerable perturbation, which often tends to move the system outside of its normal operating range. For this reason, models of managed biological systems, to retain their usefulness, must be updated as new information becomes available. An ongoing cycle of management, monitoring, and model revision therefore is prescribed, including the following activities: 9 As the modeling process progresses, the model is verified, validated, and analyzed based on available population data. 9 The model is used to assess the consequences of management decision-making (e.g., population harvest or stocking). 9 Management decisions lead to population changes and updated information about population status (and the impact of management decisions on population status). 9 The updated data base is used to revise and refine the model, through the processes of model identification, verification, validation, etc.
134
Chapter 7 Principles of Model Development and Assessment
The revised model is again used to inform decisionmaking, leading to further changes in population status and further refinements in the model. A key point in this sequence is that modeling is (or should be) an evolutionary process that is ongoing throughout the useful life of the model. At no point can one stop the iterative refinement with an assurance that, because the model now represents the past adequately, it is certain to represent the future adequately. The embrace of uncertainty, along with the iterative refinement of management-oriented models with new information as it becomes available, defines an adaptive approach to management. The term adaptive is used to characterize management approaches that account for learning, i.e., that focus on the biological information obtained through management and use this information in future decision-making. On condition that information for model refinement and improvement is simply an unintended by-product of management decision-making, the approach is described as passive-adaptive management (Waiters, 1986). If, on the other hand, management actively seeks decisions that are informative of system structure and function, the approach is described as activeadaptive management. We discuss adaptive management in the context of dynamic optimization in Chapter 24.
7.7. D I S C U S S I O N
In this chapter we have presented a systemsanalytic view of the modeling of animal populations. The framework for much of the discussion is borrowed from systems engineering, which has a long and distinguished history in the modeling and analysis of dynamic systems. Animal ecology has benefitted substantially in recent years from the approaches and techniques of systems modeling and analysis, and increasingly the ecological literature documents this work. Nevertheless, the notational density and mathematical rigor exhibited in the systems literature continues to be a barrier to understanding for many ecologists. The field of dynamic modeling is truly huge and includes quite sophisticated treatments of subjects such as systems identification, systems analysis, and optimal control (Luenberger, 1979; Palm, 1983; Flood and Carson, 1988; Dorf, 1989; Bertsekas, 1995). We have touched only briefly here on these concepts, especially in such complex areas as stochastic differential modeling and assessment and the mathematical treatment of systems control. We discuss these and other concepts in some detail in the following chapters and in the appendices, recognizing that a comprehensive treatment is well beyond the scope of this book. We leave it to the interested reader to explore further this very rich body of knowledge.
C H A P T E R
8 Traditional Models of Population Dynamics
8.1. DENSITY-INDEPENDENT GROWTHwTHE EXPONENTIAL MODEL 8.1.1. Habitat Effects 8.1.2. Harvest Effects 8.1.3. Perturbations 8.2. DENSITY-DEPENDENT GROWTHmTHE LOGISTIC MODEL 8.2.1. Incorporating Harvest 8.2.2. Incorporating Time Lags 8.3. COHORT MODELS 8.3.1. Populations with Independent Cohorts 8.3.2. Transitions among Cohorts 8.4. MODELS WITH AGE STRUCTURE 8.4.1. Life Tables 8.4.2. Discrete-Time Models with Age Cohorts 8.4.3. Continuous-Time Models with Age Cohorts 8.4.4. Characterizing Populations by Age 8.5. MODELS WITH SIZE STRUCTURE 8.5.1. Discrete-Time Models with Size Cohorts 8.5.2. Continuous-Time Models with Size Cohorts 8.6. MODELS WITH GEOGRAPHIC STRUCTURE 8.7. LOTKA-VOLTERRA PREDATORPREY MODELS 8.7.1. Continuous-Time Predator-Prey Models 8.7.2. Discrete-Time Predator-Prey Models 8.8. MODELS OF COMPETING POPULATIONS 8.8.1. Lotka-Volterra Equations for Two Competing Species 8.8.2. Lotka-Volterra Equations for Three or More Competing Species 8.8.3. Resource Competition Models 8.9. A GENERAL MODEL FOR INTERACTING SPECIES 8.10. DISCUSSION
Some familiar population models have been analyzed in the literature in considerable detail, especially as concerns the influence of model parameters, the interactions among model components, and in some cases, the projected impacts of management actions on model behaviors. In particular, there is a large literature on the logistic, Leslie matrix and other single-species models, and on certain multispecies models that account for competition and predation. In this chapter we describe some of these models, beginning with examples that are biologically and mathematically simple. Additional complexity is incorporated gradually, with the addition of environmental factors, management effects, cohort structures, and other components that enhance biological realism (and also complicate model analysis!). We restrict attention here to deterministic models and defer to later chapters the treatment of statistical uncertainties and stochastic model behaviors. To help frame the discussion, it is useful to consider the nature of density dependence in population growth. Density is described here as population size divided by the area occupied by a population (see Section 1.1 for a generic definition). If the area under consideration is constant over time, population density is proportional to population size, and the influence of density on biological processes can be expressed in terms of population size. Because our focus here is primarily on population dynamics rather than fluctuations in available area, we make the convenient assumption that the area associated with a population is constant over time. Density dependence can be introduced via the bal-
135
136
Chapter 8 Traditional Models of Population Dynamics
ance equation (1.1) from Chapter 1, which expresses population change over a unit time step in terms of birth, death, and migration. The balance equation also can be written in terms of per capita rates, by N ( t + 1) = N(t) + B(t) + I(t) - D(t) - E(t) =
[1
+
bt +
it -
d t -
et]N(t)
= )ttN(t),
which provides the simplified form N ( t + 1 ) = )ttN(t)
to describe population dynamics. The parameter )~t, called the finite rate of population growth, expresses the per capita change in a population over a unit of time by N ( t + 1) - N(t) N(t)
=
)~t - -
1.
It is clear from this equation that a population increases, decreases, or remains constant over [t, t + 1] as )kt ~ 1, ~kt ~ 1, o r )k t = 1 . Density dependence is established by the influence of population size on the population rate of growth, that is, by ~kt - - M N ( t ) ) at each point in time. On the other hand, density independence obtains when the population rate of growth is independent of N(t). A familiar form of density-dependent growth has M N ( t ) ) decreasing monotonically in N(t), so that a larger population has a lower population growth rate, and population losses--for example, through removal of individuals through harvest--lead to an increase in the population growth rate. In the latter scenario, increases in growth rate are said to "compensate" for population losses. By inducing compensatory changes in growth rate, monotonic density dependence attenuates the effects of environmental variation, harvest management, and other influences, and thereby promotes stability in population dynamics. Of course, other forms of density dependence also are possible. For instance, h(N(t)) could be monotonically decreasing in N(t) for large values of population size, but monotonically increasing for small values (the Allee effect of Chapter 1) (Allee et al., 1949; Courchamp et al., 1999; Stephens and Sutherland, 1999). In this situation an increase in the size of a depauperate population (for example, through stocking of individuals) would actually lead to an increase in the population rate of growth. Such an effect might arise through an increased frequency of mating, as a result of additional mating opportunities. On the other hand, the removal of individuals from a depauperate population would
lead to declining growth rates, ultimately resulting in population extinction. In the following sections we discuss the forms and effects of density-dependent growth, preparatory to the consideration of other factors such as population structure and interspecific interactions. We begin with the exponential model and variations of it that do not include density dependence, and compare behaviors against models such as the logistic equation that do include density-dependent factors.
8.1. D E N S I T Y - I N D E P E N D E N T GROWTH~THE EXPONENTIAL MODEL The exponential model is perhaps the simplest of all models for population dynamics. It is used to describe population growth that is directly proportional to population size at each point in time, in the absence of mechanisms for regulating growth. Typical derivations of the model incorporate a number of restrictive biological assumptions: 9 Generations are either nonoverlapping (as in annual plants), or else surviving offspring reach sexual maturity within one time step. 9 All individuals in the population have the same reproductive potential and the same survival probability. 9 The per capita rate of growth for the population is not subject to temporal variability. 9 There are no density-dependent mechanisms that alter the population rate of growth in response to changing population densities. These assumptions result in an exponential model with constant per capita growth rate over the time frame of the model (see Figs. 7.3 and 7.4). The discrete-time form of its transition equation is N ( t + 1) = N(t) + rN(t),
(8.1)
which yields a population trajectory with the elements N(t) = N(t0)(1 + r) t.
The continuous-time analog is d N / d t = rN(t),
(8.2)
corresponding to a population trajectory given by N(t) = N(to)e rt
(see Chapter 7 for mathematical associations between discrete and continuous forms of the exponential model). If the rate parameter r is positive, the trajec-
8.1. Density-Independent Growth--The Exponential Model tories of both continuous and discrete models exhibit uncontrolled, explosive population increases known as exponential population growth. For negative values of r, the population trajectory exhibits exponential declines, leading asymptotically to population extinction. For the continuous model, population change is smooth over the course of the time frame, with the rate of change determined by the initial population size N ( t o) and the maximum growth rate r. Population change for the discrete model occurs in steps as time increases in discrete increments. Again, the rate of change is controlled by N ( t o) and r. These patterns of unregulated change exemplify density-independent population dynamics (Fig. 8.1). Often the growth rate r in the exponential model is disaggregated into birth and death components" N ( t + 1) = N(t) + rN(t) = N(t) + ( F -
d)N(t)
= FN(t) + (1 - d)N(t) = FN(t) + SN(t),
where S is the single-period survival rate and F is the net reproduction rate (also referred to as fecundity). Because net reproduction includes both birth and survival, the parameter F can be further disaggregated into a product of survival and birth rate. Assuming reproduction occurs at the end of [t, t + 1], only S x N(t) individuals are available to produce b[S x N(t)] offspring, which then are recruited into the population: N ( t + 1 ) = (bS)N(t) + SN(t)
= S(1 + b)N(t),
(8.3)
137
Steady-state conditions dN/dt = 0 and N(t + 1) = N(t) for the exponential model can be attained only on condition that r = 0 or N(t) = 0. The former condition eliminates all population dynamics, reducing the population size to a time-invariant constant across the time frame of the model. The latter condition is tantamount to there being no population at all. In either case, population dynamics are trivial, with N(t) = N o. For r ~ 0 and N O ~= 0, from Eqs. (8.1) and (8.2) it is easy to see that the population grows or declines depending on the sign of the rate parameter r. Obviously, no population can continue to increase exponentially over an indefinite period; there are limits to the growth of all populations (see Chapter 1) (Malthus, 1798; Lotka, 1956). Thus, an exponentially increasing population eventually must approach the limits of the resource base needed to support it, and the tendency toward ever-increasing growth leads to resource depletion and population collapse. Thus, population trajectories for exponential populations might be expected to follow a cyclic pattern of explosive growth and population collapse (Fig. 8.2). It is necessary to modify the model accordingly, to allow not only for growth but also for periodic population crashes. Suitably modified, the exponential model sometimes is used to describe the growth of insect populations and other opportunistic species with high reproductive potential. The key biological features in most applications are short generation times, high numbers of offspring, and the lack of any mechanism to regulate population size. In the case of insect infestations, artificial means such as application of insecticides sometimes are used to limit population size or to hurry along the periodic population crashes.
with b the per capita reproduction of survivors.
1800 1600
1400
1400
1200
1200 1000 1000 800
800
600
600 400
400
200 200 10
2'0
30
40
50 20
FIGURE 8.1 Exponentialpopulation growth. At each point in time, the rate of population growth is proportional to population size.
2'5
30
t
FIGURE 8.2 Exponential population growth for a population with periodic population crashes.
138
Chapter 8 Traditional Models of Population Dynamics
8.1.1. Habitat Effects Habitat conditions can be included in the exponential model by means of a variable E(t) that is (possibly) subject to management control through time. Habitat and environmental effects can be registered either through the reproduction process or through survivorship (or both). For example, the influence of E(t) on reproduction might be expressed as a linear function b = f(E(t)) = E o + cE(t),
so that a change in the amount or quality of available habitat leads to a proportionate change in per capita birth rate. Such a formulation might be used to record the deterioration of a resource base through time, with concomitant declines in population birth rate and eventual population extinction. The model also could express the potential for population growth as a result of management efforts to increase the amount and quality of available habitat. Alternatively, habitat and environmental effects might be registered through the influence of habitat on survivorship. For example, the relationship of habitat and survival rate might be modeled by the function St =
E(t) + K
where K is the amount of habitat at which survivorship is half the maximum survivorship S0. In this case the value So is approached asymptotically as the habitat measure E(t) is increased. Again, a declining resource base through time would lead to lower survivorship and thus to declines in the population growth rate. On the other hand, efforts to improve the habitat base would result in improved survivorship and increased potential for population growth.
8.1.2. Harvest Effects
decline over time. Note that the sustainable harvest rate is independent of population size, in that the same rate applies to the population irrespective of its size. An alternative approach to the modeling of harvest is to express harvest impacts through a relationship between harvest and survival rate. The compensatory mortality and additive mortality hypotheses described earlier offer two expressions for such a relationship. If harvest mortality simply adds to other sources of mortality such as disease and predation, the relationship between survival and harvest takes an approximately linear form. Strict additivity has an increase in harvest rate leading to a corresponding decrease in survival rate: St
=
S0[1
-
h(t)],
where So denotes the probability of survival that would exist in the absence of hunting mortality. This relationship assumes that harvest and nonharvest mortality act in the manner of independent competing risks (e.g., Berkson and Elveback, 1960; Chiang, 1968) and is known as the additive mortality hypothesis (Anderson and Burnham, 1976). The effect of the relationship is essentially to add a harvest component to nominal mortality, thereby decreasing the survival rate St to S011 - h(t)]: N ( t + 1) = N(t) + bN(t)
- {1 - S011 - h(t)]}N(t). On the other hand, changes in harvest mortality may be compensated by corresponding changes in other sources of mortality (e.g., increases in harvest may bring about decreases in risks associated with nonhunting mortality). A simplified expression for compensatory harvest mortality has survival rate remaining unchanged over a range of values for harvest rate up to some compensation limit and declining thereafter as harvest rate increases:
Harvest can be included in the exponential model by means of a control variable H(t) specifying the level of harvest at time t. In discrete time a harvest model might be described by N ( t + 1) = N(t) + rN(t) - H(t),
where H(t) represents the postreproduction harvest of individuals in the population at time t. The population remains unchanged through time if H(t) = rN(t), i.e., if the harvest rate h(t) = H ( t ) / N ( t ) is identical to the population rate of increase. This is the maximum harvest rate that allows for a sustainable population; any harvest rate in excess of r causes the population to
h(t) > C h(t) - to and N ( t o) = N 0, the trajectory
40
N(t) =
K 1 + Ce -r(t-t~
20
10
2o
3o
4o
t
FIGURE 8.3 Exponential population dynamics, with periodic changes between positive and negative rates of change.
with C = K/No - 1 solves the continuous logistic equation (8.5) (see Appendix C). If N O< K / 2 , this solution has the familiar $ logistic shape, with monotonic increases in population size over t >- t 0, an inflection point for N ( t ) = K / 2 , and asymptotic convergence of
140
Chapter 8 Traditional Models of Population Dynamics Steady-state conditions obtain when harvest exactly balances population growth, that is, when
100
H(t) = rN(t)[1 - N ( t ) / K ]
80 or 60
h(t) = H(t) / N ( t )
= r[1 - N ( t ) / K ] .
40
J 2o
40
60
80
F I G U R E 8.4 Population dynamics for a logistic population. Population rates of growth are low when population size is near zero or K. The m a x i m u m population rate of growth occurs when population size is half the carrying capacity.
N ( t ) to K as t --~ oo. This pattern of change is an example
of density-dependent population growth, with populations that are attuned to their resources and population dynamics that tend toward resource-based equilibria. A key to the pattern is density dependence in reproduction a n d / o r survivorship. Indeed, the logistic model can be seen as a simple modification of the exponential model, to include linear density dependence in the rate of growth r -- b - d. Equilibrium states for the logistic model can be found by setting d N / d t = 0 or N ( t + 1) = N ( t ) . In either case the resulting equation is
This is the maximum harvest rate allowing for a sustainable population; any larger harvest rate causes the population to decline. Note, however, that the sustainable harvest rate is dependent on population size. One implication of this dependence is that, within certain limits, each level H(t) of annual harvest corresponds to an equilibrium population size that can sustain it. A question of traditional interest to managers concerns the "maximum sustainable harvest," i.e., the maximum level of annual harvest that can be maintained over time. This harvest level corresponds to the equilibrium population size that is given by dH/dN
= r-
2r(N/K)
=0 or
N* = K / 2 .
The maximum sustainable harvest then is H* = rN*(1 - N* / K)
= rN*(1 - 0.5K/K) = (r/2)N* = rK/4,
rN(t)[1 - N ( t ) / K ] = 0,
from which it follows that N* = 0 and N* = K are population steady states. The equilibrium condition N* = 0 is shown in Appendices C and D to be unstable, in that small population sizes lead to population increases. On the other hand, N* = K is a stable equilibrium, in that small deviations of population size from K* are eliminated through time. The population level corresponding to maximum growth is found by simple differentiation of the growth rate rN(t)[1 - N(t)/K]. After some arithmetic it can be shown that the population grows most rapidly at half the maximum population size, or K/2.
8.2.1. Incorporating Harvest As with the exponential model, harvest can be incorporated in the logistic model by means of a variable H ( t ) specifying the postreproduction harvest of individuals at time t. A discrete-time model that includes harvest is N ( t + 1) = N ( t ) + rN(t)[1 - N ( t ) / K ] - H ( t ) .
and the optimal per capita harvest rate is given by h*= H*/N* = r/2.
Note the following conditions: 9 The optimal per capita harvest rate h* is simply one-half the maximum rate of growth and is not influenced by the carrying capacity K. To determine the optimal per capita harvest rate it is necessary only to know the rate r. 9 The optimal sustained population size N* is onehalf the carrying capacity K and is not influenced by the rate of growth r. To determine the optimal sustainable population size it is necessary only to know the carrying capacity for the population. At a population size of one-half the carrying capacity the population grows as rapidly as possible, and the harvest of this growth maintains the population in optimal equilibrium. 9 The maximum sustainable harvest H* is the product h'N*, or H* = r K / 4 . To determine the maximum sustainable harvest it is necessary to know both the rate of growth r and the carrying capacity.
8.3. Cohort Models
8.2.2. Incorporating Time Lags Lags can be incorporated in the logistic model by including a lag parameter -r in the density-dependent term (e.g., see Hutchinson, 1948; Wangersky and Cunningham, 1957; Caswell, 1972). Thus N ( t + 1) = N ( t ) + rN(t)[1 - N ( t - "r ) / K ]
for the discrete model, with an analogous form for the continuous model: dN/dt
= rN(t)[1 - N(t - "r)/K].
The effect of a lag is to accelerate the growth of the population to the carrying capacity. To see why, assume that a population is below its carrying capacity at times t and t - r with the population size at time t - r less than the population size at time t. Then the damping effect of the carrying capacity is not as great if N ( t - T) is used in place of N ( t ) , and consequently, population growth is more robust. One result is that the population eventually grows beyond the carrying capacity, resulting in a population size in excess of K. Population growth beyond the carrying capacity continues until N ( t - ~) = K, at which time, population growth ceases and then becomes negative. This leads to a downward trajectory of the population, which reduces the population to a level below the carrying capacity. Population reductions continue until the lagged population size reaches carrying capacity, at which time the population begins to increase again. These oscillations, which are a direct result of a lag in adjustment for the carrying capacity, are larger in amplitude as both the lag time and the maximum rate of growth increase. Within certain parameter limits, they eventually damp out, approaching the population carrying capacity asymptotically over time. Example
The effects of the parameters r, K, and T can be illustrated with a sensitivity analysis of the logistic model that is parameterized by r = 0.3, K = 3000, r = 4, and N O = 10. Figure 8.5a displays trajectories for this model for a 50% proportionate change in r, from 0.15 to 0.45. Note that larger values of r lead to more rapid growth toward the carrying capacity, more extreme amplitudes in the oscillations, and a longer transition period before the oscillations damp out. Figure 8.5b displays trajectories for the model based on a 50% increase in K to 4500 and a 50% decrease to 1500. Larger values of carrying capacity again lead to larger amplitudes in the oscillations and an extended transition period until stabilization. However, the effect of a proportionate change in carrying capacity is not as severe as the same proportionate change in the intrinsic rate r. Figure 8.5c displays model trajectories for a change
141
in the lag r of one time step. Note that larger lags have a strong effect on the oscillation amplitudes and lead to substantially greater transition times until stabilization. As the lag becomes larger the oscillations essentially become nondamped and the population exhibits stable oscillatory behavior throughout the time frame of the model. The logistic equations described above represent only a few of the expressions that have been used to model density dependence in single-species population dynamics (May, 1972, 1974a,b, 1975, 1976; May et al., 1974; May and Oster, 1976). In particular, difference equation models can exhibit bifurcations, whereby the dynamical behavior moves from a stable point to stable cycles of differing period and finally to a regime of apparent chaos (e.g., May, 1976; May and Oster, 1976). Quite complicated dynamics of this sort can be produced by simple model structures, with very different behaviors resulting from different parameter values. 8.3. C O H O R T M O D E L S If there is substantial variation in reproduction or survivorship among individuals in a population, it often is useful to aggregate individuals into population segments or cohorts, with cohort-specific parameters controlling reproduction and mortality. The idea is to capture heterogeneity among individuals in a population by stratifying the population into groups of individuals that are homogeneous in reproduction and survivorship. Depending on the nature of the cohort structure, it often is necessary to account for transitions among cohorts.
8.3.1. Populations with Independent Cohorts In some cases, a population can be represented with cohorts that are independent, in that there are no transitions among them. Genotypic variation in a population of asexually reproducing individuals provides an example. Consider a population that is partitioned genetically into, say, k cohorts, each characterized by its own genetically based maximum growth rate r i and its own initial size Ni(O). A continuous-time exponential model for such a population is d N / d t = r l N 1 4- raN 2 4- ... 4- rkN k
= [~iPi(t)ri]N(t) = ~(t)N(t),
142
Chapter 8 Traditional Models of Population Dynamics
a
FIGURE 8.5 Population dynamics for a logistic population with per capita growth that includes a time lag in population size. The time lag induces oscillations that damp out over time. (a) Effect of a 50% increase and 50% decrease in r. (b) Effect of a 50% increase and 50% decrease in K. (c) Effect of an increase and decrease on one time step in the lag. The first column shows standard parameterization, the second column displays effects of decreasing parameter values, and the third column displays effects of increasing parameter values.
N(t)
b N(t)
/
/
c
N(t)
where N(t) = ~ i N i is the aggregate population size and ~ (t) = ~ i pi(t)ri is an average of cohort growth rates, w e i g h t e d by the cohort proportions Pi = Ni/N. Change in the average population rate of growth through time reflects the change in these proportions:
d~ -
piri]
of change of fitness (measured here by the rate of growth ~) in a population at any instant in time is equal to the variance of fitness a m o n g genotypes in the population at that time (e.g., see C r o w and Kimura, 1970). The cohort with the largest rate of g r o w t h eventually dominates such a population, with convergence of the population growth rate to that of the d o m i n a n t cohort.
Example
= [ N ~, rdNi i---~
--
(~i riNi,]-~j-~ ~dN11
The change in cohort proportions can be illustrated with a model consisting of only two cohorts, with initial cohort sizes NI(0) and N2(0). Let
p(t) = Nl(t)/N(t),
l
1 = N ~ r2Ni- y2N2]N2
where N(t) = Nl(t) + N2(t). The change in proportions is given by
i
= ~ pir2 _
~2.
i
This is a special case of Fisher's (1930) f u n d a m e n t a l theorem of natural selection, which states that the rate
dt
=
~L-N/ dN1
=
_ dN1 1
N--d- f- - N,-~JlCr
8.4. Models with Age Structure
= [N(rlN1) - N I ( r l N 1 + r2N2)]/N 2 = (r I - r 2 ) N I N 2 / N 2 = (r 1 - r2)p(1 - p). For r I > r 2 the proportion p(t) exhibits logistic growth, increasing asymptotically in time to 1. This confirms the tendency of the cohort with largest growth potential to dominate other cohorts. For the two-cohort model, the pattern of change is logistic, with asymptotic convergence of p(t) to unity. In the behaviors of these simple cohort models one can observe certain properties that otherwise might not be anticipated. For example, though all the cohorts exhibit exponential growth, the growth of the population as a whole is not exponential (because the population intrinsic rate of growth is not constant). This illustrates the concept of an emergent property, whereby patterns of change are manifested at one level of ecological organization but not at others.
143
what follows, we use the index a to denote age, as in the reproduction rate ba for animals of age a. When convenient we also use the index i to characterize age for discrete age classes, as in the survival rate S i for animals in age class i.
8.4.1. Life Tables A traditional approach to age structure organizes age-specific model parameters into a life table. There are two key parameters involved in life table analysis. The first is a survivorship function la, defined as the probability of survival from birth to age a. To illustrate, assume that individuals mature continuously over time, so that a continuous survivorship function can be expressed as la = e-f~ "(~)dv where l0 = 1 and bL(v) is the instantaneous risk of mortality to an animal of age v, i.e.,
ix(v)dv = Pr[death in (v, v + dv)lsurvival to age v]
8.3.2. Transitions among Cohorts Many populations have cohorts that are not independent. Perhaps the most familiar example involves models incorporating age structure in a population, wherein one age class matures into the next older class. Models that include size structure generalize this situation by allowing individuals in a size class to remain in the class or to transfer into a larger (or smaller) class over time. Models that incorporate geographic structure generalize the situation yet further, by allowing individuals to transfer among classes that lack the natural ordering of age and size. For models with interacting cohorts such as these, it becomes necessary to account for the transfer of individuals by means of cohort-specific transition equations. In the sections to follow, we investigate the behavior of models with age, size, and geographic structure. We will see that the trajectories of these more complicated models generalize the simple growth patterns discussed thus far, but at some considerable cost in mathematical complexity.
8.4. MODELS WITH AGE STRUCTURE Demographic parameters for many populations vary with the age of individuals in the population. It is useful under these conditions to model the population as an aggregation of age classes, with distinct survival a n d / o r reproduction rates for each class. In
(Caswell, 2001). Here we assume that instantaneous risk can be aggregated from birth to any particular age a of an individual. The survivorship function can be used to describe specific survival probabilities for populations having two different temporal patterns of reproduction. The first is known as birth flow (Caughley, 1977; Caswell, 2001), in which reproduction occurs continuously over the interval [t, t + 1]. Then the probability S a that an individual of age a at time t survives the interval is approximated by
fi+llvdv Sa ~
a
f
lv dv a-1
la+ 1 + la la + la-l" where the numerator and denominator approximate the average l v for animals in age classes a and a - 1, respectively (Caswell, 2001). The second pattern is known as birth pulse, in which reproduction is concentrated in a relatively short breeding season. Age-specific survival in discrete time is obtained in a more straightforward manner for birth pulse populations. Thus, age-specific survival is defined by
Si-- li+l/li. Population growth over each interval may be considered from times either immediately prior to (prebreed-
144
Chapter 8 Traditional Models of Population Dynamics
ing census) or following (postbreeding census) reproduction. We note that for postbreeding censuses, survival over the first age interval is given by So = l ( i ) / l ( O ) and covers the period from birth until age 1 year. On the other hand, this mortality component for prebreeding censuses is included in the reproductive parameters (see below). In either case, age-specific survival determines the number of animals in an age class that survive to the subsequent age class over [t, t + 1]. For example, Ni+l(t + 1) = N i ( t ) S i, where N i ( t ) is the number of animals of age class i alive at time t. As with survival, reproduction also may be considered a continuous function of age, according to a maternity function b a. Continuous forms for this function are considered in Section 8.4.2; here we consider reproduction assuming discrete time. Computation for birth flow populations is complicated by the fact that the average number of births occurring over an interval must be approximated, and several ways of doing so are described by Caswell (2001). For a birth pulse population, an age-specific reproductive or fecundity rate can be calculated as F i = S ibi+l
for a postbreeding census, where b i is the per capita number of age 0 animals born to individuals of age class i. Here F i represents the per capita number of offspring the following year, from individuals of age i in the current year. This definition reflects the fact that an animal of age i following breeding in year t must survive the year until the next breeding period in order to reproduce. In contrast, individuals of age class i in a prebreeding census reproduce and a portion of their offspring survives to the subsequent year, to become members of
TABLE 8.1 i
0 1 2 3 4 5 6 6+ a
that year's prebreeding population. Recruitment under this scenario is defined as F i = Sobi,
where the only survival rate at issue is that for the newly born animals in their first year of life. Thus F i reflects the number of young in the prebreeding period of year t + 1 per animal of age i in the prebreeding period of year t. Unless otherwise noted, in what follows we describe population dynamics in terms of postbreeding census times. Table 8.1 illustrates reproduction and cohort aging for a hypothetical cohort of 1000 newborn animals that is followed until all are dead following 6 years of age. We note that for sexually reproducing organisms, definitions of parameters such as b i and F i lead to a tendency to use age-specific population models that follow the female component of the population rather than both males and females. In most of the following discussion, we simply refer to individuals, but the reader should note that the ideas apply most naturally to females. It is possible to develop two-sex models that incorporate different vital rates for the sexes. These models will not be described here, but the interested reader is referred to Caswell (2001; also see Keyfitz, 1968, 1972; Pollard, 1973; Yellin and Samuelson, 1977; Schoen, 1988).
8.4.2. D i s c r e t e - T i m e M o d e l s with Age Cohorts
For discrete-time models of populations with age structure, one must include transition equations as above for each age cohort in the population. Thus, surviving individuals in any age class except the last automatically transfer into the next age class. The last age class can be modeled in either of two ways. In some formulations (e.g., Leslie, 1945), individuals in
Life Table Data for Hypothetical Cohort of 1000 Animals a
li
Si
N i
bi
Yi = Sibi+ 1
Bi = NiF i
1.000 0.250 0.162 0.114 0.080 0.040 0.024 m
0.250 0.650 0.700 0.700 0.500 0.600 0.000 m
1000 250 163 114 80 40 24 0
0.50 1.00 2.00 2.00 2.00 2.00
0.125 0.650 1.400 1.400 1.000 1.200 u
125 163 228 160 80 48 0
Followed from birth (age class i = 0) until all have died (age class i = 6).
8.4. Models with Age Structure the final age class k at time t are all assumed to die before reaching time t + 1. In the other formulation, the oldest age class represents all individuals in the population of age k or older, and surviving members of the cohort remain there. Recruitment for such agestructured models are given as an aggregate of agespecific reproductive efforts, based on cohort sizes at the time of reproduction (Fig. 8.6). A conventional model for this situation includes age-specific survival and reproduction rates, which are assumed for now to be constant over time. The transition equation for each age cohort except the first and last is
Ni+ 1 (t + 1) = SiNi(t) , with S i representing the probability of survival from t to t + I of individuals in age cohort i. Because surviving individuals from cohort i at time t are recruited into cohort i + 1 in time t + 1, both the subscript and time index in this equation are incremented. Updating the oldest cohort involves the addition of surviving individuals from the oldest and next oldest cohorts:
Nk(t + 1 ) = SkNk(t)
+ Sk_lNk_l(t
) .
Reproduction in each time period can be handled in one of two ways, depending on the census time. For populations censused just after breeding occurs, reproduction is based on the reproduction rates of surviving individuals from the previous time period:
145
time period as the number of surviving newborns from the previous time period:
[k
]
Nl(t + 1 ) = S O ~_, biNi(t) . i=1
Example A simple extension of the discrete model [Eq.(8.3)] allows for two age classes: a "birth-year" or juvenile class that survives at rate Sj over [t, t + 1], but does not reproduce during that time, and an "adult" class that survives at rate Sa and produces b young per adult. At the beginning of each year (in this development we assume a postbreeding census) the population is of size N(t) = Na(t) + Nj(t), where Nj(t) and Na(t) are the number of birth-year and adult (i.e., breeding age) animals, respectively. The transition equations for adults and juveniles are
Na(t + 1 ) = SaNa(t) + SjNj(t) and
Nj(t + 1) = [SaNa(t) 4- SjNj(t)]b, respectively. The finite rate of increase for each age class is )~a(t)
=
Na(t + 1)/Na(t)
= S a 4- S j [ N j ( t ) / N a ( t ) ]
and )~j(t) = Nj(t + 1)/Nj(t)
k
No(t + 1 ) = ~ biNi(t + 1)
= +(SaNa(t) \ N---~ Sj)b,
i=1 k-1
= ~ bi+lSiNi(t) i=0
4-
bkSkNk(t).
On the other hand, reproduction for populations censused just before breeding is carried forward at each
and the population rate of increase is given in terms of these cohort rates:
)t(t) = N(t + 1)/N(t) = [Na(t + 1) + Nj(t + 1)]/N(t)
= [)ta(t)Na(t) + )tj(t)Nj(t)]/N(t).
F I G U R E 8.6 A g e - s t r u c t u r e d m o d e l w i t h t w o age classes. The c o n v e r t e r s B 1 a n d B2 are g i v e n b y B i = biSiNi, i = 1, 2, w i t h S i = 1 - d i. The i n p u t f l o w for N 1 is the s u m of B 1 a n d B2, a n d the i n p u t f l o w for N 2 consists of the n u m b e r of s u r v i v o r s f r o m N 1.
Because the factor N j / N a can vary over time, the rates )k a and )tj can as well, and thus the population rate )~ changes as the population grows. A constant rate of growth for the population requires a stable age distribution, that is, a constant proportion of animals in each age class. If the population is not at stable age distribution, growth rates will change every year until a stable age distribution is achieved, even with constant survival and reproduction rates. Once a stable age distribution is attained, the growth rates of the two age classes become equal: )k a = )kj = )k. Of course, age stability is reached quickly for a simple two-cohort population (Fig. 8.7).
146
Chapter 8 Traditional Models of Population Dynamics ing at time t to postbreeding at time t + 1. Combining Eqs. (8.6) and (8.7) in sequence results in
No(t + 1) Nl(t + 1) N2(t + 1)
/
3o
(8.8)
jf J J 1
1
/
J
\
~
.Nk(t + 1).
Nl(t) N2(t)
'\\//
]
-No(t) N~(t)
= FSoblS;b 25 1. ""SkolbkS!.
,
t F I G U R E 8.7 Dynamics of a prebreeding population model with two age cohorts and constant per capita birth and survival rates. 0
For a multicohort model, both survivorship and reproduction can be expressed in terms of matrix multiplication. Assuming a postbreeding census, the product
m
0
0
0
0
0
$1 0 ...
..
No(t)
No(t) S0 0 0 . . .
Nk-l(t) | Nk(t) J
J
Sk
which tracks k + 1 age classes in N(t) = [N0(t), Nl(t), ..., Nk(t)]' through time. One also can track the transitions for a prebreeding census, simply by switching the order of the matrix multiplications shown above. Thus,
B
Nl(t + 1) N2(t + 1)
Sk-1
Nl(t) N3(t)
Nl(t) N2(t)
m
bl 1 0
b2 ... bk 0 ... 0 1
...
0
0
0
...
1
,,.
m
Nl(t) N2(t) (8.9)
(8.6) 0
Nk(t + 1)
0
0 ... Sk_ 1
Sk
Nk(t)
Nk_l(t) Nk(t)
represents survival and aging from the period immediately after breeding in year t to immediately before breeding in year t + 1. In turn,
characterizes the transition from prebreeding to postbreeding at time t, whereby No(t) newborns are added to the population. Then m
[Nl(t + 1) N2(t + 1
-No(t + 1)" Nl(t + 1) N3(t + 1)
Nk(t)
0 0...
0
fl
S1 0
0
o
9
9
. . . . . . . (8.7)
0 0
lNk(t + 1
i
."
Sk-1 k
m
bl
b2
bk- -Nl(t + 1)-
1
0
0
1
0 0
9
o
0
0
1
N2(t + 1)
accounts for reproduction of surviving individuals at time t + I and completes the transition from postbreed-
Nk(t)
m
represents survival and aging of the population cohorts until just prior to breeding at time t + 1. The application of Eqs. (8.9) and (8.10) in sequence produces
Nl(t + 1)
I
rSobl Sob2 ... Sobk Sobk
N2(t. + 1)
Nk(t + 1)
(8.10)
Nk_l(t ) n
_Nk(t + 1).
m
N0(t) Nl(t) N2(t)
LNk(t + 1)
$11 $20 =
.
O0
O0
.
.
Nl(t)
. 9 (8.11)
|Nk-l(t) | 0
sk_~ sk
LG(t)
J
There are some noteworthy differences between the postbreeding transitions of Eq. (8.8) and the prebreed-
8.4. Models with Age Structure ing transitions of Eq. (8.11). First, the biological time reference differs for the two models, with Eq. (8.8) tracking population status just after breeding each year and Eq. (8.11) tracking population status just prior to breeding. Second, the survival parameters used to compute reproductive input differ between the two models, with age-specific parameters used in Eq. (8.8) and a single survival rate So used in Eq. (8.11). Note that when survival rate Sk for the oldest age class is zero, the reproductive contribution Skbk in Eq. (8.8) vanishes and the final column of the projection matrix consists entirely of zeros. This is not the case with model (8.11). Third, the vector of age cohorts in Eq. (8.8) includes the cohort No(t) of newborns, whereas the vector in Eq. (8.11) does not. Similarly, the (k + 1)-dimension projection matrix in Eq. (8.8) accounts explicitly for newborns, whereas the k-dimension matrix in Eq. (8.11) does not. In essence, the number of young in the postbreeding model [Eq. (8.8)] is treated as a state variable, along with the other cohort counts. In the prebreeding model [Eq. (8.11)], it is treated as an intermediate variable. The matrix A of age-specific constants for survival and reproduction in Eqs. (8.8) and (8.11) is known as a population projection matrix. A standard form for the postbreeding model, Eq. (8.8), is ..
No(t + 1) Nl(t + 1) N2(t + 1)
,,.
Nk(t + 1)
F~_I ik 51... 0 0
Sk-1
9
Nk-l(t)
Nk(t)
..
LNk(t + 1)
FIs1 F20 "'" FO-1 i k Nl(t)
0
0 5k-1
skJ
= ~ hNi(t) i = )~ ~ Ni(t) i
Example (8.12)
m
-- 0052
i
= KN(t).
where the parameter Fi (for fecundity) represents the number of young produced by survivors who were in cohort i at time t. For a prebreeding census, a standard expression for model (8.11) is
FNI(tN2(t++ 11))
N(t + 1)= ~Ni(t + 1)
m
Nl(t)
SkJ
.,
animal population ecology and in applied areas dealing with management and conservation. In what follows we refer to age-specific projection matrices of the general form of Eq. (8.12), with Sk = 0, as Leslie matrices. The above matrix projection model can be applied iteratively to determine cohort trajectories. Starting with an initial vector N(0) of cohort sizes, application of Eq. (8.12) or (8.13) yields the vector N(1) at time 1. Application of Eq. (8.12) or (8.13) a second time, using N(1) for input, yields N(2) at time 2. This process can be repeated indefinitely, with cohort sizes used as input to produce new cohort sizes the next time. It can be shown that repeated application of the model in this manner eventually leads to a stable age distribution for the population, i.e., an age distribution for which Ni(t + 1 ) = )~Ni(t). The parameter )~ = 1 + r specifies the population growth rate r for the population once it has achieved a stable age distribution:
No(t)
F~ F10 "'"
=
147
IN/l(t)|
(8.13)
LNk(t) J
where F i now represents the number at time t + 1 of surviving young that were produced at time t by individuals in cohort i. Population projection matrices with this general form (but with Sk = 0, indicating a final age after which all individuals die) were developed independently by Bernardelli (1941), Lewis (1942), and Leslie (1945, 1948). These models saw little use in animal population ecology until the 1970s (Caswell, 2001), but now are widely used in studies of
Consider a population with four age cohorts and age-specific survival rates of S' = (0.5, 0.65, 0.85, 0.4). Assume that the age-zero cohort consists of (nonbreeding) immature organisms and that reproduction rates for the other three cohorts are age specific: b' = (0, 1.0, 2.0, 3.0). The corresponding projection model is
No(t+ Nl(t + N2(t + N3(t +
I
1) 1) 1) 1)
=
0 0.5 0 0
0.65 0 0.65 0
1.7 0 0 0.85
1.2 0 0 0.4
~No(t)- ] |Nl(t) | /N2(t) | " LN3(t)_]
Figure 8.8 shows the trajectories of each cohort in the population starting with initial age distribution _N(0)' = (10, 100, 200, 500). Note that the cohorts exhibit variation early on in their trajectories, but gradually a stable age distribution is attained and all cohorts expand exponentially at the same per capita rate of growth. This behavior is indicative of Leslie matrix models. 8.4.2.1. Stable Age Distribution and Rate of Growth
Convergence of projection matrix models to a stable age distribution follows from the mathematical structure of the matrix A. In particular, the lead right eigenvector of the matrix specifies the stable age distribution
148
Chapter 8 Traditional Models of Population Dynamics
Nl(t)
2000
for every age cohort but the first. Assuming a stable age distribution, dynamics for the zero-age cohort are given by
1500
No(t) - ~kaNo(t - a)
/ / / ~/"
~- 1000
500
/\/k/~ / ~N , / V \ / ~ . , / ./
"
/
___/~/ /'----- /. / ~
'X,>~'/-
/
N2(t)
~.N3(t) N,(t)
or
No(t - a) = ~k-aNo(t).
Substituting these expressions into the transition equation for the zero-age cohort leads to k
No(t ) = ~ , baNa(t) 2
4
6
8
a=l
10
t
k
F I G U R E 8.8 Dynamics of a prebreeding population model consisting of four age cohorts, with constant per capita birth and survival rates.
= ~ , l a b a N o ( t - a) a=l
or k
of a population, and the lead eigenvalue is the population rate of growth X = 1 + r, assuming stable age distribution (see Appendix B for a discussion of eigenvectors and eigenvalues). Both the lead eigenvalue and elements of the lead right eigenvector are positive (Gantmacher, 1959). The lead eigenvalue and right eigenvector of A can be determined by solving the characteristic equation A P = XP
(8.14)
m
for X and P. Starting with any nonzero vector, iterative application of Eq. (8.8) or (8.11) eventually produces numerical values corresponding to both P and X. Alternatively, X (and therefore P) can be obtained as a solution of the well-known Euler-Lotka equation [Euler, 1970 (1760); Lotka, 1907, 1956] k
1 = ~
~k-abala,
a=l
where
No(t) = ~ , X-al~b~No(t), a=l
and division of both sides of this equation by No(t) produces the Euler-Lotka equation. The Euler-Lotka equation makes explicit the influence of survivorship and reproduction on the population growth rate. For example, the same value of L can be produced by a population with high cumulative survivorship and low reproduction, or a population with low survivorship and high reproduction. Clearly, if X > 1 the corresponding population trajectory will exhibit a pattern of exponential increase as it attains a stable age distribution, whereas the trajectory will show an exponential decrease if X < 1. Of course, if = 1, the population remains unchanged after the stable age distribution is attained. The Euler-Lotka equation can be expressed in terms of the parameters of a Leslie matrix, as in Eqs. (8.12) and (8.13) with Sk = 0. In terms of the postbreeding parameters in Eq. (8.12), the Euler-Lotka equation is k
a-1 la= H Si
1 = ~ , ~k-ala_l(Sa_lba ) a=l
i=0 k
and l0 = 1. The Euler-Lotka equation is really just a combined form of the transition equations, assuming a stable age distribution. A derivation (e.g., see Mertz, 1970) is based on
= K-1 ~
K-(a-1)la_lFa_l,
a=l
from which we get k-1
Na(t)-- S a _ l N a _ l ( t - 1) -- Sa_lSa_2Na_2(t-
K = ~ , X-alaFa . 2)
(8.15)
a=0
With Eq. (8.15) we can show that a vector P with components = laNo(t - a)
Pi = )t-(i-1)li-1,
8.4. Models with Age Structure i = 1, ..., k + 1, is the lead right eigenvector for a postbreeding projection matrix with Sk = 0. Thus,
149
with a-1
l* = II Sa*
-F o So 0
0
F1 0 S1
0
... ... ...
...
Fk_ 1 0 0
Sk- 1
O0 0
i=0
"k-1 " i~,=oh-ilif i
1 U-1ll
a-1
= ca[i.o
ll h-ll2
and h* the rate of growth for a population with agespecific birth rates b a and survival rates cS~. But
u-(k--'l)lk_l u-klk
0
= Cala
m
h-(k'-l)lk ,n
k
1 = ~ , (h*)-abal'~
with the lead term in the resultant vector equal to h from Eq. (8.15). Factoring h out of each of the terms in the resultant vector produces UP and demonstrates that P is an eigenvector of A. In terms of prebreeding parameters in Eq. (8.13), an expression of the Euler-Lotka equation is obtained by multiplying both sides by So:
a=l k
-- ~
(U*) -a ba(c a la )
a=l k
-- ~_~ (U*/c)-abala, a=l
which is satisfied by the unique rate of growth h corresponding to birth rates ba and survival rates Sa. Thus,
k
S O = ~_j h-ala(Soba) a=l
(8.16)
h*/c =
k or
= ~_~ h-alaFa 9 a=l
U* = c h .
Equation (8.16) can then be used to show that a vector with components Pi = u-(i-1)li,
i = 1,..., k, is the lead right eigenvector for a prebreeding projection matrix with S k = 0: .. k ~_, h-(i-1)liFi S1
0
...
0
]
12
OoS2"" 0i il |X-~k-2'/k-1 " _ 9 iii 0
...
Sk_ 1
L ~k-(k-1)lk
Example
i=1
h - 12
.
h-(k--2)lk
This result indicates that the scaling of survival rates induces an equivalent scaling of the population rate of growth, provided birth rates remain unchanged. For example, a 50% reduction of all the cohort survival rates results in a 50% reduction in the population rate of growth.
.
Factoring h out of the lead term of the resultant vector produces SO = lI from Eq. (8.16), so that the resultant vector can be expressed as UP and thus recognized as an eigenvector of A.
Consider two populations with age-specific survival rates that are related by S* = cS a, as in the previous example. Assume that the population with reduced survivorship also has geometrically larger birth rates, according to b* = b,,/c a. The Euler-Lotka equation for the latter population is k
1 = ~ , (h**)-a(ba / Ca) (Ca Ia) a=l k
Example
Tradeoffs between survivorship and reproduction can be illustrated by the scaling of age-specific survival rates. Suppose that each of the parameters S~ in a Leslie matrix is reduced by a positive constant c < 1, i.e., Sa is replaced by Sa* -- CSa. From the Euler-Lotka equation we have
-- ~_j (U**)-abala, a=l
which again is satisfied by the unique rate of growth U corresponding to birth rates b a and survival rates Sa. We therefore have U ~ "~b ~
U I
k
1 = ~,~ (h*)-abal*a, a=l
demonstrating that the two populations have identical rates of growth and confirming the fact that a geometric
150
Chapter 8 Traditional Models of Population Dynamics
scaling of birth rates "compensates" for the constant scaling of survival rates. Note that the scalings of survival and birth rates are reciprocal, in that a decrease in survivorship requires an increase in birth rates and vice versa. For example, a 50% reduction in survival rates requires a geometric doubling of birth rates in order to maintain the population growth rate. On assumption that a stable age distribution has been attained, it is straightforward to show that the pattern of relative cohort sizes at any point in time is determined by survivorship. Thus, a stable age distribution requires that Ni+l(t 4- 1 ) = KNi+l(t) ,
which, when combined with the cohort transitions Ni+l(t 4- 1 ) = SiNi(t) ,
produces Ni+l(t) Ni(t)
_
Si
-
--.
)~
(8.17)
It follows that the relative sizes of adjacent cohorts in stable age distribution vary with survival rates but not with birth rates. As argued below, this property can be used to advantage in determining recruitment to the population based on cohort-specific harvests.
parameters in a projection matrix model. Recall from Section 8.1 that N ( t + 1) = N(t) + rN(t) - (F + S)N(t) = KN(t)
for the exponential model, with F and S the per capita net reproduction rate and survival rate for an exponential model. Thus the factor )~, which scales N(t) to produce N ( t + 1), is simply the sum of the net reproduction and survival rates. In words, an exponential population at time t + 1 consists of those organisms alive at t that survive to t + 1, along with the offspring produced by surviving organisms. A population for which the sum F + S exceeds unity expands exponentially; a population with F + S less than unity declines exponentially. Now consider an age-structured population that has achieved its stable age distribution, with Pi = N i ( t ) / N(t) the proportionate representation of cohort i in the population. From the cohort transition equations in the projection matrix model, Eq. (8.8), we have k
N(t + 1)= ~ N i ( t
+ 1)
i=0 k
= N o ( t + 1) + ~ N i ( t
Example
+ 1)
i=1
Consider two populations with the same birth rates and with age-specific survival rates that are related by S* = cSi as in the previous example. Because the scaling of survival rates by a constant induces the same scaling of the population rate of growth, we have N*+l(t) _ S* N*(t)
)~* _ cSi cK _ Ni+l(t) Ni(t ) '
so that the relative sizes of adjacent cohorts are unaffected by constant scaling of survivorship, and both populations have the same stable age distribution. Thus, the scaling of survival rates affects population growth rate but not stable age distribution. An implication is that the pattern of age distribution in a population is not diagnostic of the potential for population growth. Indeed, the same stable age distribution can apply to populations that are increasing, decreasing, or stable. It is instructive to consider the relationship between the growth parameter )~ and the birth and survival
k-1
k
= ~
bi+lSiNi(t) + ~
i=0
=
SiNi(t) + bkSkNk(t)
i=0
Pifi + ~ -
PiSi N(t)
i=O
= (F + S)N(t) = KN(t).
As with single-age exponential populations [Eq. (8.3)], an age-structured population with stable age distribution exhibits exponential growth at a rate that depends on net reproduction and mortality. However, the reproduction and mortality parameters of the single-age model are replaced here with average reproduction and mortality rates, in which age-specific values are weighted by cohort proportions in the stable age distribution. Long-term population increases occur if the average reproduction and survival rates sum to a number in excess of 1, and long-term population decreases occur if the sum is less than 1. Thus, the same patterns are found for models with and_ without age structure, and the weighted averages b and S for the projection matrix model reduce to the population reproduction and survival rates for the single-age
8.4. Models with Age Structure model. Indeed, the Leslie matrix model can be described as a multivariate analog to the univariate exponential model.
8.4.2.2. Sensitivity Analysis A matter of some interest is the sensitivity of the population growth rate k to variation in survival and reproduction rates. At issue is the change to be expected in the asymptotic population growth rate k in response to a corresponding change in one of the vital rates. This issue has been addressed numerically (e.g., Cole, 1954; Lewontin, 1965; Mertz, 1971b; Nichols et al., 1980) and, for specific characteristic equations, by implicit differentiation (Hamilton, 1966; Demetrius, 1969; Goodman, 1971; Mertz, 1971a). For purposes of illustration, define 0 to be some component of a population projection matrix, i.e., a matrix element aq, a component of a matrix element, or a parameter that appears in multiple elements. The sensitivity of k with respect to 0 is the change in k that accompanies a small change in 0, or
151
models. Note also that they are equivalent (up to a constant) to reproductive values for individuals in age class/[see Eq. (8.20)]. A formula for the sensitivity of growth rate to changes in survival and fecundity can be expressed in terms of the components Pi and qj of the right and left eigenvectors of A (Caswell, 2001). Taking the differential of both sides of Eq. (8.14) produces (dA)P + A(dP) = (dk)P + k(dP),
and multiplication by the left eigenvector Q yields Q(dA)P + Q A(dP) = (dk)Q P + kQ(dP).
Substituting Q A = kQ into this expression and simplifying, we have Q(dA)P = (dk)Q P or
dk = Q ( d A ) P / Q P.
s = 0k/00. In particular, the sensitivity of k to changes in survival and fecundity rates can be expressed in terms of the eigenvectors of A. Recall that the rate of growth k is given by the characteristic equation A P = kP,
where k and P are the dominant eigenvalue and associated right eigenvector for the projection matrix A. From above, the lead eigenvector for a prebreeding projection matrix A with Sk = 0 has components
In the case of differential change in a single element aij of A, one therefore obtains the useful formula 3k / Oaij = q iPj/ Q P,
which asserts that the sensitivity of growth rate to aij is proportional to the product of the reproductive value of the ith cohort and the relative size of the jth cohort in stable age distribution. For example, the sensitivity of k to F i is 3k
Pi = K-(i-1)li,
3Fi
i = 1..... k. It can be shown (see Appendix B) that the lead left eigenvector Q of matrix A corresponds to the same eigenvalue as does the lead right eigenvector: Q A = kQ.
(8.18)
qlPi QP K-(i-1)li Qp "
so that
Direct substitution of the components qi of Q into Eq. (8.18) produces qi = K - l ( q l F i + Siqi+l),
3Fi+1
qlPi+l = )k/S i.
and choosing ql = 1 results in qi = K-1Fi + )k-lSiqi+l
qlPi
n
3Fi
Thus, the sensitivity of k to fecundity is monotone decreasing in age (assuming k>0). On the other hand,
)ki_ 1 k
= l,
E. .
]=l
,-Jljfj.
Note that these eigenvector components apply equally for both prebreeding and postbreeding Leslie matrix
c~)k/cgSi = qi+lPi/Q P
~ E k S i j=i+l
-Jlf j/
QP,
152
Chapter 8 Traditional Models of Population Dynamics MATLAB code to compute the matrix of sensitivities for any projection matrix.
so that 0k /
0k
qi+lPi
c~Si
cgSi+1
qi+2Pi+l
J
Jlj j
Sij=i+l
j=i+2
Si+ 1//S i.
Thus, the sensitivity of k to survival is monotone decreasing if survival rates increase with age. A measure of sensitivity that is useful for some comparative purposes is proportional sensitivity or elasticity, defined in Section 7.5.2 by e =
a~/ao x/o ax/x aOlO
8.4.2.3. D e m o g r a p h i c R e l a t i o n s h i p s f o r A g e - S t r u c t u r e d M a t r i x Models
Age-structured matrix models can be used to draw a variety of inferences about the populations that they characterize. Many of these inferences can be viewed as "asymptotic" in the sense that they apply to a population exposed to the same survival and reproductive rates (i.e., the same projection matrix) every time step. Such a population can be described in terms of its stable age distribution and asymptotic growth rate k. An inference that requires survival rates that do not vary with time, but does not depend on a stable age distribution, involves the expected life-span remaining to individuals in cohort (age) a. This quantity is given by
_ 0 log k,
Ea -- ~ a l x / l a , x~a
a log 0 where 0 is either survivorship Si or a reproductive parameter (F i or b i) for a projection matrix model. Elasticity is found by dividing this expression by k/0. For example, the elasticity of population rate of growth with respect to the survivorship and fecundity parameters is given by Ok / Oa q _ q iPja q k/aij
k Q P"
with aij = S i or F i. If aij = F i this expression becomes 3k/OFi k/F i
qlPiFi k Q P = k - l l i F i / Q P,
and if aij
=
S i
the expression is cOk/3Si m qi+lPiSi k/Si k Q P
with l x the probability of surviving to age x. To see why, note that the probability of surviving to age x is also the probability of dying at age greater than x. Thus, if Px is the probability that a newborn individual survives to age x and then dies, lx can be expressed as lx = P x + 1 4- P x + 2 4- "'" 9Furthermore, the probability is Px/la that an individual of age a survives to age x (with x > a ) and then dies (see Section 4.1.6). It follows that the average number of years remaining to individuals of age a is E(x - a]a) = ~ (x - a ) P x / l a x~a
= [Pa+l 4- 2Pa+2 4- 3Pa+3 4- "'" ]/la --[Pa+l 4- Pa+2 4- Pa+3 4- "'" 4-
Pa+2 4-
Pa+3 4- "'"
4-
Pa+3 4- "'" o
j=i+l
Although the above formulas are relevant to the Leslie matrix model, general expressions for stagebased projection matrix models (see Section 8.5; also see Lefkovitch, 1965; Caswell, 2001) have been derived for sensitivity (Caswell, 1978, 2001) and elasticity (Caswell et al., 1984; de Kroon et al., 1986; van Groenendael et al., 1988; Caswell, 2001). These expressions permit simultaneous changes in several life history parameters, but they simplify considerably for the case of changes in a single parameter. Caswell (2001) provides
.]/l a .
Thus, the average life-span remaining to individuals in cohort a can be expressed rather simply, in terms of the survival factors l x 9 E a = E(x - a]a)
- (/a+l + la+2 4- la+ 3 4- " " ) / l a -- ~ a lx/la" x~a
8.4. Models with Age Structure
153
so that
Example Assume that the survival rate for a cohort transition between successive ages is S, irrespective of cohort age. The probability of an individual surviving to age a under this assumption is la = ~
E(a) = ~
aP a
a~O ~laba~ : a>O ~ a \ x ~ o lxbx '1
Px
x~a
= ~
sx-l(1-
S),
x~a
which, after some algebra, simplifies to la = S a. The expected life-span remaining to individuals in cohort a is therefore Ea--
~lx/l
a
x~a
-- ~_j S x / S a x>a =
s/(1
-
s).
Thus, the average life-span remaining to an individual is S/(1 - S), no matter what its age. On reflection, this apparently counterintuitive result makes sense. Whether young or adult, an individual is assumed to survive from one year to the next with probability S. Under these circumstances all individuals alive at a given time are equivalent in their survival probabilities and therefore have the same expected life-span from that time on. A related demographic measure is the average age of reproduction, denoted here by E(a). E(a) characterizes the mean age at which individuals reproduce, based on age-specific birth rates and assuming stable age distribution. Leslie (1966) noted that E(a) provides a measure of generation time (cohort generation time) of a population characterized by a specific projection matrix. Let Bo(t ) = ~
Na(t + a)b a
a>O -- ~ No(t)laba a>O
represent the total number of offspring produced by individuals born at time t. Then the proportion of total offspring that are produced at age a is Pa =
Na(t + a)b(a) Bo(t) No(t)laba
~, No(t)lxb x
x>O
laba
~, lxb x"
x>O
is the average age of reproduction based on these proportions. Reproductive rate, another important measure for age-structured matrix models, focuses on the amount of age-specific reproduction rather than age. Consider the future production of offspring for Na(t) individuals in cohort a at time t, Ba(t ) = ~, Na+x(t + x)ba+ x x>>_0 = ~, No(t - a)la+xba+ x x>>_o
= No(t-
(8.19)
a) ~, lxbx. x>_a
Then the per capita future production of offspring or reproductive rate for individuals in cohort a is R(a) = Ba(t)/Na(t) = N o ( t - a) ~ lxbx/Na(t) x:> a
= No(t - a) ~, lxbx/No(t - a)l a x>__a
= ~, lxbx/la. x>~a
In particular, the net reproductive rate is the average per capita production over the lifetime of offspring: R o = R(O)
= ~ lxbx. x-O A comparison of R0 and ~,x K-Xlxbx shows that the net reproductive rate must exceed unity for the population to grow (i.e., for )~> 1). Conversely, the population declines ()~a represents the die-off over (t - a, t) of individuals born into the population at time t - a. To determine a solution in the latter case, the number of births N(0, t - a) at time t - a is required. Reproduction is given by the renewal equation N(0, t) =
f
oo
b(a)N(a, t) da.
(8.28)
o
This equation is analogous to age-specific reproduction in the Leslie matrix, except that summation over a
156
Chapter 8 Traditional Models of Population Dynamics
discrete number of age cohorts is replaced by integration over a continuous age distribution. The limits of integration encompass all possible ages, shown here as ranging from 0 to oo. At any particular point in time the actual range of ages for the population is limited by the initial age distribution and the value t. As with the Leslie matrix model for discrete age cohorts, the age distribution for continuous time stabilizes to a pattern of exponential growth for all age cohorts, with the same growth rate across cohorts. For population size
and foo
= bN(t), where birth rates b(a) = b and death rates d(a) = d are independent of age. From Eqs. (8.26)-(8.28) a solution is given by
N(a, t) = N ( a - t , O)e -dt if ta. Under the condition of stable age distribution, survivorship has the form l(a) = exp(-da), so that the Euler-Lotka equation is
at time t, asymptotic stability is characterized by
1 = b f ~ e -a(r+d) da
lim ft(a) = f(a).
= b/(r + d)
t - + oo
Under conditions of stable age distribution, the intrinsic rate of growth, the age-specific life-spans, and the reproductive value all can be expressed in forms that are similar to those in the Leslie matrix population model. For example, the continuous-time analog of the Euler-Lotka equation is 1 = f ~ e-rab(a)l(a) da, where
and the population intrinsic growth rate is r = b - d. Thus, the population increases if birth rate is greater than death rate, declines if death rate is greater than birth rate, and remains constant if birth and death rates are equal. These patterns are equivalent to patterns described earlier for the simple exponential model and demonstrate the rather obvious fact that if mortality and birth rates are age independent, accounting for age in a population model is unnecessary.
Example l(a) = exp [ - f oad(x) dx ]
is the probability of a newborn individual surviving to age a, and r is the instantaneous rate of growth for the population in stable age distribution. For any agespecific birth and mortality functions b(a) and d(a), it can be shown that only one value r satisfies the Euler-Lotka equation. As with the Leslie matrix model, this parameter governs the exponential growth of each cohort in the population, once the stable age distribution is attained.
Example Consider a continuous-time population with constant birth and mortality rates for all ages at all times in the time frame. The transition equations in this situation are
ON(a, t)/Ot + ON(a, t)/Oa = - d N ( a , t)
Let the birth function in the previous example be given by b(a) = be ka, with b the rate of birth for 0-aged individuals and k a constant that takes positive or negative values. As above, trajectories N(a, t) for the age-specific cohorts are given by Eqs. (8.26)-(8.28). However, the Euler-Lotka equation now takes the form 1= b f ~ e -a(r+d-k) da
= b/(r + d -
k),
so that r = b + k - d. A comparison with the previous example shows that age-specific increases in birth rate result in a larger value for r than is the case with constant birth rate. Thus, if birth rates increase with age (k>0), the constants b and k jointly compensate for the effect of mortality, leading to an increased value of r. However, age-specific decreases in birth rate result in a smaller value for r, because the difference (rather than the sum) of the constants b and k is used to compensate for the effect of mortality.
8.5. Models with Size Structure As with the Leslie matrix model, age-specific harvest can be added to the continuous-time model. To illustrate, consider a model with instantaneous harvest rates h(a) that (possibly) vary among age cohorts. Then the model can be expressed as aN(a, t)/at + ON(a, t)/aa = -[d(a) + h(a)]N(a, t) and N(0, t) --
f
oo
b(a)N(a, t) da. o
In this formulation the age-specific harvest rate h(a) adds to other sources of mortality, resulting in the survivorship curve
Lf a[d(x) + h(x)]dx ] .
l(a) = exp -
0
This in turn leads to a stable age distribution and intrinsic rate of population growth that reflect the impact of harvest, through its depressing effect on the survivorship curve l(a). For example, assuming a birth function of the form b(a) = beka and constant harvest rate leads to 1=
b f ~ r -a(r+d+h-k) da
= b/(r + d + h -
k),
so that r = b + k - d - h. This makes explicit the depressing influence of harvest on the population rate of growth. It follows that a population of size N can be maintained in stable-age distribution by choosing the harvest rate h = b + k - d. Harvest rates of smaller magnitude allow for population growth, and rates of larger magnitude lead to population declines.
8.4.4. Characterizing Populations by Age The examples above illustrate that the dynamics of age-structured populations are determined by agespecific reproduction and survivorship parameters, with each combination corresponding to a different stable age distribution for the population. For both discrete and continuous models, population dynamics typically can be divided into two phases. Thus, a transition phase is characterized by transient dynamics in which the distribution of age cohorts differs from, but converges to, the stable age distribution. Convergence to the stable age distribution is irrespective of initial distribution; however, the cohort rates of growth in the transition phase, and the rate of convergence to stable age distribution, all depend on initial population structure. After stable age distribution is attained, the population grows at a constant rate that is given by
157
the solution of the Euler-Lotka equation. The age distribution is maintained throughout this phase, with each age cohort growing at the same rate. Thus, the population as a whole, and every cohort in it, exhibits unregulated exponential growth. Even with the added flexibility and realism that age structure can bring to a population model, for a number of reasons it may not be advantageous to characterize population structure in terms of age. In many instances, size is a more important factor than age in expressing demographic variability within a population. This is the case whenever survival and reproduction are closely associated with size, but not necessarily with age, a situation that appears to exist in a number of animal populations (e.g., Hughes, 1984; Crouse et al., 1987; Hughes and Connell, 1987; Sauer and Slade, 1987a,b; Nichols, 1987). In other cases, both size and age are relevant to variation in vital rates (Slobodkin, 1953; Law, 1983; Law and Edley, 1990). Of course, factors other than size can introduce heterogeneity in the structure of a population--for example, the simultaneous occurrence of multiple reproductive a n d / o r survival strategies in the population (e.g., see Schaffer and Rosenzweig, 1977; Pugesek and Wood, 1992; McNamara and Houston, 1996). It is common for some animals in a population to forego breeding in a particular year, such that some individuals of a particular age breed and others do not (e.g., Nichols et al., 1976b, Newton 1989; Pugesek and Wood, 1992; Cam et al., 1998). Age alone is not sufficient to characterize variation in such situations. The point here is that in many cases it is sensible to characterize animals by state variables other than age (Houston and McNamara, 1992; McNamara and Houston, 1996), and more general models may be needed to deal with such state specificity of vital rates. Stage-based projection matrix models were introduced by Lefkovitch (1965; also see Goodman, 1969; Houllier and Lebreton, 1986) and are now widely used in animal population ecology (Caswell, 2001).
8.5. M O D E L S W I T H
SIZE STRUCTURE As noted above, in many cases it is the size of an individual rather than its age that determines reproductive success and survivorship. If the size of individuals is determined exclusively by age, whereby size y is given in terms of age x through a monotonic growth function y = f(x), then a population model based on size cohorts has essentially the same structure as one based on age (up to the labeling of the cohort index). Typically, however, size is not uniquely determined by
158
Chapter 8 Traditional Models of Population Dynamics
age. Thus, a given age cohort may contain individuals of more than one size, and, similarly, a particular size cohort may contain individuals of more than one age. If demographic factors are associated more directly with size than with age, then size structure and the parameters necessary to account for transitions among size classes provide a more appropriate structure for the population transition matrix.
8.5.1. D i s c r e t e - T i m e M o d e l s w i t h Size Cohorts Consider a population in which the sizes of individuals can be divided into discrete categories, ranging from 0 (the smallest cohort) to k (the largest cohort). Assume also that individuals can either remain in their size class or grow into the next larger class over the course of one unit of time. If Pi(t) is the proportion of individuals in cohort i at time t that grow into cohort i + 1 at time t + 1, then the cohort transition equation is
where bi(t) is the per capita reproduction rate for cohort i at time t. With constant parameters, the model is written in matrix form as No(t + 1) Nl(t + 1) N2(t + 1) N3(t + 1)
.Nk(t + 1).
(1 - Po)So + PoSobl
$162
S2b3
...
Sk_l"bk
PoSo 0 0
(1 - P1)S1
0
...
0
0
Nk(t + 1 ) = Sk(t)Nk(t) + Pk_l(t)Sk_l(t)Nk_l(t). On condition that the transition and survival probabilities are constant through time, the transition equations for populations with size structure reduce to
Ni+l(t 4- 1) = [1 - Pi+l]Si+lNi+l(t) 4- PiSiNi(t) and
Nk(t + 1) = SkNk(t) 4- Pk_lSk_lNk_l(t). As with populations with age structure, reproduction is modeled simply by aggregating the reproductive contribution from each cohort: k
No(t + 1) = ~ biNi(t 4- 1) + (1 - Po)SoNo(t) i=1
k-1
= ~ bi+l{Pi(t)Si(t)Ni(t) i=0
+ [1 - Pi+l(t)]Si+l(t)Ni+l(t)}4- bkSk(t)Nk(t) + (1 - Po)SoNo(t),
(1
0
-- P 2 ) 5 2
...
0
P252
... ...
0
... ...
(1 - P k _ l ) S k _ l Pk_lSk_l
0
I
i! I
Sk
where bi represents the average reproduction at time t + 1 for surviving individuals from size class i - 1 at time t:
Ni+l(t 4- 1) = [1 - Pi+l(t)]Si+l(t)Ni+l(t) 4- Pi(t)Si(t)Ni(t) for all but the smallest and largest cohorts. Note that if Pi(t) = 1 for all cohorts, this transition equation has the same form as the age-structured model. As with one of the age-structured models, updating the largest cohort involves the addition of surviving individuals from the largest and next largest cohorts:
PIS1 0
Skb k" r N n ( t ) l i Nl(t) I 0 I N-,(t) i 0 I N~(t) I 0
m
bi =
(1
-
Pi_l)bi_l
4- P i _ l b i .
Note that the principal differences between this model and the model for age-structured populations [Eq. (8.8)] are the averaging of reproduction rates, along with the occurrence of transition elements on both the diagonal and lower off-diagonal of the matrix. If the transition probabilities Pi in the size model are mall unity, then the average reproduction rates reduce to b i = b i + l , all diagonal elements except the first and last vanish, and the mathematical form of the size model is identical to that of model (8.8) with age structure.
Example Consider a population with four size classes, for which the reproductive rates are (1.0, 2.0, 3.0), survival rates are (0.5, 0.65, 0.85, 0.4), and transition probabilities are (0.75, 0.55, 0.35), respectively. The transition equations for this population are
No(t+ 1) Nl(t 4- 1) N2(t + 1) = N3(t + 1)
I
0.5 0.375 0 0
1.0 0.2925 0.3575 0
2.0 0 0.5525 0.2975
1.2 0 0 0.4
FNo(t)-] /Nl(t)/ /N2(t)/" LN3(t)j
These reproduction and survival parameters were used in a previous example of an age-structured population. Here we simply redefine the cohort index to represent size rather than age and incorporate the nonzero probability of individuals remaining in a cohort longer than one time period. A comparison of the behavior of this
159
8.6. Models with Geographic Structure model (Fig. 8.9) and that of the corresponding agestructured model (Fig. 8.8) reveals that a principal effect is to reduce the transition phase of the model with size structure, and retard growth of the largest size class below that of the oldest age class. However, the model with size structure shows the same general pattern of convergence to a stable distribution among cohorts, followed by exponential growth for each cohort in the population.
As with age-structured models, there is a continuous-time analog for population models in discrete time (see Sinko and Streifer, 1967; Streifer, 1974). We may think of N(s, t) as characterizing the number of individuals of size s at time t, with a population size of
~
oo
N(s, t) ds
o
and a distribution ft(s)
N(s, t) / N ( t )
=
of sizes in the population at time t. As above, two transition equations are required, one for reproduction and one for physiological development and survivorship. Transitions among size classes are given by dN(s, t)/dt = -d(s)N(s, t) or
ON(s, t)/Ot + g(s, t) 0IN(s, t)]/Os = -d(s)N(s, t),
Nl(t)
2000
/
/
1500
Ndt) ~- looo ~/ 500 ,, "k~
Z\ 0
/"- ~
~~" ~
.
~
/
N3(t)
~
. ~ "
/
_. N,(t)
i
i
I
2
4
6
8
N(O, t) =
b(s)N(s, t) ds. o
The function g(s, t) in the first equation is the growth rate for individuals at time t, i.e., ds/dt = g(s, t). Thus the term g (s, t) 0IN(s, t)]/Os represents the growth into and out of cohort s. Size-specific harvest can be added to the model by including an instantaneous harvest rate h(s), so that ON(s, t)/Ot + g(s, t) 0IN(s, t)]/Os = -[d(s) + h(s)]N(s, t) .
8.5.2. Continuous-Time M o d e l s with Size Cohorts
N(t) =
and reproduction is given by
1'0
t
F I G U R E 8.9 Dynamics of a prebreeding population model consisting of four size classes, with constant per capita birth and survival rates and constant transition probabilities among size classes.
8.6. M O D E L S W I T H GEOGRAPHIC STRUCTURE In both the age-structured and size-structured models discussed above, there is a natural order in the cohort indices. Thus, cohort 0 is the youngest (or smallest) cohort, and cohorts increase in age (or size) with increasing indices. However, the cohort structure of a population need not follow such an ordered pattern. In some cases a natural progression is indeed carried in the indices for stage structure, as in larval and instar stages of development in the life cycles of certain insect species. In others the cohort index may not denote a sequential process of physiological or morphological development, so that a natural progression in indices is absent. For example, there is no natural ordering for populations consisting of geographically identified cohorts. A well-known application is in the field of island biogeography, in which migration rates among geographic cohorts are modeled as functions of island size, distance to a mainland, and sizes of mainland and island population units (MacArthur and Wilson, 1967). Migration rates and sources of variation in these rates are relevant to modeling in population genetics [e.g., island versus stepping stone versus more general isolation-by-distance models; see Crow and Kimura (1970)]. Current interest in metapopulation dynamics (Hanski and Gilpin, 1997; Hanski, 1999), source-sink models (Pulliam, 1988), and the general topic of dispersal (Clobert et al., 2001) have also sparked interest in estimating and modeling migration rates. As is the case with many aspects of populationdynamic modeling, human demographers were the first to incorporate multiple locations into projection matrix models (Rogers, 1966, 1968, 1975, 1985, 1995; Le Bras, 1971; Schoen, 1988). These so-called multiregional matrix models now are being applied in animal ecology (e.g., Fahrig and Merriam, 1985; Lebreton and Gonzalez-Davilla, 1993; Lebreton, 1996; Lebreton et al., 2000). To illustrate, consider a population consisting of three
160
Chapter 8 Traditional Models of Population Dynamics
age cohorts in each of two regions, with N 1, N 2, and N 3 representing juveniles, subadults, and adults in region 1, N 4, N 5, and N 6 representing juveniles, subadults, and adults in region 2. Here the transition among cohorts consists of the processes of aging and movement of subadults between regions. For individuals remaining in region 1, the transition matrix is
Nl(t) ~ooo a
/
/
800 ~
600
/ N3(t)
/
//
/
I
Fs0~ F2
0
N2(t)
F3 ]
400
$3
200
/
0 ,
52
and the corresponding matrix for individuals remaining in region 2 is
[i:
~ooo b
0 $5
N4(t)
S6J
Combining both subpopulations in the absence of migration produces a transition matrix of the form
800
600
200 7 /
hL
N3(t + 1) = (1 - P2)S2N2(t) -F PsS5Ns(t) + S3N3(t) and
N6(t + 1) = (1 - Ps)S5Ns(t) + P2S2N2(t) -I- S6N6(t), and the transition matrix now has the form F3 0
0
(1 - P2)$2
0
0
0 _0
0 0
0 0
00
53
0
P5S5
0
0
F4
F5
F6
0
0
P2S2
0
$4 0
0 0 (1 - P5)$5 56.
Note that the cohort indices for this model carry a natural sequence only within each region, but not across regions. The behavior of this model is displayed in Fig. 8.10 for F' = (0, 0.8, 1.6, 0, 0.8, 1.6), S' = (0.5, 0.65, 0.85, 0.4, 0.55, 0.75) and (P2, P5) - (0.4, 0.15). Note that both B
/./"
400
Migration of subadults between regions is incorporated by, e.g., incorporating a parameter P2, for the proportion of subadults migrating from region 1 to region 2, and a parameter P5, for the proportion of subadults migrating in the opposite direction. The corresponding transition equations are
F2 0
/
/
-F 1 F 2 F 3 0 0 0 S1 0 0 0 0 0 0 S2 53 0 0 0 0 0 0 F4 F5 F 6 0 0 0 54 0 0 _ 0 0 0 0 S5 $6_
"F 1 51
/ Ns(t)
/
~,,/"
,.,......_.-__ ~
...-- /
/
/
N6(t)
.v
,
2
4
'6
'8
t
FIGURE 8.10 Dynamics of a prebreeding population model consisting of two subpopulations with three age classes each. Each subpopulation has constant per capita birth and survival rates, and rates of migration between subpopulations are constant. (a) Cohort dynamics for subpopulation 1. (b) Cohort dynamics for subpopulation 2.
subpopulations exhibit "Leslie matrix" behaviors, in that both show a nonequilibrium transition phase with rates of change that are specific to cohort age, followed by an equilibrium phase with constant rates of change within the subpopulations. Note also that the two subpopulations have the same asymptotic rates of growth, even though subpopulation I has higher survival rates. This is a direct result of migration rates linking the two subpopulations. The asymptotic rate of growth for the whole population is in some sense an average of the rates of growth for the subpopulations considered separately, weighted by the migration rates. Thus, migration from subpopulation I to subpopulation 2 compensates for lower survival rates in subpopulation 2 and results in a higher asymptotic growth rate for subpopulation 2 than would be the case in the absence of migration. On the other hand, the loss of animals from subpopulation 1 via differential migration retards its
8.7. Lotka-Volterra Predator-Prey Models growth to a rate below what would be the case in the absence of migration. Eventually the growth rates for the two populations become identical, as additions and losses between the subpopulations from reproduction, mortality, and migration come into balance. This pattern occurs irrespective of the migration rate from subpopulation 1 to subpopulation 2, as long as there is some movement from one area to the other: large numbers of migrating animals, produced by a large subpopulation 1 with its higher growth potential, contribute to the growth of subpopulation 2, elevating the growth of the latter to that sustained by the former. We note in closing that the addition of cohort structure adds substantially to the burden of identifying the parameters controlling cohort transitions. This burden increases as one includes size structure with the attendant cohort growth functions, and geographic structure, which requires migration rates among geographically distinct cohorts. In Part III we describe statistical models, field protocols, and data requirements for these situations. In particular, we highlight some of the advances in areas such as sample survey methodology, tag-resighting approaches, and other estimation techniques, which provide enhanced capabilities for model development and analysis. Nevertheless, it will be clear in later chapters that data requirements and mathematical complexities can quickly overwhelm an investigation of these parameter-rich models.
8.7. L O T K A - V O L T E R R A PREDATOR-PREY MODELS Lotka-Volterra models (Volterra, 1926, 1931, 1937; Lotka, 1932) explicitly incorporate predation via state variables for both predators and prey, and two transition equations are necessary to track changes in both predator and prey population levels. The models assume that predators influence prey populations through prey mortality, whereas the prey influence predator populations through predator reproduction. Predator-prey interactions can be addressed in both discrete and continuous time, though there are important differences in mathematical behaviors between the two models.
8.7.1. Continuous-Time Predator-Prey Models If Nl(t) and N2(t) are prey and predator population levels, respectively, the continuous form of the Lotka-Volterra transition equations is d N 1 / d t = [r I - dlN2(t)~Nl(t )
161
and d N 2 / d t = [b2Nl(t) - d2~N2(t ).
Here r I represents the (constant) per capita rate of growth for prey in the absence of predation, whereas the mortality rate d i N 2 is a linear function of the number of predators. On the other hand, the per capita mortality rate d 2 for predators is constant, and the birth rate b2N 1 is linear in the number of prey. Thus the coefficient d I expresses the (negative) impact of predators on prey, and b2 expresses the (positive) impact of prey on the predators (Fig. 8.11). Here the predation rate is assumed to be proportional to the rate of encounter of predators and prey, with predation modeled as a simple product of population sizes, scaled by a speciesspecific parameter d 1. The coefficient d I in the transition equation for prey represents the proportion of prey taken by each predator, whereas the coefficient b2 in the transition equation for predators represents the "efficiency of conversion" of prey into predators. In the absence of predators [N2(t) = 0], the prey population grows exponentially according to the equation d N 1/dt = rlNl(t),
and in the absence of prey [Nl(t) = 0], the predator population declines exponentially according to d N 2 / d t = -d2N2(t).
Equilibrium conditions for the Lotka-Volterra predation model are given by setting the transition equations to 0 and solving the resulting equations. A quick inspection indicates that there are two equilibrium conditions: (N~, N~) = (0,0) and (N~, N~) = (d2/b 2, rl/dl). Thus, the model has the rather odd property that the nonzero equilibrium size for the predator population is independent of the birth and death rates for predators, and the equilibrium population size for prey is independent of prey growth and death rates. As argued in Appendix C, _N* = _0 is an unstable equilibrium, in that deviations in a neighborhood of 0 exhibit growth away from 0. On the other hand, population dynamics near (N~, N~) = (d2/b 2, r I / d 1) exhibit stable oscillations m
k2,,,'
FIGURE 8.11 Lotka-Volterrapredator-prey model. Output flow for prey N 1 is influenced by predator population size N2. Input flow for predators N2 is influenced by prey population size N 1.
162
Chapter 8 Traditional Models of Population Dynamics
about the equilibrium. This allows one to partition the "phase plane" of points (N1, N 2) into four quadrants defined by the perpendicular lines N1 = N~ and N 2 = N~_,with different population behaviors in each quadrant (Fig. 8.12). The patterns of population change for predators and prey are distinct in each quadrant:
populations, and the oscillations are stable, i.e., there is no tendency for the populations to converge to equilibrium (Appendix C). The recurring pattern of oscillation in population dynamics shown in Fig. 8.12 is known as neutral or cyclic stability (Edelstein-Keshet, 1988) (see also Section 7.5.4 and Appendix C.2.2).
Quadrant h This region of the phase plane is defined by Nl(t) > N~ and N2(t) > N~. Under these conditions the per capita growth rate b2N 1 - d 2 of predators is positive [because N 1 > N~, b 2 N 1 - d 2 > b 2 N ~ - d 2 (-- 0)], and the per capita growth rate of prey is negative [because N2(t) > N'~, r 1 - d i N 2 < r I - d i N ~ (= 0)]. Therefore the predator population increases in quadrant I, whereas the prey population decreases. Quadrant Ih This region is defined by predator and prey values such that Nl(t) < N~ and N2(t) > N~. Under these conditions, the per capita growth rates of predators and prey are both negative, and therefore both populations decline in quadrant II. Quadrant III: This region is defined by predator and prey values such that Nl(t) < N~ and N2(t) < N~_.Here the per capita growth rate of predators is negative, and the per capita growth rate of prey is positive. Therefore the predator population decreases in quadrant III, whereas the prey population increases. Quadrant IV: This region is defined by predator and prey values such that Nl(t) > N~ and N2(t) < N~. Here the per capita growth rates of predators and prey are both positive, and therefore both populations increase in quadrant IV.
Example
Consider the dynamics for populations governed by the Lotka-Volterra predator-prey equations, with an initial predator population of 210 individuals and initial prey population of 900 individuals. Per capita reproduction rates are (0.0001)N 1 for the predator population and 0.25 for the prey population, and per capita mortality rates are 0.1 and (0.001)N2, respectively. Figure 8.13a displays the population dynamics for this system. Both populations exhibit stable oscillations, with the same oscillation period but different amplitudes and phase shifts. Figure 8.13b exhibits a phase
a
1000
800
600
prey predators
400
These behaviors induce continuous oscillations about the nontrivial equilibrium point. The magnitude of the oscillations depends on initial conditions for the
200
400lb 300
350
I I I
II
i
I
280
300 or) 0
r r L
260
~
240
250
t_ 200
150 220
IV
III
100 400
600
800
10;0
1200
prey
200
800
900
1000
11;0
12;0
prey F I G U R E 8.12 Phase diagram of a Lotka-Volterra predation system. Predator and prey populations oscillate in a stable pattern,
without any trend toward equilibrium.
F I G U R E 8.13 Dynamics of a Lotka-Volterra system with one predator and one prey species. Population dynamics for both predator and prey are characterized by stable oscillations. (a) Time series trajectories for predator and prey populations. (b) Phase diagram of predator-prey dynamics.
8.7. Lotka-Volterra Predator-Prey Models diagram of these same population dynamics, with quadrant-specific increases and decreases in population sizes. The oscillation amplitudes for both predators and prey are determined by the degree of displacement of initial population sizes from equilibrium.
Both the size and shape of the oscillations in a predator-prey system are dependent on the location of the equilibrium point (N~, N~_) in the phase plane and the initial population sizes NI(0) and N2(0) relative to (N~, N~). The influence of the location of N O relative to N* is shown in Fig. 8.14, where initial population sizes close to equilibrium result in oscillations of small amplitude, and initial sizes that are distant from the equilibrium result in large fluctuations about N*. The "shape" of the oscillations also is influenced by the relative positions of N O and N*, with a more nearly elliptical phase diagram for N O close to N* (Fig. 8.14). The influence of the absolute position of N* on oscillation size and shape is shown in Fig. 8.15, which displays the phase diagrams for two predator-prey systems with different equilibria. The two systems differ only in that growth and death parameters r I and d 2 for one system in Fig. 8.15 are twice those for the other, so that steady-state population sizes are twice as large. Initial population sizes were chosen so that the distance between (NI(0), N2(0)) and (N~, N~_)is the same for both systems. Note that the phase diagram corresponding to the larger equilibrium is more nearly
bL )
200
150 t~ 0 L Q.
8.7.1.1. Oscillation Size and Shape
163
100
~oo
.oo
,'oo
~oo
~ooo
prey
FIGURE 8.15 Phase diagrams for two Lotka-Volterra predator-prey systems corresponding to different equilibria. (a) Equilibrium condition N* = (N~I,N~2)is D units from the origin. The distance between equilibrium N* and initial condition NOdetermines the amplitudes and period of oscillations. (b) Equilibrium state is 2D units from the origin; initial condition NOchosen to maintain the same distance to the equilibrium state N* as in (a). _
m
elliptical, as indicative of symmetric oscillations about (N~, N~). On the other hand, oscillations for the system nearer the origin are less symmetric, with larger amplitudes. The oscillation period is determined by the factor (rid2) 1/2, so that high prey growth rates and predator death rates accelerate the cyclic changes in population status. Thus, the oscillation period of the system with larger equilibria is 50% that of the system with smaller equilibria. 8.7.1.2. L o g i s t i c E f f e c t s
Density dependence can be incorporated into the Lotka-Volterra equations by modifying the prey a n d / o r predator reproduction functions. For example, logistic effects in prey reproduction lead to the system of equations
100
dN1/dt
= r1N1(1 - N 1 / K ) - d l N I N 2
and dN2/dt
0
200
400
600
800
prey
FIGURE 8.14 Phase diagram of predator-prey dynamics for a Lotka-Volterra predator-prey system, starting at different levels of initial population size. Trajectories correspond to an initial predator population size of 30 and initial prey population sizes of 100, 150, and 200. Trajectories exhibit stable oscillations with differing periods and amplitudes.
= b 2 N I N 2 - d2N2,
with the per capita rate of prey reproduction decreasing logistically with prey population size. As before, equilibria for this system are defined by d N / d t = 0, or r1N1(1 - N 1 / K ) - d l N I N 2 = 0 and b 2 N I N 2 - d 2 N 2 -- O.
164
Chapter 8 Traditional Models of Population Dynamics
These equations are satisfied for N* = 0 and 1200
:
LNt/
bI
d2/b2 1 b,a2 l,
1000
bdlKl
with d2 < b2Ka necessary condition for N~ to be positive. It is shown in Appendix C.2.2 that N* = 0 is an
unstable equilibrium, near which the prey population grows and the predator population declines. However, the system no longer exhibits neutral stability about the nontrivial equilibrium N* = (d2/b2, b l / d I - bld2/ b2dlK)'. As a result of the logistic modification, this equilibrium is now stable, with small deviations resulting in d a m p e d oscillations as population sizes return to N*. The stability of the nontrivial equilibrium obtains no matter how minor is the logistic adjustment. However, the time required to approach equilibrium very much depends on the size of the carrying capacity K.
8.7.2. Discrete-Time Predator-Prey Models The transition equations for the Lotka-Volterra predator-prey system also can be expressed in discrete time: Nl(t + 1) = Nl(t) + [r I - dlN2(t)]Nl(t)
and N2(t + 1) = N2(t) + [b2Nl(t) - d2]N2(t).
Equilibrium conditions are found by equating population sizes in successive time periods, which leads again to the equilibrium conditions N~ = d2/b 2 and N~ = r 1/d 1. As with the continuous-time model, this results in a partition of the plane of predators and prey into four quadrants, defined by the lines N 1 - N~ and N 2 = N~ that intersect at the point (N~, N~). The patterns of population change for predators and prey are specific in each quadrant, leading to oscillatory behavior. However, oscillations for the discrete-time system are unstable, with steadily increasing population sizes and more dramatic population reductions through time (Fig. 8.16). The cause of this instability is tied directly to the discrete nature of the time step, which effectively induces a time lag into the transition equations. Thus, the population at time t + 1 is determined by growth rates that are based on population sizes at time t. As with lag effects in the logistic model, lags in the Lotka-Volterra system cause predator and prey populations to "overshoot" what would otherwise be their maximum and minimum population sizes. Assume, for example, that the prey population is large and increasing slowly and the predator population is small and increasing rapidly. The depressing effect of preda-
800
~.._
600 400 200
20
40
6'0
80
100
120
140
FIGURE 8.16 Trajectoriesfor a Lotka-Volterra predator (---) and prey (m) system in discrete time. Oscillations increase in magnitude over time, in contrast to the continuous model.
tors on the growth of prey over [t, t + 1] is based on predator population size at time t, which is substantially lower than the predator population size at t + 1. This allows the prey population to continue to increase above what would otherwise be the case with a smaller time step, in turn inducing more rapid growth in predators as more prey are available. On the other hand, if the prey population is small and decreasing slowly whereas the predator population is large and decreasing rapidly, then prey population reductions over It, t + 1] are driven by larger predator population sizes at time t than would be the case with a smaller time step. The result is more dramatic reductions in both predators and prey than would be exhibited with a smaller time step. These effects are manifested at each cycle of oscillation, leading to ever-increasing population sizes at their peaks and ever-decreasing population sizes at their nadirs (Fig. 8.14).
8.8. M O D E L S OF COMPETING POPULATIONS In this section we introduce two models for competition among populations, one that is appropriate for interference competition and one that applies to exploitation competition. In the former, the competitive impact of one species on another is registered directly, through the use of "competition coefficients" that essentially depress the population rate of growth in the manner of a carrying capacity. In the latter, competitive impacts are registered through the exploitation of a shared resource, whereby resource consumption by one species leaves a reduced resource base for the other. The distinguishing feature for these models is whether
8.8. Models of Competing Populations there is mediation of species interactions through a shared resource. 8.8.1. Lotka-Volterra Equations for Two Competing Species
We first consider a system of two competing species with density-dependent population growth rates in the absence of competition. Competition between the species influences growth rate by adding to the effect of density dependence. The continuous-time model is
165
of each is retarded by the presence of a competitor (Fig. 8.18). The logistic form of growth and the damping effect of the interaction for both species distinguish the Lotka-Volterra competition model from the LotkaVolterra predator-prey model. Equilibrium conditions for this system are given by setting both of d N i / d t to 0, which results in N'; = K 1 - a12N~2, N ~ = K 2 - a21N ~.
A rearrangement of terms leads to the matrix equation dN1/dt
= rlNI[K 1 - N 1 - a12N2]/K 1
dN2/dt
= r2N2[K 2 - N 2 - a21N1]/K2,
where r i and K i are the growth rate and environmental carrying capacity for population i in the absence of competition. The coefficients a12 and a21 represent competitive interactions between species, whereby the growth rate of one species is depressed because of the presence of the other. The coefficient a12 is a nonnegative competition coefficient specifying the per capita impact of species 2 on species 1, whereby the carrying capacity of species 1 is effectively reduced to K1 a 1 2 N 2 in the presence of N 2 individuals of species 2. Similarly, the coefficient a21 specifies the per capita impact of species 1 on species 2, so that the carrying capacity of species 2 is effectively reduced to K2 a21N1 in the presence of N1 individuals of species 1 (Fig. 8.17). This model is known as the Lotka-Volterra competition model, in reference to the fact that it characterizes direct competition between two competitors through linear terms in the transition equations. Here the competition is couched in terms of interference competition, wherein two species, through direct contact, negatively interact or "interfere" with each other. Under the model, each population is assumed to grow in a logistic fashion in the absence of the other, and the effect of competition is essentially to lower the carrying capacities of both species. Thus, the growth
[K1] = I 1 a121 [ g ~ l K2 a21 1 LN~__I with solution
[1 a12]IK1] --1
LN~_J
a21
1
K2
1000
800
fl
7
600
II I / / ' / 400
N~(t) N2(t) 200
1000
b
/
800
f
600
400
Nl(t) N2(t)
200
0
FIGURE 8.17 Lotka-Volterramodel for two competing species. Input flow for each population is influenced by the size of the other population.
,0
do
~0 t
do
.0
6o
FIGURE 8.18 Population dynamics for two populations described by the Lotka-Volterra competition equations. (a) Dynamics of population one with the competition coefficient a21 = 0 and a12 = 0.25. (b) Dynamics of population two with a12 0 and a21 0.25. =
=
166
Chapter 8 Traditional Models of Population Dynamics
or
N~
l
= 1/D
E--a21 1 lI1 ll a12
K2 '
where D = 1 - a12a21 (see Appendix B.4 for a discussion of matrix inverses). Thus the equilibrium population sizes are K1 N~ = 1 -
-
a12K2 a12a21
and K2 -
N~ = 1
-
8.8.1.2. Competitive Exclusion
a21K 1 a12a21
For analysis of population dynamics it also is useful to describe population sizes for which dNi/dt = 0 for one but not both of the populations. Setting dN 1/dt = 0 yields N 1 = K 1 - a12N2,
which describes combinations (N 1, N 2) of population sizes with 0 growth rate for population 1. The corresponding line describes a null cline (see Appendix C.3), so called because of the absence of growth in population I along it. Similarly, setting dN2/dt = 0 yields the null cline N 2 = K2 -
a21N 1
for population 2. The point of intersection of the null clines defines equilibrium population sizes, because the growth rates for both populations are 0 there. The numerators of the equilibrium formulas above define conditions for population coexistence, and graphs of the null clines can be used to highlight directions of population change toward equilibrium. Three possibilities arise: stable coexistence, competitive exclusion, and unstable population equilibrium. 8.8.1.1. Stable Coexistence
If the numerators of the equilibrium formulas are both positive, i.e., if K 1 > a12K 2
rying capacities K1 and K2, reduced by amounts a12K2 and a21K1, respectively, and scaled by 1 - a12a21. Both populations converge to the equilibrium population levels irrespective of the initial population sizes. Initial population sizes larger than N~ and N~ lead to population declines toward the steady-state values, and initial population sizes lower than N~ and N~ lead to increases toward the steady-state values. These tendencies are shown in Fig. 8.19a, which displays the population equilibria and null clines and indicates with arrows the direction of population change at any point in the phase plane.
(8.29)
and
If only one of the two conditions in expressions (8.29) and (8.30) is met, the corresponding population eventually approaches its carrying capacity and the other population is driven to extinction. Thus, if K1 > a12K2 and K2 ~ a21K1, then species 2 is excluded and species 1 converges to K1 (Fig. 8.19c). Convergence to the carrying capacity is independent of population initial conditions. If K 2 ~ a21K 1 but K 1 ~ a12K2, species 1 is excluded and species 2 converges to K2 (Fig. 8.19b). Again, convergence of population 2 to its carrying capacity is irrespective of population initial size.
8.8.1.3. Unstable Population Equilibrium If both conditions in expressions (8.29) and (8.30) are met, the equilibrium population sizes describe an unstable equilibrium. It is easy to show that K 1 ~ a12K2
and K 2 ~ a21K 1
are equivalent to a12a21 ~ 1, or D = 1 - a12a21 ~ 0. In this case, populations with initial population sizes of N~ and N~_ will be maintained at equilibrium levels indefinitely, but initial population sizes other than N~ and N~ result in the extinction of one of the populations. Thus, K2 - a21K1 N2(0) > Kll K 2a12 ~NI(O) leads to the extinction of population 1, and K2 a21K 1 N 2 ( 0 ) < Kll - a12K---~2NI(O) -
K 2 > a21K 1,
(8.30)
then the populations can coexist in equilibrium. In this case, steady-state population levels are simply the car-
leads to the extinction of population 2. Null clines and direction arrows indicating population changes for this situation are shown in Fig. 8.19d.
8.8. Models of Competing Populations
167 F I G U R E 8.19 Phase plane diagram for the Lotka-Volterra competition equations for two species, exhibiting isoclines and zones of coexistence, competitive exclusion, and unstable equilibrium. (a) Population coexistence. (b) Extinction of species 1. (c) Extinction of species 2. (d) Species extinction depends on population initial conditions.
K~
K1/a12~
Kl/a12
K2 N2*
~.
N1*
K1
K1 K21a21
K2/a21
K1/a12 c K~
K11a12 N2*
K2/a21
K1
Example The Lotka-Volterra competition model can be illustrated with a discrete-time model of two competing species, with transition equations
Nl(t + 1) - Nl(t) + 0.2 Nl(t)[700 - Nl(t) - 0.25 N2(t)]/500 and
N2(t + 1) = N2(t) + 0.3 N2(t)[1000 - N2(t) - 0.5
Nl(t)]/lO00.
Species 2 in this model has a higher rate of growth than species 1 (r 2 > r 1) and higher carrying capacity (K2 > K1). On the other hand, species 1 has a stronger competitive effect on species 2 (a21 > a12). Figure 8.20a displays trajectories for each species under these parametric conditions, starting with initial population sizes of NI(0) = 25 and N2(0) = 25. For comparative purposes the population trajectories in the absence of competition are shown in Fig. 8.20b. Note that competition lowers the effective carrying capacity of each population. However, the impact of competition is disproportionate between the populations, as a result of differences in the population parameters. The effect of competition on species 1 is to lower the effective carrying capacity by about 20% and to slow the rate of
NI*
K2/a21
convergence to this size limit. Similarly, the effect of competition on population 2 is to reduce its carrying capacity by about 20%, even though there are large differences between the two populations in the sizes of their competition coefficients and carrying capacities.
Example If the competition coefficients a12 and a21 in the previous example are increased to a12 -- 0.75 and a21 = 1.5, the system becomes unstable, with equilibrium conditions depending on initial population sizes. Instability results from the strong competitive interactions, expressed by a12a21 -- (0.75)(1.5) > 1.0. Under these conditions, one or both of the populations are driven to extinction, depending on the initial sizes of the populations. For NI(0) - 700 and N2(0) = 800, population declines result in the extinction of population 1, after which population 2 converges to its carrying capacity. For N 1(0) = 900 and N2(0) = 800, population 2 is eliminated and population I converges to its carrying capacity. For N 1(0) = 800 and N2(0) = 800, both populations are driven to extinction. 8.8.2. Lotka-Volterra Equations for Three or More C o m p e t i n g S p e c i e s The Lotka-Volterra competition equations for two species can be extended to include three or more spe-
168
Chapter 8 Traditional Models of Population Dynamics
1000 /
..,.,.._ ~
.
.
.
As before, rearrangement of these equations leads to the matrix equation
.
/
Nl(t) N2(t)
800
I
i
..
B
600
/
/
/
/
K1 K2
/
1
a12
a13
...
alto
a21
1
a23
...
a2m
B
J
I]
m
N~ N~d .
I
|
I
400
//
am1
1. LN*m_
am2
200
with solution .,.,
~o
do
30
40
m
I> NI~ N2(O
6 0 0
/
/
/
t
Nm*
.=
..
1
a12
a13
...
a21
1
a23
...
am1
am2
alm] -1 -K 1a2m I K2.
This matrix equation defines m equilibrium conditions, one for each species. If m = 3, for example, steadystate population sizes are
........--t
1
m
N~
1000
800
..,
N~
50
I
N~ = [(1
-
a23aB2)K1 -
a12(K 2 -
aagK3 )
(8.32)
a 1 3 K 3)
(8.33)
a12K2 )
(8.34)
400
-- a 1 3 ( K 3 - a B 2 K 2 ) ] / D ,
/
//
N~ = [(1
200
.j/
-
F I G U R E 8.20 Trajectories for two species with d y n a m i c s given by the Lotka-Volterra competition equations. (a) Population trajectories for a12 = a21 = 0. (b) Population trajectories for a12 = 0.25 a n d a21 = 0.5.
cies. The competition equations for, say, m competing species are
dt
= riNi
Ki -
Ni -
~
j=l
j,i
a Nj q
/Ki,
(8.31)
with species index i = 1.... , m. Equilibrium population sizes N~, ..., N* are given by the solution of
N~ =
K1 -
a12N'~ -
a13N ~ .....
almNr~ ,
N~ =
K2 -
a21N ~ -
a23N ~ .....
a2mNr~ ,
9
N *m = K m -
~
amlN'~ -
o
am2N ~ .....
am,m_lNr~_ 1 .
-
a13a31)K2 -
a23(K 3 -
a21(K 1 -
a31K1)]/D ,
and N~ = [(1 -
-
a12a21)K3 -
a32(K 2 -
a31(K 1 -
a21K1)]/D,
where D is the determinant of the competition matrix. As with a two-competitor system, a stable equilibrium is assured if D > 0. In addition, species coexistence is assured by the three equilibrium conditions N* > 0, i = 1, 2, 3. Thus, the populations either coexist or are driven to extinction, depending on the sizes of the competition coefficients and population carrying capacities. The following patterns are noted: 9 The equilibrium population sizes are given by the corresponding carrying capacities, reduced by amounts that account for competitive interactions among species. For example, the equilibrium size N~ for population 1 is the carrying capacity K 1, scaled by 1 -a23a32 and reduced by a sum of terms for species 2 and 3. These terms are products of the appropriate competition coefficients and respective carrying capacities, with the latter adjusted to account for competitive interactions between species 2 and 3. 9 Under certain conditions, the steady-state equilib-
8.8. Models of Competing Populations rium for a population is expressed in an additive form that is analogous to the two-species case. If, for example, a23 = a 3 2 - - 0 , the steady-state size for population 1 is N~
=
K1
-
a12K2
1 - a12a21
-
a13K3
-- a13a31
9 If m = 2 the steady-state population sizes, and the conditions for positive equilibria, reduce to the equilibrium conditions described above for two competing species. Example
Consider three competing populations with population dynamics specified as in Eq. (8.31). Assume that the populations each have the same carrying capacity (K i = K, i = 1, 2, 3) and all competition coefficients are identical (aij = a, i = 1, 2, 3 and j = 1, 2, 3). The determinant D = (1 - a) 2 (1 4- 2a) of the competition matrix is positive for all a ~ 0 except a = 1, so the system possesses stable equilibria for all levels of competition except a = 1. The coexistence conditions (8.32)-(8.34) reduce to 0 < [(1 - a 2) - a(1 - a) - a(1 - a)] = (1 - a) 2, which again is satisfied for values a ~ 1. Thus, nonunity competition coefficients lead to coexistence of all three species. For example, a = 0.2 and K = 1000 yields an equilibrium size for all three populations of 714, whereas a = 0.8 yields equilibrium population sizes of 385.
8.8.3. Resource Competition Models A second class of competition models expresses competition through the sharing of a resource by two or more competitors. To illustrate, consider a community of herbivores that utilize the plant biomass in an area for food (Tilman, 1980, 1982). Let R represent the available biomass of the food resource, subject to herbivory and regeneration over time. Here it is assumed that herbivore reproduction rates are influenced by availability of the food resource, but mortality rates are not: d N i / d t = b i N i [ R / ( R + Hi)] - mini,
with H i the amount of food resource necessary to sustain a reproduction rate for species i that is one-half the maximum reproduction rate b i. Thus, the reproduction rate for each herbivore population increases asymptotically from 0 to b i as R increases. In contrast, the popula-
169
tion mortality rate m i is constant irrespective of the availability of food. The dynamics of the food resource reflect the fact that each herbivore population depletes the resource according to the food requirements of individuals in that population. Regeneration of food is modeled in terms of growth to a maximum supply of food, with the rate of regeneration dependent on the difference between actual and potential supply: dR dt = a(S - R) - ~ ,
i
Ci f(Ni),
where S is the maximum amount of food that is potentially available and the parameter a represents the rate at which food supply is replenished. Depletion of the resource is a function of consumer populations through a consumption function f ( N i) that is scaled by speciesspecific terms c i. Exploitation competition is effected through the second term of this equation, wherein interspecific exploitation of food resources results in fewer resources and thus in lower growth rate than would be the case in the absence of competing species. Equilibrium conditions for this system are obtained by setting the transition equations to 0 and solving for the equilibrium levels of R and Ni:
biN~[ R* ] N*= mi R* + Si and
R*- S-
~ ciX~/a i
for f ( N i) = N i. From these equations an equilibrium value of R can be defined for each population considered in isolation. The solution for the complete system of equations leads to an equilibrium value R* such that biR*/(R* + H i) - m i vanishes for one competitive population and is negative for all others. This in turn leads to the eventual elimination of all other consumer species except that corresponding to the 0 growth rate. Example
Exploitation competition is illustrated in Fig. 8.21 with a discrete-time model involving two species that are competing for a common forage resource. Mortality losses for both competitors are described by simple death processes, with per capita death rate of 0.5 for each species. Reproductive success is modeled as a simple birth process (intrinsic birth rate is 1 in both cases) that is scaled by the factor R / ( R + H i) expressing the availability of food resources (H 1 = 400 and H2 = 500 for species I and 2, respectively). Thus, the amount
170
Chapter 8 Traditional Models of Population Dynamics
16
N~(t)
14
the population sizes. By extension, we can express a general model for m interacting species as -dN1/dt"
12
aN2/at
10 8 6 ,~ ~.. /
dNm/dt
\
4
"~
2
- all(N)
a12(X)
a13(X)..,
a21(X)
a22(X)
a23(X)..,
alm(X)- -Nl(t)N2(t) a2m(~
-.._.
40
9
F I G U R E 8.21 Population dynamics of two species that are competing for a common resource. Depending on the relative consumption efficiencies Hi, one species eventually is driven to extinction.
of food resource at which species 1 grows at one-half its maximum rate is less than the amount required for species 2. The stock of food resources is depleted in proportion to the sizes of the competitor populations, with species-specific scaling factors of cI = 10 and c2 = 10. The level of food resources is constrained by the maximum resource level of S = 1000, with replenishment occurring at a rate that is one-half the unmet potential. Note that species 2, with a lower efficiency in the transformation of resources into reproduction, declines asymptotically to extinction. On the other hand, species 1 and the resource asymptotically approach nonzero equilibrium states.
8.9. A GENERAL M O D E L FOR I N T E R A C T I N G SPECIES As indicated above, both predation and competition can be modeled with linear combinations of terms that express the species interactions. Thus, the Lotka-Volterra competition equations express competition by scaling the per capita population rates of growth with linear functions
E
Ki
-
-
N i - "= aq
K---~
that incorporate population sizes of the competitors. Similarly, the Lotka-Volterra predator-prey model expresses predation by scaling per capita rates of birth and death with scaling factors that also are linear in
~
_aml(~ arn2(~ am3(~ ... a m m ( ~ .
Nm(t)
where aij(N) is a (usually differentiable) function expressing the impact of population j on population i. A useful special case of the general model is defined by the linear forms (8.35)
a i j ( ~ = bij + c ijNi
for the model coefficients. By restricting the choice of the coefficients cij and bij, the model can be used to characterize interspecific interactions as described earlier. For example, if
i=j ir
and cij =
-ri/K i riaq/K i
i = j i ~ j
then Eq. (8.35) is identical to Eq. (8.31) describing multispecies Lotka-Volterra competition. Though some models discussed in this chapter satisfy a linearity requirement as in Eq. (8.35), in general, population models do not. Obvious extensions include quadratic, cubic, and higher degree equations, sinusoidal and other periodic relationships, multispecies functions aij(~_ that include nonlinear terms in the population sizes, and other mathematical characterizations. Population dynamics for these more complicated systems can be described by a linear model in a neighborhood of system equilibrium, via a linear approximation of the transition equations. Under fairly mild differentiability conditions, the function F i (-~ in the model d N / d t = F(N) can be expressed as Fi( ~
-- Fi(N* ) -t-- ~ L a N j ( N * ) 1
] (Nj-
N 7)
8.10. Discussion in a neighborhood of an equilibrium _N* (see Appendix C). Thus, the system dynamics of d N / d t = F(N) in a neighborhood of N* can be approximated by the linear system dt = F(N*) +
(N*)
( N - N*),
(8.36)
where the matrix d F ( N * ) / d N is defined as in Appendix B.10. On condition that species interactions are symmetric [i.e., 3Fi/ON j = 3 F j / 3 N i for every species pair (i, j)l, d F ( N * ) / d N can be expressed in terms of the singular value decomposition dF
aN(N_*) = e
e',
with _ha diagonal matrix of eigenvalues for C, and P an orthonormal matrix (i.e., P P' = P ' P = / ) with columns consisting of the corresponding eigenvectors (see Appendix B.7 for a description of singular value decomposition and other matrix procedures). Recognizing that d N / d t = d ( N - N * ) / d t and that F(N*) = 0 for the equilibrium point N*, we therefore can express Eq. (8.36) as dn/dt
= [P )~ P']n,
where n(t) = N(t) - N*. Multiplication of both sides of this system of equations by P' yields d m
d~ [P'n(t)] = k P ' n ( t ) or
dZ/dt
= k Z(t),
with Z(t) = P ' n ( t ) and n(0) = N(0) - N*. This reduces to m independent equations d z i ( t ) / d t = hiz(t)
in the synthetic variables zi(t) , with solutions zi(t) =
Zi(O) exp()~it). Back-transformation of Z ( t ) by P Z ( t )
= P P ' n ( t ) = n(t) then produces the population dynamics
for each population. A general solution is given by the solution based on zi(t) - Zi(O) exp ()~it), combined with the particular solution Ni(t) = [Fi(N*)]t , i = 1, ..., m (Rainville et al., 1996).
8.10. D I S C U S S I O N In this chapter we have described some models that traditionally have been used in ecological and biologi-
171
cal sciences. We began with simple expressions for the exponential and logistic models and added structure throughout the chapter to account for various biological features. In the appropriate context, each of these models can provide useful insights about population dynamics. As discussed in Chapter 7, there is an inevitable tradeoff between the generality provided by relatively simple models lacking specific, detailed mathematical structures, and the realism and precision that can be attained by more complicated and biologically rich models. We have seen that the incorporation of additional biological structure and function into a population model quickly leads to difficulties in interpreting model behaviors. For example, a complete sensitivity analysis of a Leslie matrix model with four age cohorts would involve the assessment of eight reproduction and survivorship parameters (four per capita birth rates and four survival rates), requiring a sensitivity analysis for each parameter at a minimum of 27 different combinations of values of the other parameters (assuming only two values for each parameter). Thus, even without accounting for the influence of system initial conditions, this relatively simple model requires a total of 256 different sensitivity assessments. Clearly, there is strong incentive to include only the features in a model that are essential in meeting its objectives. As indicated in earlier chapters, the biological justification for a model and the interpretation of evidence for it are key to its usefulness. Consider the observation of a sequence of population abundances, to be fitted with a model of population growth (Fig. 8.22). These data appear to support the assumptions of the logistic model, which imply that per capita birth rates, death rates, or both decrease with increasing density. Assume for now that abundance at each time step is estimated perfectly, so that statistical sampling error is not at issue. As shown in Fig. 8.22a, a discrete logistic model with parameters r = 0.2, K = 500 seems to fit the data, so we might conclude that the logistic model "explains" temporal variation in abundance. But other biologically reasonable models fit the data as well. For example, an alternative model assumes the population is growing in a density-independent fashion up to an absolute limit K (e.g., as determined by available space), and then all excess individuals above K either die or migrate. The predicted trajectory for this model (Fig. 8.22b) is similar to that for the logistic, yet the model invokes no assumptions about density dependence. A second alternative might involve growth that is density independent up to K, but that K, rather than being fixed, varies randomly (e.g., because of annual precipitation factors). Again, the predicted trajectory
172
Chapter 8 Traditional Models of Population Dynamics FIGURE 8.22 Comparison of three models againstfield data. (a) Discretelogisticmodelwith r = 0.2 and K = 500. (b) Exponential model, truncated at K = 500.(c) Exponential model, truncated at randomly varying K.
N(t)
-%-.
a
500
._
oooeeee
c
00000
o 9
450
400
350
3OO
0
;
10
is
20
fs
0
5
10
-- 15
(Fig. 8.22c) is similar to that of the logistic model and resembles the observed abundances. In fact, the available field data are unuseful in discriminating among these three candidate models for population growth. Any of the three models could have generated the observations, and thus it is not possible to validate one particular model based on the data. In Chapter 9 we consider the identification of models with time-series data, and Part III deals in considerable depth with the use of sampling data to estimate specific model parameters. Here we simply emphasize that the use of biologically based models, combined with good experimental designs and careful inference, can help to avoid an unjustified connection between observations and underlying premises. In this example, improper inferences about density dependence could lead to erroneous predictions about the impacts of harvest on population growth and thus to faulty management recommendations. Well-designed monitoring efforts and careful assessments help to avoid such errors. We also emphasize the need to account for sampling variablility when population inferences rely on vital rates that are sample-based estimates rather than exact parameter values. Suppose we use demographic data from two populations to obtain estimates of survival and reproductive rates, which in turn are used in Leslie matrix models to determine the population growth rates of 1.05 and 0.99. Because these growth rates rely on estimates of survival and reproduction rates that are based on sampling data, the growth rates inherit randomness from this sampling variability. Replicated sampling of the populations would produce different values for the growth rates, according to a probability distribution that depends on the sampling scheme. A
0
5
10
1'5
20
2'5
number of questions therefore arise about the magnitudes and differences between the population growth rates, assuming one of the underlying models is appropriate. Inferences about population vital rates, and thus the growth rates, must account for sampling variation in estimates that are based on sample data. Methods for estimating sampling variances of asymptotic growth rates include both bootstrap approaches and delta method approximations (e.g., see Lenski and Service, 1982; Lande, 1988; Alvarez-Buylla and Slatkin, 1991, 1993, 1994; Brault and Caswell, 1993; Franklin et al., 1996; Caswell, 2001). This variation must be considered when one asks questions about the magnitude and direction of population changes and about differences in growth rates between the two populations. The usefulness of modeling procedures, especially in decisionmaking, is likely to be improved by smartly designed sampling efforts supporting models that avoid unnecessary complexity and by continual comparison of model predictions to observations, where possible in an experimental framework. Finally, it should be emphasized again that, though none of the models discussed in this chapter is "correct" in the sense of capturing all the features of a population, no model is necessarily "incorrect." In fact, no model is capable of a comprehensive characterization of a real biological population. Put differently, all models are "wrong," in that all models leave out far more about population structures and functions than they incorporate. It is the role of the biologist, modeler, and analyst to determine what level of biological detail is necessary (and feasible!) for a model to meet its objectives and, having made that decision, to find informative ways to investigate model behaviors pursuant to those objectives.
C H A P T E R
9 Model Identification with Time Series Data
9.1. MODEL IDENTIFICATION BASED ON ORDINARY LEAST SQUARES 9.2. OTHER MEASURES OF MODEL FIT 9.3. CORRELATED ESTIMATES OF POPULATION SIZE 9.4. OPTIMAL IDENTIFICATION 9.5. IDENTIFYING MODELS WITH POPULATION SIZE AS A FUNCTION OF TIME 9.5.1. Model Identification in One Dimension 9.5.2. Model Identification in Two Dimensions 9.5.3. Model Identification in Three or More Dimensions 9.6. IDENTIFYING MODELS USING LAGRANGIAN MULTIPLIERS 9.7. STABILITY OF PARAMETER ESTIMATES 9.8. IDENTIFYING SYSTEM PROPERTIES IN THE ABSENCE OF A SPECIFIED MODEL 9.9. DISCUSSION
ulation model, based on available data and other relevant information. Typically the information used in model identification comes from a wide range of sources, including laboratory experiments, field studies, anecdotal information, historical information that is documented in the published literature and in field notes, and other sources. The process of identification involves the use of this information in recognizing model features and estimating model parameters. Three elements in the process can be recognized (Ljung, 1999): 1. An available set of information, including extant data bases and data collected in the field pursuant to the identification of model structures and parameters. 2. A set of candidate models, from which to identify that model which is most appropriate for its intended uses. 3. A rule for comparing and contrasting models, to serve as an aid in selecting a "best" model as guided by data and other information.
In Chapter 8 we discussed a number of population models with parameters such as initial population size, survivorship, and reproduction rates, having outlined in Chapter 7 a process of model development that incorporates these parameters and assesses their importance in influencing model behavior. However, the actual procedures by which model features can be identified a n d / o r estimated with data are yet to be developed. In this chapter we describe some techniques to "identify" a model of a particular mathematical form, utilizing data to guide the identification process. This is preparatory to a comprehensive treatment in Section III of statistically based approaches to parameter estimation. The overall objective of model identification is to specify the structural features and parameters of a pop-
Model identification can be recognized in the approach described in Chapter 7 for model development, especially in the processes of verification, validation, and identification of system features and boundaries. Several kinds of activities are involved, including the following approaches: 9 Initial specification of model equations, parameters, operating constraints, the model time frame, and so on. Initial model development is based on biological theory, mathematical analysis, intuition, expectations about model performance, and other sources of infor-
173
174
Chapter 9 Models and Time Series Data
mation. The process of model development was described in Chapter 7. 9 The "fitting" of models to time series data, in the sense of identifying structural features a n d / o r parameters of a model through a comparison of model trajectories against sequences of field observations. In this case, model identification occurs indirectly, through, e.g., the choice of parameter values that provide a good "match" between the model trajectory and field observations. 9 The estimation of parameters such as population size, density, and survivorship with data collected for the purpose of estimating particular population parameters. Estimation in this context typically involves data collection according to a sampling design that targets the parameters of interest and data analysis based on probability models that include these parameters. In this chapter we focus on the "fitting" of models to time series data, through a comparison of model trajectories against a sequence of field observations. This activity is closely associated with model verification, which also involves a comparison of model performance with observed patterns in population dynamics. Model identification also can be seen as part of an adaptive process of model development, application, and refinement (see Chapter 24). A conceptual framework for model identification includes a population model
N(t + 1) = N(t) + f(N, Z, U), N(O) = N O and data that are collected at various discrete times (typically at each time) over the course of the time frame. The model describes population dynamics in terms of population size N(t), environmental influences Z(t), and (possibly) management actions U(t). Its mathematical form is assumed to be well defined, but some of the parameters in the model are not known and therefore must be identified. Here we use N(t) to indicate a model-based prediction of population size, with the actual population size to be estimated with data. The data collection focuses on population size and possibly other population attributes at various times in the time frame. Let S = {tI .... , tk} represent times at which an estimate IQ(t i) of population size is available. The notation N(t i) is used for estimated population size to emphasize that these values are based on data that are subject to sampling~ variability. The amount of variation in N(t i) depends on the sampling design and sampling effort, as discussed in Chapter 5. Assuming the samples are representative of the population and the estimation procedure is unbi^
ased, the accuracy of 1Q(ti) increases asymptotically with sampling effort.
9.1. M O D E L I D E N T I F I C A T I O N BASED ON ORDINARY LEAST SQUARES For illustrative purposes, we begin with a simple exponential model over a discrete time frame T, along with a set {N(ti): t i ~ S} of estimates of population size. The estimates in this "observation set" correspond to a set S = {t1, ..., t k} of times that are distributed over the time frame. In the absence of additional information about birth/death rates or population growth rates, model identification consists of choosing model parameters based on the set of population estimates. From Section 8.1 we know that the behavior of an exponential model with net per capita growth r is given by
N(t + 1) = N(t) + rN(t), N(O) = N 0, with a trajectory determined by the two parameters N Oand r. To identify the model, it therefore is sufficient to estimate these parameters. A method for doing so consists of comparing predicted and estimated (or observed) population sizes at each point in the time frame for which data are available and of choosing values for N Oand r to ensure the best possible match of model output and data. A standard index by which to measure such a match is "mean squared error:" F(N 0, r) = ~
[N(t i) - ]Q(ti)]2/k,
ti~S
where N(t i) is the predicted population size from the model and N(t i) is the estimated (or observed) population size based on field data (Rawlings, 1988). Because the parameter values N Oand r influence the predicted values, they also influence the mean squared error function. We describe 1Q(ti) in what follows as an observed population value at time t i, recognizing that the "observations" are based on data with which population size is estimated. Note that mean squared error is small to the extent that the model represents the observed population values, with a limiting value of zero in case the model fits the observations exactly. On the other hand, mean squared error is large to the extent that the model fails to represent the observations. The effect of squaring the deviations between observed and predicted values is to give very large weight to large deviations. Thus the mean squared error, which is greatly inflated by
9.1. Model Identification Based on Ordinary Least Squares large deviations, can be reduced dramatically by reduction (or elimination) of these deviations. Of course, this reduction is obtained through the choice of values for the parameters No and r. Because the mean square error depends on the pair (N 0, r) of parameter values, the fitting of the model to data can be seen as an optimization problem, wherein the pair (N 0, r) is to be chosen to minimize F(N0, r): minimize F(N o, r) = ~ [ N ( t i) - l~](ti)]2/k NO, r
175
in Fig. 9.1. Mean squared error, represented by the vertical axis, is shown as a function of the parameters N O and r, represented by the two horizontal axes. At some point in the parameter plane, the mean squared error assumes a minimum value. Provided the error function is minimum for positive values of N o and r, its partial derivatives both are zero at the minimum point. Geometrically, this means that the tangent plane for the error function is horizontal at (N~, r*) (see Appendix H).
tieS
Example
subject to N(t + 1) = N(t) + rN(t), N(O) = No.
Because N(t) = N0(1 + r) t for the exponential model, the transition equations can be incorporated directly into the objective function: F(N o, r) = ~ [ N o ( 1
+ t") ti -- l ~ ( t i ) ] 2 / k .
ties
Necessary conditions for a nonzero solution to this optimization problem are
aF/aNo
Consider a population of rodents that were introduced into a previously uninhabited habitat. Resource managers are concerned about the rapid growth of this population and need to predict population size as they consider potential control programs. Population growth has been tracked each year since the time of introduction with population surveys, producing the estimates [/~/(1),/~/(2), N(3), N(4)] = (20, 35, 68, 121) of population size. Because introduction of the species occurred only recently and as yet there are no indications of declining population rate of growth, an exponential model is used to describe the population. Identification of the model involves estimation of per capita growth rate r and initial population size N 0, based on the available survey data. Optimal estimates of these parameters can be obtained by minimizing the
where the partial derivatives are given by OF~ONo = 2~[No(1
+
F) ti --
/Q(ti)](1 + r)ti/k
ties
and OF~Or = 2 N 0 ~ ti[No(1
+ r) ti -
/~/(ti)](1
+ r)ti-1/k
ties
(see Appendix H). Thus the optimality conditions are equivalent to gl(N0, r ) = ~[(N0(1 + r) ti - N ( t i ) ] ( 1
+ t') ti = 0
ties
and g2(No, r ) = ~
ti[No(1 + y ) t i _ ]Qti](1 + r ) t i - l =
O,
tieS
and fitting the exponential model to the data set {/~(ti)" t i ~_ T} reduces to a problem of finding zeros for the two functions gl(No, r) and g2 (No, r) that are defined by partial derivatives of the mean squared error function. Model identification through the minimizing of an error function is illustrated for the exponential model
F I G U R E 9.1 Geometry of mean square error for the exponential model N(t) = N0(1 + r) t. The error function F(N 0, r) = ~i[N(ti) /~(ti)] 2 is minimized for values (N~, r*) of the model parameters
(No, r).
176
Chapter 9 Models and Time Series Data
mean squared error, subject to the model transition equations. As above, this is equivalent to finding the zeros of gl(N0, r ) =
~[N0(1
+
r) ti -/Q(ti)](1 + r) ti
ti*S
= [N0(1 + r ) - 2 0 ] ( 1 + r) + IN0(1 + r) 2
_
35](1
+ r) 2
4- [N0(1 4- r) 3 - 68](1 + r) 3 + [No(1 + r) 4 -
121](1 + r) 4
and g2(X0,
r) = X tiN0(1 + r)ti
__
Xtil(1 + r)ti-1
ti~S
= EN0(1 + r) - 20] + 2IN0(1 + r) 2
_
35](1 + r)
+ 3[N0(1 + r) 3 -
68](1 + r) 2
+ 4[N0(1 + r) 4 -
121](1 + r) 3.
Application of a gradient search procedure (see Appendix H) yields the values N~ = 11 and r* = 0.83 that minimize the mean squared error for this population. Thus, the population model is
N(t) = N~(1 + r*) t = 11(1.83) t, and the predicted population size for year 5 is 226. Though uncharacteristically simple, this example of model fitting is nevertheless informative of a general approach to model identification. Key components of the approach are as follows: 9 Description of the problem in terms of constrained optimization, with mean squared error as the objective function to be optimized and the system transition equations representing constraints on the choice of parameter values (see Chapter 21). A general statement of the problem is minimize F(a) = ~ [ N ( t i) - N(ti)]2/k a
tieS
subject to
N(t + 1) = N(t) + f(N: a), N(0) = No, where _a is a vector of model parameters (perhaps including N 0) that are to be identified, and f(N:a) specifies the predicted change in population size through time. As a matter of notational convenience, environmental
and control variables Z(t) and U(t) are suppressed in this formulation. 9 Incorporation of the transition equations into the mean squared error objective function. For simple systems, this sometimes can be accomplished by actually solving the transition equations, so that N(t) can be expressed as a function of the model parameters. In the example above, N(t) is given in terms of the parameters N O and r, by N(t) = N0(1 + r) t. In most cases an analytic expression for N(t) cannot be obtained, and the transition equations must be incorporated by means of "Lagrangian multipliers." The use of Lagrangian multipliers is described in some detail in Section 22.1 and Appendix H. 9 Differentiation of the objective function (as modified by incorporation of the transition equations) with respect to the model parameters. This defines a system of functions in the model parameters. 9 Determination of the zeros for these functions (i.e., the parameter values for which the functions have a value of 0). The zeros can be determined by numerical methods or, in a few instances, by mathematical analysis.
9.2. O T H E R M E A S U R E S OF M O D E L FIT Though mean squared error is the most common measure by which to judge the fit between data and a model, it is by no means the only measure. Another that sometimes is used is mean absolute error:
F(a) = ~,lN(ti) - /~(ti)l, tieS
where IN(t) - ~l(t) I is the absolute value of the difference N(t) - ~l(t). This measure is less sensitive than mean squared error to large deviations between predicted and estimated population sizes. Nevertheless, its value is large when deviations are large and small when deviations are small, with a lower limit of zero as deviations approach zero. Other mathematical forms can be used to measure the importance of deviations, and other factors can be included in the objective function to account for, e.g., patterns of variation in the estimates N(t). For example, both the mean squared error and mean absolute error functions can be modified so that the deviations are scaled with weights that decrease with increasing variation in the estimates /x/(t). The logic for such a weighting scheme is that population estimates with large variance are not as informative of the true population size as estimates with small variance. Under these circumstances, N(t) - ~l(t) only imprecisely represents
9.2. Other Measures of Model Fit the deviation between actual and predicted population sizes, making it more difficult to ascertain the "best" parameter values with which to represent the population. It therefore is reasonable to weight deviations with small variance more heavily, because they better represent differences between actual and predicted population sizes, and to weight deviations with large variance less heavily (Rawlings, 1988). A generalized expression for the identification problem is minimize F(a) = ~ g[N(t i) - ~l(ti)]/k a
ti*S
subject to
N(t + 1) = N(t) + f(N: a), N(0) = N 0, where a is the vector of model parameters to be identified anti giN(t) - / Q ( t ) l is a monotone increasing function of the deviations N(t) - [q(t). In the case of mean squared error,
g[N(t) -/~/(t)] = wt[N(t) - /~/(t)]2, where w t is the weight assigned to deviation N(t) /Q(t) and Xt~s wt = 1. In the case of mean absolute error,
ity). An experiment to investigate this hypothesis involves several populations of fruit flies that are subjected to different temperature regimes under controlled experimental conditions. A small (but unknown) number of fruit flies is released at the beginning of the experiment into each of several growth chambers that are regulated for temperature, and daily estimates of population size are recorded for 5 consecutive days thereafter. The data subsequently are used to fit a series of logistic models of continuous population growth under the different temperature regimes. Model parameters for each of the populations are identified by means of an error function with components wt[N(t) - 1Q(t)]2, with the weights based on (1) an indication from the data that variation in the population estimator increases with population size, and (2) improvement of the investigators' counting skills through the course of the experiment, so that later counts are less subject to counting error than are earlier counts. Identification of model parameters for each experimental population is obtained through an optimization process that accounts for these features: 5
minimize F(N 0, r, K) = ( 1 / 5 ) ~ wt[N(t) -/Q(t)] 2 No,r,K
g[N(t) - / ~ ( t ) ] = [N(t)
-/Q(t)]
3/2,
which is influenced by large deviations to a lesser degree than mean squared error, but to a greater degree than mean absolute error. It also would allow for weighting schemes that include other factors besides variation in the population estimates. For example, it often is reasonable to emphasize the fit of the model to data of more recent vintage. A weighting scheme that emphasizes the value of more recent data over [0, 1.... , T] is w(t i) = ti/~, j tj, for which weights decline linearly with the age of the data.
Example Temperature is hypothesized to influence the growth of fruit fly (Drosophila spp.) populations through its effect on both the rate of population growth and the population potential (i.e., the carrying capac-
t= 1
subject to
g[N(t) -/Q(t)] = wt]N(t) - ~l(t) I. This formulation of the identification problem is quite general, in that it can accommodate any deviation function, so long as it is monotone increasing in N(t) - /Q(t), and any weighting scheme, so long as the weights are nonnegative and their sum is 1. For example, the formulation would allow for the deviation function
177
dN/dt = rN(1 - N/K), N(0) = N 0, where {var[/Q(t)]} -1 W t
s
1{var [/Q(t)l }-1
with var[1Q(t)] the sampling variance of the estimator N(t). In this particular case, var[l~(t)] can be approximated by N(t)/t, so that the weights are
t/1Q(t)
wt
-
~t=15 t/1Q(t)"
The continuous logistic model has solution
N(t) = 1 +
K Ce -rt'
with C = K/N o - 1 (see Section 8.2), which can be incorporated directly into the objective function:
5 [ F(N 0, r, K) = ( 1 / 5 ) ~
Wt
N(t) = 1 +
Ce -rt
t
9
t=l
As before, necessary conditions for a nonzero solution to this problem are given by partial differentiation of
178
Chapter 9 Models and Time Series Data
this function, so that the problem reduces to finding the zeros of a system of equations in N 0, r, and K.
9.3. CORRELATED ESTIMATES OF POPULATION SIZE Depending on the sampling and estimation procedures, the estimates of population size used in model identification can be correlated, in that the estimates in successive (and possibly other) time periods have a nonzero sampling covariance. Both the sampling variances and covariances can be accounted for in the weighting scheme of the objective function (Seber and Wild, 1989). Assume, for example, that estimates of population size are obtained in successive years of a multiyear study, and the estimates are subject to both sampling variability and covariation. Assume also that the estimator variances and covariances are known (or can be estimated). Let 0-i2 represent the variance of the ith estimate and 0-/j represent the covariance between estimates for periods i and j. An appropriate form for the error function that "adjusts" for the correlation structure is k
F(a)
=
-
,Y__, 0-q[N(i) -
l~( i) ][N( j)
-/~(j)],
i,j=l
where/q' = [/~(1),/~(2) .... ,/~/(k)] is a vector of population estimates in successive years, and 0-zjis the element in the ith row and jth column of the inverse of the dispersion matrix of variances and covariances. . .
Example
Assume that the estimates of population size in k successive time periods all have the same variance and that the correlation of the estimates decreases exponentially with the time between estimates" 0-i2 -- 0 -2 and 0-q = O"213 Ii-jl. Under these circumstances, the inverse of the dispersion matrix is composed of the elements ~1/(1 - [32)0- 2 .. J(1 + 132)/(1 -o"] = ~-p/(1 - 132)0.2
Lo
132)0"2
i = j = 1 or k j = 2 .... , k - 1 [i- jl = 1
i =
otherwise
(Graybill, 1969) and a quadratic error function that adjusts for the correlations is given by F(a) = [N(1) -/~/(1)] 2 + [N(k) -/~/(k)] 2 k-1
+ ~ (1 + p2)[N(t) - /~/(t)]2 t=2 k-1
- 2p ~, [N(t + t=l
1) -/~/(t + 1)][N(t) -/~(t)].
On examination, similarities can be seen between this somewhat complicated function and the simpler error function F(a) = ~ t [ N ( t ) - N ( t ) l 2 for uncorrected data. For example, both retain a sum of squared error terms. However, the more complicated quadratic error function also includes cross-product terms that are associated with the nonzero correlation p.
9.4. OPTIMAL IDENTIFICATION There is a large, mathematically sophisticated literature on optimization of multidimensional functions. Here we discuss three general approaches to the problem of finding points at which a smooth (differentiable) function assumes a minimum value. To restrict attention to optimization methods that are appropriate for model identification, we assume the following: 9 The only feasible parameter values are positive. Because population initial conditions, rates of growth, carrying capacities, competition coefficients, predation coefficients, and the like are positive (or can be reparameterized to be positive), this assumption is not biologically limiting. 9 The error function is everywhere differentiable over the set of feasible parameter values. Again, this assumption is unlikely to be limiting for the usual measures of identification error. 9 The error function has a minimum value for some unique point in the set. Geometrically, this means that the error function is "downward sloping" toward a single minimum value over the range of parameter values (e.g., Fig. 9.1). 9 The population transition equations are incorporated into the objective function either directly, by solving for the population size as a function of time, or indirectly, by adding the transition equations to the objective function by means of Lagrangian multipliers (see Appendix H). In either case the problem is one of minimizing a function (either the objective function or the Lagrangian function) of the parameters of interest. We illustrate three data-based approaches to the identification of model parameters, each of which is distinguished by its computational and analytic requirements. We first discuss their application when population changes through time are incorporated directly into the objective function and then consider applications when population change is accommodated by means of Lagrangian multipliers. The approaches first are described in terms of models containing a single unknown parameter and then are generalized to account for two or more parameters.
9.5. Identifying Models with Population Size as a Function of Time 9.5. I D E N T I F Y I N G M O D E L S W I T H P O P U L A T I O N SIZE AS A FUNCTION OF TIME We begin with a description of methods for which parameterized forms of population size can be incorporated directly into an objective function, utilizing a closed form for population size as a function of time. For example, population dynamics for the exponential model can be described by N(t) = N0(1 + r) t, and this function can be substituted directly into the error function prior to its being minimized. The result is a minimization problem involving the two parameters r and N 0. Similarly, population dynamics for the continuous logistic model can be described as above by
N(t) =
1 +
Cr - r t
which can be substituted directly into the objective function to be minimized. The result is a minimization problem in the three parameters N 0, r, and K.
9.5.1. M o d e l Identification in One D i m e n s i o n A biological example of model identification with a single parameter might involve the fitting of an exponential model for which initial population size is known with certainty. The problem then reduces to finding a value for intrinsic rate of growth so that the model optimally fits a set {l~(ti): t i ~_ S} of data. In general terms, the optimization problem is
179
9 In the event that the derivative of F can be derived but zeros of the resulting equation cannot be obtained analytically, numeric procedures can be used. A standard approach is Newton's method, in which the derivative of the error function is used in an iterative search procedure (see Appendix H). Newton's method utilizes the derivative dF/da at some starting value a 0 to determine the tangent line of the objective function at a 0. The zero of this line is used as an updated value a 1 for a, and the derivative of F at a I is used to determine a new tangent line with a zero that defines yet another value for a (Fig. 9.2). The updating process continues iteratively until no further change is found in the value of a. Note that the derivative of the error function must be evaluated at each iteration. 9 In case the error function a n d / o r transition equations are so mathematically intractable that derivatives cannot be obtained or Newton's method is computationally burdensome, one can use directed search procedures, in which some initial value a is updated through evaluation of the error function at points in either direction from a (Appendix H). The initial value is replaced with a new value that gives the largest reduction in the value of the objective function. This process continues iteratively until reductions in the error function cease.
9.5.2. M o d e l Identification in Two D i m e n s i o n s An example of model identification in two dimensions is the fitting of an exponential model in both
minimize F(a) = ~, g[N(t i) - l~l(ti)]/k a tieS subject to
N(t + 1) = N(t) + f(N: a), N(0) = N 0, where a is a single model parameter to be identified. Approaches to this problem include the following considerations: 9 Solving the equation obtained by equating the derivative of the objective function to zero. Because an optimal value a* must satisfy dF/da = 0 for a > 0, solving the equation identifies candidates for minim u m error identification. A sufficient condition for minimization is d2F/da 2 > 0 (see Appendix H). Note that this approach requires F to be differentiable, and the equation dF/da = 0 must have a solution. Either or both these requirements may fail to be met for a particular problem.
(.9
6
/
a2
a1
80
F I G U R E 9.2 N e w t o n ' s m e t h o d for finding the m i n i m u m of a differentiable function F(a), given that dF/da = G(a) v a n i s h e s at a m i n i m u m . Starting at an initial v a l u e a 0, the zero of the t a n g e n t line to G(a) at a 0 is u s e d as an u p d a t e d value a 1. This v a l u e is u s e d in t u r n to d e t e r m i n e a n e w t a n g e n t line at a], w i t h a zero that defines yet a n o t h e r value a 2. The u p d a t i n g process continues iteratively until the values of a cease to change.
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its parameters, as discussed above. The corresponding optimization problem is expressed by
function. This process continues iteratively until reductions in the objective function cease.
minimize F(a) = ~, g[N(t i) - l~(ti)]/k a
--
ti*S
9.5.3. M o d e l Identification in Three or More D i m e n s i o n s
subject to
N(t + 1) = N(t) + f(N: a), N(O) = N 0, where the vector _a consists of the two parameters r and N O. Approaches to this problem include analogs to those for the one-dimensional problem: 9 Solving the system of equations obtained by equating the partial derivatives of the objective function to zero. A necessary condition for a positive value of a* is that a* must satisfy OF/Oa = 0. Thus, solving the system of equations identifies candidates for m i n i m u m error identification. Sufficient conditions for minimization are
32F/3a 2 < 0 for i = 1, 2, and
32F/ Oa2
c32F/ Oa13a2
32F/Oa10a2
c92F/Oa2
3. Approaches to the multidimensional problem include analogs to those for the two-dimensional problem: 9 Solving of the system of equations obtained by equating the partial derivatives of the objective function to zero. A necessary condition for a positive value of a* is that a* satisfy OF/a_ = 0. Solving the system of equations identifies candidates for m i n i m u m error identification. Sufficient conditions for minimization involve some rather complicated expressions in the second partial derivatives of F (Appendix H). Note that the number of equations in the system increases with the number of parameters to be identified, and the analytic requirements for differentiability of the error function do as well. This makes it increasingly difficult to construct and solve the system of equations. 9 If differentiability conditions are met, gradient search procedures can be used to find the m i n i m u m value of F. As in the two-parameter case, a standard approach is to search for candidates to update an initial value a 0 in the direction of the gradient OF/Oa_ of the objective function at a 0. A new value for _a is chosen that minimizes the error function along the gradient. At the new value, the partial derivatives for F are computed and a new search is initiated along the resulting gradient. This process continues iteratively until no further change is found in the value of a. Note again that the partial derivatives of the error function must be computed at each point in the iteration. 9 Derivative-free search procedures can be used in the multiparameter case, in which the initial value for a is updated through evaluation of the error function at other points near a. Because of the increased dimensionality of the parameter space, it is necessary to evaluate the error function at a large number of different parameter values in each iteration. The value of a is replaced at each iteration with a new value that most reduces the error function. This process continues iteratively until reductions in the objective function cease. Obviously, the amount of computation with this method increases dramatically as the number of parameters increases.
9.7. Stability of Parameter Estimates
181
ables )~t that were not required for the exponential model. This example displays the key features of model identification with Lagrangian multipliers:
9.6. I D E N T I F Y I N G MODELS USING LAGRANGIAN MULTIPLIERS The approaches described above require a closed form for population size as a function of time. In the more usual situation in which the population trajectory cannot be determined in closed form, model identification must be modified to allow the transition equations to serve as constraints on the minimization of the error function. This is accomplished with Lagrangian multipliers, by means of which the objective function is modified to include the transition equations. The use of Lagrangian multipliers is most easily described by an example. Consider the rodent population of the previous example and assume that, prior to implementation of any management strategy, a population size of 162 is recorded for the fifth year after introduction. Because of the much reduced rate of population growth between years 4 and 5, a logistic model now is thought to be appropriate for the population. However, resource managers remain concerned about the potential for additional population growth and continue to need a prediction of population size. Identification of the model now involves specification of the initial population size N 0, intrinsic growth rate r, and population carryin~ capacity K, based on the observations [N(1), N(2), N(3), N(4), N(5)] = (20, 35, 68, 121, 162). As before, model identification involves the minimization of mean squared error subject to the model transition equations. However, in this case the logistic transition equations can be incorporated into the objective function by means of Lagrangian multipliers:
9 The mathematical form of a model characterizing population dynamics must be assumed, with model identification described as a minimization problem constrained by the model transition equations over the time frame. 9 The transition equations are incorporated into the objective function by means of Lagrangian multipliers, with a distinct multiplier ~'t for the transition equation at each time t in the time frame. This extended objective function, called the Lagrangian function, is influenced not only by the parameters of interest, but also by the Lagrangian multipliers. 9 The Lagrangian function is minimized with respect to both the parameters of interest and the Lagrangian multipliers. In case the Lagrangian function is differentiable, setting its derivatives with respect to )~t equal to zero reproduces the transition equations. These equations, along with analogous equations based on the derivatives of the Lagrangian function with respect to the parameters of interest, constitute a system of equations that can be solved numerically for the parameter estimates. For example, a two-parameter model would involve a gradient search utilizing the gradient (OL/cOa 1, OL/3a 2, 3L/OK_) of the Lagrangian function. Lagrangian procedures for constrained optimization are discussed in some detail in Section 22.3 and Appendix H.
4
L(N 0, r, K, K) = F ( N o, r, K) + ~, ht{N(t + 1) --
t=0
9.7. S T A B I L I T Y O F PARAMETER ESTIMATES
- N ( t ) - r[1 - N ( t ) / K ] } 4
= ~ ([N(t + 1) -/~/(t + 1)]2/5 t=0
+ )~t{N(t + 1) - N ( t ) - r[1 - N ( t ) / K ~ } ) ,
where K t is a "Lagrangian multiplier" for the corresponding transition equation, and the extended objective function L(N 0, r, K, ,~) is the "Lagrangian function." The problem now is to minimize the Lagrangian function by choosing the parameters N 0, r, K, and h. Differentiation with respect to N 0, r, and K results in three equations in the parameters, and differentiation with respect to _h reproduces the transition equations. As before, this reduces to the problem of finding the zeros of functions defined by the derivatives; however, the problem is complicated by the need to consider the additional logistic parameter K and additional varim
An important consideration in model identification is the size of the data set {/~/(ti)" t i ~_ S} relative to the number of model parameters to be identified. In essence, the size of the data set used to define the error function should be substantially larger than the number of parameters; otherwise, variation in the data may lead to identification of parameter values that are unreasonable a n d / o r highly unstable. This can be seen with a simple example involving identification of the exponential model N ( t ) = N0(1 + r) t, based on a data set {N(ti): t i ~ S}of k observations. Here we assume that the exponential model is structurally correct, in that the form of the underlying process for the data is exponential in its mean: E[N(t)] = N0(1 + r) t. Thus the objective is to estimate the parameters r and N O by fitting the model to the data set. The problem can be
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simplified greatly by transforming the data and the model with a logarithmic transform and by using logarithms for model fitting. The net effect of the logarithmic transform is to "linearize the model," wherein the exponential model is transformed into log[N(t)] = log(N 0) + [log(1 +
r)]t.
Thus the model can be expressed as
Yt = a + bt, with Yt = log[N(t)], a = log(N0), and b = log(1 + r). The transformed model, which now is linear in the parameters a and b, can be expressed in matrix form as
where 1 is a vector of ones, t is a vector of observation times, and Y consists of transformed model values. Model fitting based on mean square error is equivalent to least-squares estimation from linear regression, with
9 The greater the amount of data with which to identify the model, the easier it is to ensure that the model is structurally correct. With only a few data points, it is difficult to tell whether the model actually characterizes population dynamics. With additional data, the parameter estimates are more precise, and it becomes possible to assess the "goodness-of-fit" of the mathematical form as well as the parameterization of the model. We note that these descriptions apply to the estimates/~/0 and f as well as the estimates l/and/~. Because /~0 and f are obtained from ~ and/~ by an exponential transform that is monotonic and mathematically well behaved, the same patterns of variation hold for/~/0 and f. Indeed, these same patterns apply for a broad range of models and estimation procedures.
D
k
kt
-1
[~]-= [k-[ ~i t2]
[~,itkYiYi]
= ~,i(ti- -t)(yi- ~t) , ~,i(ti - i)2 where -t = ~ i ti/k. Back-transformation of the estimates ~ and/;, then produces the estimates/~0 = exp(~) and f = 1 - exp (b). It can be shown (Graybill, 1969) that the dispersion of the estimates ~ and b is given by ~;~(t~- i)2
9.8. IDENTIFYING SYSTEM PROPERTIES IN THE ABSENCE OF A SPECIFIED MODEL
-
with the diagonal elements representing variances of ~ and/~, respectively, the off-diagonal element representing their covariance, and oa representing the variance associated with the transformed observations Yt = log[/~(t)1. Three points are particularly germane here: 9 The variances of the parameter estimates decrease with increasing^amounts of data. This is easily seen for the estimate b, because the variance of b is inversely related to ~ i ( t i - i) 2 in the dispersion matrix. This sum of squares increases with the number of observations, so that the variance decreases. Similarly, the variance of ~ decreases with increasing amounts of data. 9 The variances of the estimates decrease with the spread of the observations over the time frame. The influence of spread again is seen in the t e r m ~ i ( t i i)2 which increases with increasing dispersion of the data over the time frame.
The preceding methods have dealt with estimation of parameters from time series data, conditional on an underlying model structure. During the past two decades, efforts have been directed at the problem of drawing inferences from time series data about system dynamics in the absence of an assumed model structure (e.g., see Schaffer, 1985; Sugihara and May, 1990; Abarbanel, 1996; Ellner and Turchin, 1996; Kantz and Schreiber, 1999). The absence of a priori knowledge about the underlying system model extends to uncertainty as to the appropriate number of state variables that are needed to describe a system of interest. Remarkably, Takens (1981) has shown that it is possible to draw certain inferences about a multidimensional system based on analysis of time series data for a single state variable of the system. Ecological examples might involve a local food web with predator and prey populations, or a system of competing species, or a system of interacting populations of the same species. In fact, ecologists often are interested in making inferences about a system of interacting species or populations, based on an analysis of a time series of abundance estimates for a single member population of the system (e.g., see Schaffer, 1985; Sugihara and May, 1990; Turchin and Taylor, 1992; Pascual and Levin, 1999). One approach to identification of system properties is based on the concept of system attractors. Strogatz (1994) defined an attractor informally as a closed set of points A in the state space of a system that possess the following properties:
9.8. Identifying System Properties in the Absence of a Specified Model 1. A is invariant, in that any trajectory beginning on A will remain on A from that time on. 2. A attracts an open set of initial conditions (termed the attracting set) such that if a trajectory begins in this open set (i.e., if the initial values of the system state variables are located within the attracting set), then the trajectory will tend toward A with time (as t ~ oo). 3. A is minimal in that there is no subset of A that satisfies properties 1 and 2. Simply put, A "attracts" a system in the sense that a trajectory starting on or near the attractor (within the attracting set) will converge to the attractor [see Milnor (1985) for more detailed definition]. The attractor A of a system may thus be thought of as a phase diagram of the asymptotic trajectory of a system, with the attracting set composed of A along with the set of system initial conditions for which system dynamics converge to A. Ecological examples of an attractor include limit cycles of the Lotka-Volterra predator-prey model and equilibria of the Lotka-Volterra competition model. It is not difficult to find simple models with similar structural features but fundamentally different attractors. Indeed, Caswell (2001) showed that a simple bivariate projection model with density-dependent elements can produce attractors with widely differing geometries, simply by changing the magnitude of one parameter in the projection matrix. Takens (1981; also see Packard et al., 1980) demonstrated that it is possible to identify the geometry of an attractor with data for a single system state variable. That is, we can use data from a time series trajectory of one state variable to produce a "reconstructed attractor" that is topologically equivalent to the true attractor. With this univariate assessment it then is possible to identify certain properties of the complete system that are useful in system analysis and prediction. The numerical methods used in attractor reconstruction from a single time series are fairly involved and will not be described here, but the interested reader is referred to Abarbanel et al. (1993), Abarbanel (1996), Ellner and Turchin (1996), Kantz and Schreiber (1999), and Nichols and Virgin (2001). Applications of these and related methods to biological problems include Schaffer (1985), Sugihara and May (1990), Turchin and Taylor (1992), Ellner et al. (1998), Pascual and Levin (1999), and Nichols and Nichols (2001). A number of measures can be obtained via attractor reconstruction that convey information about the nature of a dynamical system. One such measure is system dimension, which can be viewed in various ways but basically is a metric reflecting the geometry of the attractor. A dimensional metric computed from time
183
series data provides information about the number of state variables or system components that are active determinants of system dynamics and thus are needed to describe system dynamics adequately (also see Schaffer, 1981). If the metric for a natural system is relatively low (e.g., 2-3), then it may be possible to reconstruct attractors accurately based on only a few dimensions. On the other hand, if a system is of high dimension, then attractor reconstruction from observed time series is likely to be impossible (e.g., see Schaffer, 1985). Other system measures that are useful in analysis of system structure and dynamics are the Lyapunov exponents. A Lyapunov exponent ~kn quantifies the behavior of trajectories (e.g., stretching or contracting) with respect to the nth principal axis of the attractor as a system trajectory evolves through time. In simple terms the idea is to track a measure of the difference x 1 (t) - x 2 (t) of neighboring trajectories xl(t) and __x2 (t) as they evolve through time, with the Lyapunov exponents characterizing the rate of trajectory divergence (or convergence) in each dimension. Local Lyapunov exponents are computed using local neighborhoods of the time series data, whereas global Lyapunov exponents are computed as the average of local ~kn computed over the attractor. The signs of the global exponents (positive, negative, 0) provide information on both the shape of the attractor and the dynamics of the system (e.g., characterized as periodic or quasiperiodic, chaotic, or by the absence of posttransient dynamics). In systems subject to exogenous inputs (e.g., relevant environmental fluctuations), the distribution of local Lyapunov exponents characterizes short-term transient dynamics following exogenous perturbations (Ellner and Turchin, 1996; Ellner et al. 1998). When based on an appropriate choice of the dimension and delay parameters, a reconstructed attractor can be useful for prediction. That is, prediction algorithms utilizing reconstructed attractors can be used to project system changes into the future. These predictions can serve as forecasts of system behavior (e.g., Sugihara and May, 1990) and also can be used for other purposes such as identification of the appropriate spatial scale for the aggregation and study of ecological systems (Rand and Wilson, 1995; Keeling et al., 1997; Pascual and Levin, 1999). Methods for attractor reconstruction appear to work well in practice with physical and mechanical systems, for which the time series data are characterized by large numbers (e.g., tens to hundreds of thousands of points) of very precise measurements with little noise. But ecological time series typically include many fewer data and much more noise from sampling variation as well as environmental and other influences. Thus, the
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ultimate utility of attractor reconstruction for the investigation of ecological problems is not known at this time. We believe that the investigation of system attractors will be an active area of research over the next decade.
9.9. D I S C U S S I O N There is a strong association between the statistical estimation procedures described in Chapter 4 and model identification as described in this chapter. For example, maximum likelihood estimation relies on maximization of a "likelihood function" as the basis for parameter estimation. If the underlying distribution is normal, this reduces to the minimization of a quadratic form in the distribution parameters, in analogy to the process of model fitting via minimization of a mean squared error function. In fact, the problem of statistical estimation can be seen as an application of (statistical) model fitting. Indeed, the objective is to choose estimates of distribution parameters that best "fit" an assumed statistical distribution, and in the case of maximum likelihood estimation, this means choosing parameter values for which the likelihood function is maximized. The selection of a statistical model from, say, two alternatives is facilitated by choosing the "best" parameter estimates from the two corresponding distributions, determining the "goodness" of fit of the models based on these estimates, and selecting the model with the better fit (see Chapter 4, especially Section 4.2 on parameter estimation and Section 4.4 on model selection). Indeed, the preceding discussion on the effects of observation data on estimator stability is indicative of the strong association between statistical estimation and model identification. Their similarities notwithstanding, we note that in general, dynamic model identification and statistical estimation are not identical. Recall that the process of identification was developed in terms of an error function and a weighting scheme for its components. These attributes are in some sense arbitrary, in that the analyst has very wide flexibility in the choice of both. This flexibility distinguishes model identification from statistical estimation, which is tied to the form of an assumed underlying statistical distribution of the data. This distribution influences the choice of both the metric by which goodness of fit is measured and the weighting scheme of the metric. Recall that the objective of model identification is to represent (time series) data as well as possible with a dynamic model, by appropriate choice of parameter values. The notion of stochastic variation, and the need to account formally
for random variation, is not necessarily a part of the process. On the other hand, the stochastic nature of statistical data, and the need to account for, measure, and model stochastic effects, are at the heart of statistical modeling, i.e., the modeling of components of random variation in a system. Model identification also shares many attributes with dynamic optimization, as described in Chapter 21. Both involve the optimization of an objective function over a range of values for some decision variable. Both incorporate the transition equations of a dynamic system as constraints on the optimization. Both involve (or can involve) initial conditions and possible boundary conditions on the optimal solution. However, there are substantive differences between dynamic optimization and model identification, involving the nature of the objective function, the character of the decision variables, and differences in the models that are used. Whereas model identification seeks with temporally referenced data to identify parameter values in a dynamic model, dynamic optimization seeks to identify a trajectory of controls to optimize an objective function in the control and system state variables (see Chapter 21). The identification process involves an iterative refinement and revision of structural and parametric model features, whereas dynamic optimization typically involves the use of a developed model (or set of models) to guide management a n d / or research. Indeed, one result of model identification is to produce models that can be used for dynamic optimization. It should be noted that it is not uncommon for the effort to identify a model to fail, i.e., for one to fail to construct a model that is adequate for its intended purposes. Several potential reasons for this failure can be recognized (Ljung, 1999), which tie directly to the key elements of identification that were articulated above. For example, the suite of models under consideration may focus inadequately on system features of interest to the investigator or may fail to incorporate structural features (e.g., age or stage structure in a population) that are needed describe system behaviors of particular interest. Another common failure in model identification occurs when the information set is inadequate for identification. A case in point is a mismatch between the extent of the data and the range of biological conditions intended for the model. In this situation one might identify a model that fits the data but nevertheless fails to perform adequately over the biological range of interest. Yet another source of potential failure is a poor choice of the selection criterion by which to compare, contrast, and select the most appropriate model. As mentioned above, the choice of a model fitting criterion influences the weights given
9.9. Discussion to data points entering into the identification process and thereby influences the fitting of models to the data. For example, the least-squares criterion of Section 9.1 allows data at the extremes of the data set to influence heavily the fitting of a model, whereas an absolute difference criterion (Section 9.2) weights the data equally across the data range. Depending on the intended use of the model, the choice of a fitting criterion can potentially result in a model of marginal value. Finally, model identification can fail simply because the numerical procedure used to recognize optimal values of model parameters fails. Finding optima can be quite difficult for complicated models with nonlinear features, discontinuities, complicated constraints, and other features. For such models a search procedure may "home in" on a suboptimal parameterization for
185
the model or may simply fail to recognize any optimum whatsoever (see Appendix H for further discussion). We note in closing that one usually is less than certain about the mathematical structures describing biological process, yet it nonetheless is necessary to make decisions in the face of this uncertainty. One approach is to seek optimal decisions that recognize management objectives, while also accounting explicitly for structural uncertainty in the decision-making process. Such an approach essentially integrates system identification and system control into a single optimization problem, with decision-making pursuant to the dual goals of management and improved system understanding. In Chapter 24 we describe the combination of system identification and optimization under the rubric of adaptive resource management (Waiters, 1986).
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C H A P T E R
10 Stochastic Processes in Population Models
10.1. BERNOULLI COUNTING PROCESSES 10.1.1. Number of Bernoulli Successes 10.1.2. Bernoulli Waiting Times 10.2. POISSON COUNTING PROCESSES 10.2.1. Extensions of the Poisson Process 10.2.2. Poisson Interarrival Times 10.3. DISCRETE MARKOV PROCESSES 10.3.1. Markov Chains 10.3.2. Classification of States in a Markov Chain 10.3.3. Stationary Distributions in Markov Chains 10.4. CONTINUOUS MARKOV PROCESSES 10.4.1. Birth and Death Processes 10.4.2. The Kolmogorov Differential Equations 10.5. SEMI-MARKOV PROCESSES 10.5.1. Stationary Limiting Distributions 10.6. MARKOV DECISION PROCESSES 10.6.1. Discrete-Time Markov Decision Processes 10.6.2. Objective Functionals 10.6.3. Stationary Policies 10.6.4. Semi-Markov Decision Processes 10.7. BROWNIAN MOTION 10.7.1. Extensions of Brownian Motion 10.8. OTHER STOCHASTIC PROCESSES 10.8.1. Branching Processes 10.8.2. Renewal Processes 10.8.3. Martingales 10.8.4. Stationary Time Series 10.9. DISCUSSION
tion according to a sampling or experimental design, could be used to estimate population parameters based on the rules of statistical inference as described in Chapter 4. Stochastic factors arising in the investigation as a result of r a n d o m sampling were included in the corresponding statistical models and accounted for via statistical treatments outlined in these chapters. With some exceptions (e.g., see Sections 6.5-6.6), the patterns of r a n d o m n e s s were a s s u m e d to be absent any covariation across time. An extension to this f r a m e w o r k that is particularly useful in population modeling includes sequences of r a n d o m variables that are temporally indexed. Probability structures for sequences of r a n d o m variables that are temporally indexed constitute the subject matter of stochastic processes. In simplest terms the joint distribution of a set {X(t): t ~ T} of r a n d o m variables over a time frame T describes a stochastic process over T. One elementary example of a stochastic process is the sequence of statistically independent r a n d o m variables produced by r a n d o m sampling of a population over time. The probability structure of a stochastic process typically is defined in terms of the distribution of X(t) at each point in time, as well as the statistical associations of these r a n d o m variables across time. If the potential values for X(t) are countable [e.g., if X(t) takes only integer values] then the process is said to be a discretestate process; otherwise, it is a continuous-state process. Stochastic processes also can be characterized as discrete time or continuous time, d e p e n d i n g on the discrete or continuous nature of the time frame. A particular sequence of observed values of the r a n d o m variables of a stochastic process constitutes a realiza-
In our development of statistical procedures in Chapters 4-6, we focused primarily on the treatment of r a n d o m variables that lack an identifiable reference to time. The idea there was that r a n d o m samples of observations, collected over the course of an investiga-
187
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Chapter 10 Stochastic Processes
tion of the process. A realization is essentially a time trace of the process, a particular manifestation from the collection of all possible time traces defined by it. Example
Consider a sequence of counts {N(t): t e T}, for which N ( t ) is a random variable of nonnegative integers at each time t in a time frame T. If T consists of discrete points in time, the process is a discretestate and discrete-time process; otherwise it is a discrete-state and continuous-time process. Because populations frequently are characterized by counts at discrete points in time, such counting processes often are used to model populations with stochastic components. Example
Consider a continuous-state, continuous-time process with bell-shaped distribution at each point in time, the variance of which increases proportionately with time. Under certain conditions involving temporal correlation in the process, this describes the well-known Brownian motion process (see Section 10.7). Because Brownian motion characterizes continuous change in systems, it sometimes is used in continuous-time population models. Example
Figure 10.1 displays realizations for a discrete-time process for which X(t) --- N ( ~ t , or) and corr[(X(tl), X(t2)] = 0. A process with this statistical structure is known
FIGURE 10.1 Discrete-timewhite noise process with X(t) --- N ( ~ t 1).
,
as Gaussian white noise. For white noise processes the transitions between times are independent of process history; i.e., the state of the system at time t is independent of all previous states. In essence, the future state of the process is not influenced by past or present states, and except for temporal variation in its means, process behaviors in the future look statistically like process behaviors at each time in the past. Because all random variables in a white noise process are statistically independent, this class of processes possesses the simplest possible stochastic structure. Example
Except for white noise processes, the simplest probability structure for a stochastic process is one in which the process state at time t is influenced only by its state immediately prior to t. Figure 10.2 exhibits realizations for a discrete-time process such that the process state at each time is the previous state plus a random component: X ( t + 1) = X(t) + Y(t), where Y(t) --- N(0,1). Processes with the property that future process behavior is influenced only by the present state of the system are known as M a r k o v processes. In the following discussions we describe these and other stochastic processes that arise in the modeling of biological populations. Because they are especially prevalent in the modeling of populations, we focus on Bernoulli and Poisson counting processes, along with some probability distributions that are derived from them. We then describe Markov processes and Markov decision processes, a large class of stochastic processes that play an important role in later chapters on decision-making. Then we deal with Brownian motion for
FIGURE 10.2 Realizationof a discrete-time Markov process, with transfer equation X(t + 1) = X(t) + Y(t) and Y(t) a white noise process with Y(t) ~-- N(0, 1).
10.1. Bernoulli Counting Processes continuous time and finish with brief descriptions of a few other processes that can arise in population biology. In what follows, we use the index t to designate the time at which an event occurs and k to designate the temporal order of events. For discrete-time processes like the Bernoulli, the sequential order of an event and the timing of its occurrence can coincide, d e p e n d i n g on the nature of the event. For most stochastic processes there is no such coincidence, and it therefore is convenient to include indices for both temporal order and time.
10.1. B E R N O U L L I COUNTING PROCESSES We focus here on the class of white noise stochastic processes k n o w n as Bernoulli processes, which are defined by i n d e p e n d e n t binary r a n d o m variables at each point in a discrete time frame. As described in Chapter 4, a binary r a n d o m variable X(t) can be assigned a value of X(t) = 1 if the outcome of the process at time t is a "success" (however defined), and X(t) = 0 if the outcome is a "failure." Success or failure occurs at each point in time with probabilities p and 1 - p, respectively. A formal definition for Bernoulli processes is as follows: The stochastic process {X(t): t = 1, 2, ...} is Bernoulli with probability p if 1. The r a n d o m variables X(1), X(2), ... are independent.
2. P[(X(t) = 1] = p and P[(X(t) = 0] = 1 - p for all t. Realizations of a Bernoulli process consist of sequences of unit-height rectangles, corresponding to the sequence of failures and successes. Figure 10.3 displays a realization with p = 0.5, consisting of a series of
FIGURE
10.3
R e a l i z a t i o n of a Bernoulli process w i t h p = 0.5.
189
unit increases and decreases d e p e n d i n g on outcomes at each point in time.
Example Consider a hunter check station at which the success of waterfowl hunters is determined during the waterfowl hunting season. As hunters come through the station, each harvested bird is checked for species, sex, and age. Let k designate the temporal order of birds that are checked; thus k = 1 corresponds to the first bird checked, k = 2 corresponds to the second bird checked, and so on. Let X(k) = 1 if the kth bird is a mallard and X(k) -- 0 if it is not a mallard. Provided the probabilities for harvesting and reporting mallards and nonmallards are invariant over the time frame (i.e., the probability is p that the kth bird is a mallard for all k), this situation defines a Bernoulli process.
Example Assume that a cartographically correct m a p of the state of Vermont is overlaid with a fine-grained grid system. Assume that grid plots are chosen sequentially by some r a n d o m process and each plot is field checked for forest vegetation. Let k represent the kth plot that is chosen in sequence and assign X(k) = 1 if the kth plot consists of greater than 50% forest cover. Provided the grid sampling is r a n d o m with replacement, this defines a Bernoulli process, for which the Bernoulli parameter p represents the proportion of Vermont that is forested.
10.1.1. N u m b e r of Bernoulli Successes Other processes can be derived from the Bernoulli-for example, the n u m b e r of successes N(t) = X(1) + .-. + X(t) by time t. This is again a discrete-state, discretetime process, but now the state space varies with t. Thus two values are possible at t = 1 [N(1) = 0 or 1], three values are possible at t = 2 IN(2) = 0, 1, or 2], and so on. Realizations of the process consist of unitlength step increases at those times for which X(t) = 1 (Fig. 10.4). It can be s h o w n that N(t) is binomially distributed with parameters p and t:
Because t is a parameter in this density function, the distribution of N(t) varies over time. The r a n d o m variables {N(t)lt = 1, ...} also are correlated over time: because N(t 1) and N(t 2) share r a n d o m elements in their sums, corr[N(tl), N(t2)] ~ 0.
190
Chapter 10 Stochastic Processes hunter successes are independent, then nonoverlapping periods of the record represent sequences of indep e n d e n t Bernoulli r a n d o m variables, and accumulated successes for these periods represent i n d e p e n d e n t binomial r a n d o m variables parameterized by the record lengths and the success rate. This allows one to test via m a x i m u m likelihood procedures the assumption of stationary hunter success over time.
10.1.2. Bernoulli Waiting Times
FIGURE 10.4 Realizationof a process consisting of the number of Bernoulli successes over time.
Example Consider a hunter check station at which the success of deer hunters is determined as they exit a hunting area. A s s u m i n g that hunters are independent and identical in their probability p of hunting success, a sequence of k hunters coming through the check station corresponds to k Bernoulli r a n d o m variables, the sum of which is binomial with parameters k and p. Sequential sampling of hunters provides an opportunity to test the assumption that hunters are identical in their success rates. For instance, the hypothesis that success rate depends on the age of the hunter is easily tested via m a x i m u m likelihood theory (see Chapter 4), based on the binomial distributions for samples of y o u n g and mature hunters coming through the station. It can be s h o w n that interval counts for Bernoulli processes are binomially distributed; i.e., the n u m b e r N(t 2) - N(t 1) of successes in the interval t 2 - t I is binomially distributed with parameters p and t 2 - t 1. Because N(t 2) - N(t 1) includes only the r a n d o m variables X(t) for times between t I and t 2, this count is i n d e p e n d e n t of interval counts for periods prior to t I and after t 2. This property, k n o w n as independent increments, holds for any process for which the elements X(t) are independent. Interval counts also are stationary, in that the distribution of N(t 2) - N(t 1) is independent of the starting time t I of the interval. Again, this property holds for any process with independent, identically distributed elements X(t).
Example Again using the check station, assume that records are kept of hunter success over a period of several weeks during the hunting season. On assumption that
We have a s s u m e d thus far that the time t is fixed and the n u m b e r of successes is random. It is useful to consider a role reversal for these indices, whereby the n u m b e r of successes is assumed given and the time required to achieve that n u m b e r is random. It seems intuitive that if the n u m b e r of successes over a given a m o u n t of time is random, then the time required for a given count also is random. For example, consider the time Z(1) required to record the first success in a Bernoulli process. For the first success to occur at time t, no successes can have occurred by time t - 1, and a success must occur at time t; i.e., the event {Z(1) = t} is equivalent to the joint event N(t 1) = 0 and X(t) = 1, with probability of occurrence -
P{Z(1) = t} = P{EN(t - 1) = O] n [X(t) = 1]}. Because N ( t - 1) = X(1) + ... + X ( t - 1 ) a n d X(t)are independent, their joint probability is given by P{Z(1) = t} = (1 - p)t-lp, which describes a geometric distribution (see Appendix E). Thus, the waiting time for a single occurrence of a Bernoulli process is a geometrically distributed r a n d o m variable, the value of which can be any positive integer. The distribution is parameterized by the probability p of success in any trial, with the average waiting time given by 1/p. This supports one's intuition that the time required for a success ought to increase as the probability of success declines (and vice versa).
Example A team of biologists is investigating the feeding behaviors of an endangered species. Observation stations have been set up in several k n o w n feeding areas; however, individual animals are only rarely observed there and only during the hours just before and just after dawn. In planning the team's field season, it is important to have some idea of the a m o u n t of time required at each observation station in order to observe feeding behaviors there. From previous studies the probability of sighting individuals on a given day is
10.1. Bernoulli Counting Processes about p = 0.1. Based on the geometric distribution for waiting times, the expected time for an observation at any particular station is therefore 1 / p = 10 days. A study design that requires observations at each station thus should anticipate at least 10 mornings of observations per station. By extension, n o w consider a r a n d o m variable Z ( k ) characterizing the time of the kth success in a Bernoulli process. For the kth success to occur at time t, k - 1 successes m u s t have occurred by time t - 1, and a success must occur at time t. Thus, the event {Z(k) = t} is equivalent to the joint event { N ( t - 1) = k - 1} and {X(t) = 1}, with probability of occurrence P { Z ( k ) = t} = P { [ N ( t -
1) = k - 1] A IX(t) = 1]}.
the a m o u n t of time beyond k that is required for k successes. In that case Eq. (10.1) can be written as P [ Z ( k ) = k + x] = ( k + -x - 1
-
1) = k -
= [(;-1) pk-l(ll
1 ] P [ X ( t ) = 1] _p)t-k]p
(10.1)
(;1) -
1 pk(1 - p ) t - k
which describes the negative binomial distribution (see A p p e n d i x E). Note that k in the derived process {Z(k): k = 1, ...} is the n u m b e r of s u c c e s s e s , rather than an index of time. Indeed, k is n o w a distribution parameter, and the time index t actually has become a value that the r a n d o m variable Z(k) can take. The average waiting time for k successes is given by k / p , which indicates that the waiting time increases with the number of required successes and declines with greater success rate. Example
A study of small m a m m a l s involves the capture of individuals with drop traps. Traps are visited twice a day, and individuals found in the traps are tagged and released. From a pilot study the probability of a trap being occupied on a given visit is p = 0.25. A s s u m i n g i n d e p e n d e n t trapping events, the n u m b e r of visits needed to record a specific n u m b e r of captures is given by the negative binomial distribution. A design that targets four captures per trap therefore should anticipate k/p = 16 visits to each trap and so should call for a study period of at least 8 days. Note that at least k units of time are necessary to achieve k successes; thus, the negative binomial distribution is defined for values of t such that t >- k. Sometimes the index t is written as t = k + x to emphasize
pk(1 _ p ) k + x - k
( ,xl t x
pk(1
p)X
a standard form of the negative binomial distribution (see A p p e n d i x E). It can be shown that the process describing time to success is Markovian, in that the distribution of Z(k) depends only on Z(k - 1) but not on the r a n d o m variables prior to k - 1" P i g ( k ) = tlZ(1) = t I . . . . , Z ( k -
By virtue of the independence of N ( t - 1) = X(1) + 9.. + X ( t - 1) and X(t), their joint probability is given by P [ Z ( k ) = tl = P [ N ( t -
191
= P[Z(k)=
tIZ(k-
1) = tk_l]
1 ) = tk_l].
Thus, in order to predict the time of the kth occurrence in the process, one need not keep track of the history of previous occurrences. Once one accounts for the most recent occurrence, all previous occurrences are of no value in predicting the timing of the next occurrence. This property simplifies enormously the task of modeling the time to success. With the Markovian property, one can show that the waiting time Z ( k ) - Z ( k - 1) between successive occurrences is i n d e p e n d e n t of previous waiting times and is geometrically distributed: P[Z(k)-
1) = tIZ(1) = tl, ..., Z ( k -
Z(k-
= P[Z(k)-
= p(1
-
Z(k-
1) = tk_l]
1 ) = t]
p)t-1.
It follows that the waiting times Z(1), Z(2) - Z(1), ..., Z ( k - 1) are all i n d e p e n d e n t and identically distributed r a n d o m variables. Of course, their sum
Z(k) -
k-1
Z(k) = Z ( 1 ) + ~ [ Z ( j + j=l
1)-Z(j)]
is simply the time required for k successes, which from Eq. (10.1) is distributed as a negative binomial: k-1
P{Z(1) + ~ [ Z ( j + j=l
=
(;1 t
1)-Z(j)]
1 pk(1 -- p) t
=t}
,
The independence of the increments Z(k) - Z(k - 1) also guarantees that the times m
Z(k + m) - Z ( k ) = ~ [ Z ( k + j) - Z ( k + j - l ) ] j=l
192
Chapter 10 Stochastic Processes
between multiple successes represent independent increments, and these increments also are distributed as negative binomial, with parameters p and m.
process {N(t): t -> 0} is defined under the following four conditions:
Example
2. The process has stationary and independent increments, i.e., P[N(t + At) - N(t)] is the same for all t and N(t 1 + At 1) - N ( t 1) and N(t 2 + At 2) - N ( t 2) are statistically independent for t2 ~ t I + At 1. 3. For an arbitrary time t, the probability of exactly one occurrence in a "small" interval [t, t + &t] is approximately )~At:
As part of an investigation of waterfowl movements during migration, bait traps are located in a wetland complex and checked daily for waterfowl. Trapped birds are weighed, banded, and released each morning during the course of the study. Trapping is conducted over a limited period during the peak of migration, so that the number of birds exposed to traps is not expected to vary systematically during the study. Let X(t) be a random variable representing daily trapping success: X(t) = 1 if the trap is occupied on day t, and X(t) = 0 if the trap remains unoccupied. On assumption that trapping effort remains constant over the course of the study and trapping success is not influenced by previous trapping success (i.e., there is no behavioral response to being trapped; see Chapter 14), the sequence {X(t): t = 1, 2, ...} constitutes a Bernoulli process, with parameter p representing the probability that a trap is occupied on any given day of the study. The number N(t 2) - N(t 1) of occupied traps during an interval [tl, t2] has the binomial distribution B(t 2 - tl, p), with expected value (t 2 - tl) p. Thus, the average number of captured birds can be increased either by increasing the duration of the study or by increasing the probability of capture (or by increasing both factors). Furthermore, changes in the average number of captures are directly proportional to changes in either the duration of the study or the capture probability. The waiting time between successive captures in the study has a geometric distribution, with expected value 1/p. The waiting time for, say, five captures is distributed as a negative binomial, with expected value 5/p. Thus, the average length of time required for a given number of traps to be occupied can be decreased by increasing the probability of capture, with the required time decreasing from oo to 5 as the probability of capture increases from 0 to 1.
1. N ( 0 ) = 0.
P{~N(t + a t ) - N(t)] = 1} = )~At + o(at),
where o(&t) is some value with a limiting magnitude that is of degree less than &t: lim at--*0
o(At) - 0. At
4. The probability of more than one occurrence in It, t + &t] is negligible when compared to the probability of a single event: P{[N(t + a t ) -
N(t)] > 1} = o(At).
If these four conditions are satisfied, then {N(t): t ~ 0) is a Poisson stochastic process. As illustrated in Fig. 10.5, realizations of a Poisson process exhibit unit increases at random points in time. The rate at which these increases occur is influenced by the parameter )~, as discussed below. Example
An experiment involves the maintenance of minnows in individual aquaria over an extended period of time. When a minnow dies, its aquarium is replaced
10.2. P O I S S O N COUNTING PROCESSES Poisson processes are discrete-state, continuous-time processes that often are applicable to counts over continuous time frames. The idea behind a Poisson process is that events occur at random times over a continuous time frame, subject to certain stationarity and independence conditions. The process records the total number of occurrences throughout the time frame, which typically is assumed to start at t = 0. Formally, a Poisson
FIGURE 10.5 Realizationof a Poisson process with ~ = 0.25.
10.2. Poisson Counting Processes with another containing a live m i n n o w of the same species, age, and genetic stock. Let N ( t ) represent the n u m b e r of m i n n o w s that have died by time t. If the flow-through water system is maintained properly, the death rate of m i n n o w s should remain constant through time and the sequence {N(t): t -> 0} can be modeled by a Poisson process. Clearly, the total number N ( t ) of m i n n o w deaths at any point in time will be greater or less d e p e n d i n g on the experiment-wide rate of mortality. Example
Reproduction for a panmictic endangered species occurs as a result of the r a n d o m encounter of males and females, which is indexed by the parameter ~. The likelihood of one such encounter in a unit of time is directly proportional to K, and the likelihood of k encounters declines as a p o w e r function of K. Because the probability of more than one reproduction event in a unit of time is negligible w h e n compared to that for a single event, one can model total reproduction over time as a Poisson process. Again, reproduction will be greater or smaller d e p e n d i n g on the parameter K. A probability structure for Poisson processes can be obtained through a decomposition of the event { N ( t + &t) = n} into {N(t + At) = n} = {IN(t) = n]
U{[N(t) = n - 1] and
and
[N(t + At) - N(t) = 0]}
[N(t + at) - N ( t ) = 1]}.
(10.2)
Equation (10.2) asserts that there are essentially two ways in which N ( t + At) can assume a value of n: no events are a d d e d to a count of n over At [the lead term of the union in Eq. (10.2)]; or one event is a d d e d to a count of n - 1 over At [the second term of the union in Eq. (10.2)]. On assumption that increments are stationary and independent, the probabilities for these events can be added, to produce the probability
193
where Pn(t) is the derivative of Pn(t) with respect to t. The solution of this differential equation can be s h o w n to be P I N ( t ) = n] = e - ~ t ( K t ) n / n !
(see Appendix C). Thus, the probability structure for the Poisson process {N(t): t ~ 0} is given by a Poisson distribution for N(t), with Poisson parameter Kt. Because this distribution is parameterized by t, it obviously varies as t takes different values. Note that the Poisson parameter is directly proportional to t, so that both the mean and variance of N ( t ) are proportional to the time since initiation of the process (see Appendix E). This is illustrated in Fig. 10.6. Example
In the a q u a r i u m study described above, the total n u m b e r N ( t ) of m i n n o w deaths by time t has a Poisson distribution with Poisson parameter Kt. Because both the mean and variance of a Poisson distribution are given by the Poisson parameter, the average n u m b e r of deaths and the spread in these n u m b e r s increase linearly as the study progresses. For example, if K = 0.1, the average n u m b e r of deaths after the first d a y is 0.1; after the second day it is 0.2; and so on. If an average of Kk mortalities is recorded in the study over k days, we can expect twice that n u m b e r to be recorded in a replicated study lasting twice as long. It should be noted that the Poisson distribution provides an alternative but equivalent definition for Poisson processes. Thus, a counting process {N(t): t -> 0} is Poisson with rate parameter ~ if the following conditions are met:
P[N(t + at) = n] = P[N(t + at) - N(t) = 1]P[(N(t) = n - 1] + P[N(t + at) - N(t) = O]P[N(t) = n].
Using assumptions (3) and (4) listed above for the Poisson process, this equation can be expressed as Pn(t + At) = Pn_l(t)[KAt + o(At)] + Pn(t)[1 - KAt - o(At)]
with Pn(t) = P [ N ( t ) = n], or Pn(t + a t ) - Pn(t) = [Pn_l(t) -- Pn(t)][KAt + o(&t)].
Dividing the equation by At and allowing &t --~ 0, we get the differential equation Pn(t) = [ - P n ( t ) + Pn_l(t)]K,
FIGURE 10.6 Probabilitydistribution for a Poisson process at a particular time t. The rate parameter Kt is a linear function of the process time t, and therefore the distribution evolves over the time frame.
194
Chapter 10 Stochastic Processes
1. N(O) = O. 2. The process has independent increments. 3. The number of events occurring in any interval of length s is Poisson distributed with parameter Ks: P[N(t + s) - N(t) = n] = e
-as ( h s ) n
n!
,
n = 1,....
It is instructive to compare these two definitions for the Poisson process. Both assume a starting value of zero for the process, and both assume independent process increments. One definition leads to the Poisson distribution by considering incremental (single-step) changes in process magnitudes, whereas the other starts with an assumed Poisson distribution for interval counts. However, both lead to the same stochastic framework, with Poisson distributed counts over specified intervals of time. On assumption that {N(t): t -> 0} is a Poisson process, the probability that no event occurs over [0, t] is given by P{N(t) = 0} = e -at. It follows that the probability of at least one occurrence over the interval is P{N(t) > 0} = 1 - e -at. The assumption of stationary increments ensures that this probability holds for any interval of length s in the time frame, irrespective of its starting points: P{N(t + s ) - N ( t )
>0}=
1 -e-aS.
(10.3)
10.2.1. E x t e n s i o n s of the P o i s s o n P r o c e s s
A number of stochastic processes can be derived from the Poisson process. Here we emphasize some of the more common processes that incorporate additional structural features into the Poisson stochastic framework.
10.2.1.1. Poisson Superposition It is possible to combine two Poisson processes into a single process with an identifiable probability structure. Suppose that P1 = {N(t): t -> 0} and P2 = {M(t): t >- 0} are two independent Poisson processes, with rates h I and h 2, respectively. It can be shown that the process P1 + P2 defined by P1 4- P2 = {N(t) + M(t):t >-0} is a Poisson process with parameter ~'1 4- )k2" The combined process P1 4- P2 is called the superposition of P1 and P2. Example
Consider a wildlife population that is subjected to hunting mortality over an extended period of time. Assume that the process P1 records the number of male
deaths that occur through time as a result of hunting, and P2 records the number of female deaths from hunting. If the number of deaths )k 1 and h 2 per unit time for these processes are stationary, then the total mortality for both sexes is tracked by the superposition P1 4P2 with parameter )k1 4- ~k2.
10.2.1.2. Compound Poisson Processes In addition to recording the time at which a Poisson event occurs, it often is useful to record some value associated with the event. For example, both the time of death and the weight at death might be recorded for each mortality event in the aquarium study described above. Such data form the basis of a c o m p o u n d Poisson process. More formally, let {N(t): t -> 0} be a Poisson process with parameter h and {Y(i): i = 1, ...} be a sequence of independent and identically distributed random variables. Assume that {N(t): t -> 0} and {Y(i): i = 1, ...} are statistically independent. Then the process {X(t): t -> 0} with N(t)
X(t) = ~, Y(i) i=1
is a compound Poisson process. In essence, X(t) accumulates values of Y(i) as the process progresses through time. Thus, X(t) assumes a value of 0 until the first Poisson event, at which time X(t) is updated by the value for Y(1). X(t) retains this value until the second Poisson event, at which time the value for Y(2) is added to that for Y(1). X(t) retains the value of this sum until the third Poisson event, at which time the value for Y(3) is added to the sum, and so on as time advances. Basically, the compound Poisson process {X(t): t -> 0} evolves like the Poisson process {N(t): t -> 0}, except that the unit steps of {N(t): t -> 0} are replaced by steps of size Y(t) (Fig. 10.7). It can be shown that the expected value of a compound Poisson process is the product E[X(t)] = (ht)ix
of means for the random variables N(t) and Y(t), and the variance is the product var[X(t)] = (ht)0-2 of their variances, where tx and 0 -2 a r e the expected value and variance of the random variables in {Y(i)" i = 1, ...}. In addition, the compound Poisson process inherits the property of independent increments from the underlying process {N(t)" t >- 0} and the independence of the random variables in {Y(i)" i = 1, ...}.
10.2. Poisson Counting Processes
F I G U R E 10.7 Realizations of a compound Poisson process. The unit step of the Poisson process is replaced by a step of size Y ( t ) when a process event occurs, with Y ( t ) --- N(0, 1) for this particular example.
195
occurrences at time t is distributed as a Poisson random variable with parameter Kt. Assume also that at the time of the nth occurrence there is a Bernoulli trial, with outcomes that are independent of the Poisson process. The stochastic structure thus includes both Bernoulli and Poisson processes, albeit with different temporal indices: the index for the Poisson process represents time, whereas the index for the Bernoulli process characterizes the temporal sequence of Bernoulli trials. N o w let X 1 be a new process that records the accumulated number of Bernoulli successes at each point in time and X 2 be a process that records the number of failures. Then X 1 and X 2 are compound Poisson processes with N(t)
Xl(t) = ~ Y(i) i=1
and N(t)
Example
X2(t) = ~ [1 - Y(t)],
Biologists retain records of the timing and weights of catch for each species of fish in a fishery. For planning purposes it is thought important to predict the size of the bi-catch of an infrequently caught species that exhibits considerable variation in individual sizes. Assuming that catch events are independent, the accumulated catch over time can be modeled by a Poisson process, with Poisson parameter )~t expressing the expected catch in an interval of length Kt. Based on historical records, one can estimate both the rate parameter and the mean ~ and variance 0 -2 o f the size of individual fish. If N ( t ) is the total catch over an interval of length t and Y(i) represents the weight of the ith fish at the time of its capture, then the accumulated weight N(t)
X(t) = ~, Y(i) i=1
of the catch over the interval is distributed as a compound Poisson distribution. Thus, the expected weight of the bi-catch is simply the average number )~t of individuals caught in the interval, times the average weight ~ of an individual fish. Likewise, the variance associated with the bi-catch weight is the variance for the number individuals caught, which for the Poisson distribution is also )~t, times the variance of the weights of individual fish. This information can be used by biologists to adjust the fishing season length appropriately to target an amount of bi-catch to maintain stocks while allowing for fishing opportunities. A useful example of a compound Poisson process results from the combination of Poisson and Bernoulli processes. Assume that events occur according to a Poisson process {N(t): t >- 0}, so that the number of
i=1
respectively. Both can be shown to be Poisson processes, with parameters Kp and M1 - p), respectively. Furthermore, they can be shown to be independent of each other. This particular example of a compound Poisson process is sometimes referred to as Poisson decomposition (Cinlar, 1975).
10.2.1.3. Nonstationary Poisson Processes A key assumption thus far is that the Poisson parameter )~ is constant for all t -> 0. Seasonal, diurnal, and other variations in many biological processes can combine to make this an unrealistic assumption. A generalized Poisson process allows the Poisson parameter to vary with time: ~ = )t(t). The counting process {N(t): t -> 0} is said to be nonstationary (or nonhomogeneous) if all the Poisson assumptions except stationarity remain valid when )~ is replaced with Mt): 1. N ( t ) = O.
2. {N(t): t >- 0} has independent increments. 3. P [ N ( t + & t ) - N ( t ) = 1] = K(t)At + o(&t). 4. P [ N ( t + At) - N ( t ) >- 2] = o(&t). Under these conditions it can be shown that the increments {N(t + s) - N(s)} are Poisson distributed, based on the parameter t
m(t) = f
Ms) ds. o
That is, P{N(t + s) - N(s) = n} = e-lmr162
f o r n ->0.
+ s) - m(s)]n/n!
196
Chapter 10 Stochastic Processes
A useful application of nonstationary Poisson processes involves a stationary process for which the recording of an event is less than certain. Assume that an event at time t is detected with probability )~(t)/)~. Then P[one event is counted in (t, t + At)] = P[one event occurs in (t, t + At)] P[event is detected[event occurs] =
[(),at)
X(t)
o ( a t ) ] ~x
+
o'(at),
= Mt)At +
which demonstrates that a stationary Poisson process, with follow-up sampling to confirm event occurrences, can be described as a nonstationary Poisson process. Example
10.2.2. Poisson Interarrival Times Just as the times between successes can be investigated for Bernoulli processes, so can the times between occurrences for Poisson processes. As before, let the r a n d o m variable Z(k) represent the time of occurrence of the kth event in a Poisson process, with Z(0) = 0. The derived process {Z(k): k = 0, 1, ...} records the waiting times for occurrences of events in a Poisson process. Let Y(k) = Z(k) - Z(k - 1) in turn represent the a m o u n t of time between the k - 1st and kth occurrences. The derived process {Y(k): k = 1, ...} records the interarrival times between occurrences of the Poisson process. Note that the index k is an ordering index for arrival times and not an index of time. A probability structure can be ascribed to realizations of interarrival times by noting that {Y(1) -< t}, {Z(1) -< t}, and {N(t) > 0} all describe the same event, so that P[Z(1) -< t] = P [ N ( t ) > 0]
A s s u m e that in the absence of r a n d o m influences, a continuously growing population can be described by the exponential model d N / d t = r N ( t ) , with solution N ( t ) = No eFt. However, the effect of r a n d o m influences alters this pattern in such a w a y that population dynamics are described by a nonstationary Poisson process, for which Mt) Noert/r. Increments for this process are distributed as Poisson r a n d o m variables, with Poisson parameter =
t )~(s) ds
m ( t + s) - re(s) = f s
= No[er(t+s)-
e rs].
=l-e
-at
from Eq. (10.3). Thus, the waiting time for the first event of a Poisson process has an exponential distribution, with exponential parameter )~ inherited from the Poisson parameter )~t. Because Z(0) = 0 by assumption, this means that the first interarrival time Y(1) = Z(1) - Z(0) is exponentially distributed. By extension, the probability for the second interarrival time can be obtained by conditioning on the first, recognizing that the events {Y(2)- 0} are equivalent: P[Y(2) -< t Z(1) = s] = P[N(t + s) - N(s) > 0IN(s) = 1]
Because = P[N(t + s) - N(s) > 0] t
N ( t ) = ~] [N('r + 1) - N('r)],
= P[Y(2) -< t],
it follows that the average population size at time t is
which holds by virtue of the independence of the increments N ( t + s) - N ( s ) and N ( s ) - N(O). It follows that
"r----1
t
E [ N ( t ) ] = ~,~ E[N('r + 1 ) -
N('r)]
P[Y(2) -< t] = P [ N ( t + s) - N ( s ) > 0]
"r--1
=l-e
-at
t
__ No ~_jEer(r+ 1 ) _
err]
~=1
from Eq. (10.3). A similar a r g u m e n t can be used to show that in general
= N o e rt. P [ Y ( k ) < t] = 1 - e -at
Thus, a nonstationary Poisson process with Poisson parameter Mt) = Noert/r provides a model for population growth with r a n d o m l y timed events, while maintaining exponential growth in the m e a n population size.
i n d e p e n d e n t of previous interarrival times, and therefore the process {Y(k): k = 1, ...} of interarrival times consists of i n d e p e n d e n t exponentially distributed rand o m variables with exponential parameter )~; i.e., {Y(k):
10.3. Discrete Markov Processes k = 1, ...} is an exponential white noise process. Therefore the sum k
Z(k) = ~ [ Z ( j ) - Z ( j j=l k = ~, Y(j)
1)]
j=l
of interarrival times, i.e., the time Z(k) of occurrence of the kth event, is g a m m a distributed with parameters k and k (see Appendix E). Because Z(i) and Z(j) share some of the same interarrival times in their sums, they are not statistically independent r a n d o m variables. Thus the waiting time process is not a white noise process. Example Returning again to the m i n n o w experiment, assume that a stock of 100 m i n n o w s is obtained for the experiment. To reserve the use of laboratory facilities for an appropriate a m o u n t of time, it is useful to predict the time required to exhaust the stock of minnows. If minn o w deaths follow a Poisson process with Poisson parameter k, then the time until 100 deaths is distributed as F(100, k). The mean 100/k of this distribution therefore is the expected length of the experiment. Because the variance of the distribution is 100/h 2, a conservative schedule for laboratory time of 120/k would allow the experiment to continue two standard deviations beyond its expected length.
197
This definition asserts that, conditional on the value of the value of X(t), the sequence {X(s): 0~s 0. y/---) oo
(10.5)
202
Chapter 10 Stochastic Processes
Thus, successive steps in an ergodic Markov chain will eventually stabilize on a stationary state distribution, irrespective of the initial system state. Furthermore, this stationary distribution can be shown to be unique, a property that offers a simpler method for its identification than finding the limit in Eq. (10.5). To identify the limiting distribution of an ergodic Markov chain, consider a probability distribution defined by {pj'j = 1.... , N}, with pj the probability of initially being in state j and p' = (Pl .... , PN) the vector of these probabilities. For p to be stationary, it must be reproduced after each transition of the process. Thus,
They also have been used to model colony site dynamics (Erwin et al., 1998) and movement probabilities of individual animals (e.g., Hestbeck et al., 1991; Brownie et al., 1993; Nichols, 1996). Hestbeck et al. (1991) assumed a stationary distribution for a Markov movement process for Canada geese in order to compute the stationary distribution of geese over three wintering regions. In addition, stationary distributions for Markov chains factor importantly in applications of Markov chains to conservation and management. We discuss these applications in greater detail below, when we introduce Markov decision processes.
N
Pj = i~-'1 PiPij,
10.4. C O N T I N U O U S M A R K O V PROCESSES
or, in matrix notation, P ' = p'P,
(10.6)
where P is the transition matrix of the Markov chain. Because an ergodic Markov chain possesses a unique stationary distribution, the distribution satisfying Eq. (10.6) also must satisfy Eq. (10.5), i.e., pj = "rrj for every state j. Thus, the stationary probability for state j can be obtained either by repeated transitions of the Markov chain starting at some arbitrary state i or by solving the system of equations represented by Eq. (10.6). Either approach yields the unique stationary distribution guaranteed by the ergodic property of the Markov chain. Example
Consider the Markov matrix P=
0.3 0.6 0
0.5 0 0.4
Up to now we have focused on discrete Markov processes, specifically Markov chains. Recall that the defining characteristics of a Markov chain include Markovian independence, process stationarity, and a discrete state space and time frame, the latter consisting of equal-length time intervals. These conditions give rise to the Markov matrix, which captures the stochastic structure of a Markov chain. The key attributes of a Markov chain are inherited from patterns among the transfer probabilities in the Markov matrix. In this section we continue to focus on processes that have a discrete-state space, and we retain the Markovian assumption that the future state of the process is influenced by its current state but not its history: P[X(t + at) = x t + •
0.2] 0.4 0.6
for a Markov chain with transfers among three states. It is straightforward to show that the states are all recurrent aperiodic, and therefore the chain is ergodic. Its limiting distribution is given by ~' = (6/23, 7/23, 10/23), as shown by (6 7 10)(6 7 10)[00~~ 0"5 0"2] 23,23,2-3 = 23,23,2--3 0 0.4. 0.4 0.6 Starting with any nonzero initial distribution p' = (Pl, P2, P3), repeated application of P ' t + l = P'tP ultimately will produce the limiting distribution w. Discrete Markov processes have been used for a variety of applications in population ecology. For example, in population genetics, discrete Markov processes have been used as a way of modeling gene frequency dynamics under genetic drift (e.g., Roughgarden, 1979).
= P[X(t + at)
= Xs; s ~
= xt+at]X(t)
tl
-- xt].
We also retain the assumption that the process is stationary, i.e., Pij(At[t) = Pij(&t) for all values t >- 0. However, we relax the assumption that &t is fixed over the time frame of the process and instead allow for continuous and random waiting times between process transfers. It can be shown that the length of time in which a memoryless process stays in a particular state is exponentially distributed (Ross, 1996). This property gives us an easy way to model continuous Markov processes; thus, a discrete-state process is Markovian over continuous time if (1) the amount of time the process remains in state i before making a transition to another state is exponentially distributed with rate parameter v i that depends on the current system state, and (2) the transfer from state i to state j occurs with probability Pij, with ~,jPij = 1. Thus, a continuous Markov process is simply a Markov chain in which transfers between states can occur
10.4. Continuous Markov Processes at r a n d o m times. Stated differently, it is a stochastic process with transfers between states in accordance with a discrete Markov chain, except that the a m o u n t of time between transfers is exponentially distributed. Note that the a m o u n t of time between transfers must be i n d e p e n d e n t of the terminal state of the transfer; otherwise, the duration of time prior to the transfer w o u l d inform the transfer probability, in violation of the Markovian assumption. We let Pij(s) = P [ X ( t + s) = jlX(t) = i] represent the probability that the process in state i at time t will be in state j at time t + s. This probability is a function of a discrete distribution (for the transition between states i and j) and a continuous distribution (for the length of time the process resides in state i before the transition). The product qij = viPij, k n o w n as the transition rate from i to j, parameterizes the joint distribution (see below). Note that
qij
viPij
~,jqij
vi~'jPij
203
for intervals of length s. Thus, row 1 of the matrix records the probabilities e -~s()~s)k Plk(S) = k! that a step of size k - 1 _> 0 will be taken after s units of time, starting at state i = 1. Row 2 records the probabilities that a step of size k - 2 - 0 will be taken, starting at i = 2. And so on. The subdiagonal elements of 0 indicate that the Poisson process takes only nonnegative values, so it is not possible to transfer to a smaller state. An equivalent model for the Poisson process focuses on interarrival times rather than the Poisson counts. From Eq. (10.3) the interarrival times of a Poisson process are exponentially distributed. Thus, a model for the dynamics of N(t) allows for a unit increase in N(t) at times given by the exponential distribution. The corresponding transition matrix is simply B
= Pij, so that the transition probabilities Pij reproduce the transition rates, after the latter are scaled to unity. Thus, w h e n a transition occurs, the process transfers from state i to state j with probability Pij, and these probabilities are directly proportional to the process transition rates. Note also the aggregate of transition rates reproduces the exponential parameter vi:
~ , qij = v i ~ J
0
0
0 0
1 0
0 1
m
with the u p p e r off-diagonal elements of unity indicating that w h e n the time for a transition arrives, the transfer from state i to state i + 1 is certain.
10.4.1. Birth and Death Processes
j
A simple example of a continuous Markov process is the Poisson process. If {Nt: t >- 0} is a Poisson process, then
Pij(s) = P ( N ( t + s) = j I N ( t ) = i) 0 e -~'S(Xs)J-i ( j - i)!
1
o 9
Pij = vi.
Example
=
[Pi3 =
i
0 0 0 .
if
j
if
j->i.
An important class of continuous Markov processes in biology represents transition rates in terms of birth and death events. We consider here that the process represents population size, with transfers that allow only for unit changes in state, i.e, Pij = 0 w h e n e v e r Ii - jl > 1. Thus, a population of size i can only increase to size i + 1 or decrease to i - 1. An increase obviously corresponds to a birth event, whereas a decrease represents a death event. Let birth and death rates be represented by ~'i = qi,i+l and ~i - qi,i-1, respectively. Because the two nonzero transition probabilities are related by Pi,i-1 q- Pi,i+l -- 1, w e have
This probability structure can be described at each point in time by the transition matrix
ki if- ~i = qi,i+l q- qi,i-1 --- viPi,i+ 1 q- viPi,i_ 1
-P11(S) 0 0
P(s) =
P12(S)
P13(S)
...-
P22(S) 0
P23(S) P33(S)
... ...
-- vi
and Ki
viPi,i+l
~.i q- ~i
viPi,i+l q- viPi,i-1
m
9
,.,
0
oo
o
o
9
= Pi, i+l"
204
Chapter 10 Stochastic Processes
Thus, the transition probabilities Pi,i+l and Pi,i-1 c a n be expressed in terms of the birth and death rates )k i and ixi. We may think of a birth and death process in terms of two independent Poisson processes, such that whenever there are i individuals in the population, the time until the next birth is exponentially distributed with rate parameter h i and is independent of the time until the next death, which also is exponentially distributed but with rate parameter ixi.
represents the probability that a process in state i at time t will be in state j at time t+s. With the aid of certain limiting relationships involving the probabilities Pij(s), it is possible to derive an equation for the instantaneous rate of change in the probability distribution. Given the continuous-time Chapman-Kolmogorov equation,
Example
[Eq. (10.4)], one may write
Consider a process describing the number of individuals in a population. Individuals are added to or subtracted from the population at times that are exponentially distributed with exponential parameters Xi and ixi, respectively. Then if j = i + 1
~-i
Pij(t + h ) = ~ , Pik(t)Pkj(h) k
Pij(t + h) - Pij(t) = ~ , Pik(t)Pkj(h) -- Pij(t) k = ~
Ix;
if j = i -
1
)ti q- ~i
0
lim
h--,oo
otherwise,
0 ql [Pij] =
.
9
o
0
9
0
Pl
9
9
q2
0
P2
9
~
0 w h e r e Pi = )ki/()ti q- ~l,i) a n d
9
.
o
9
1
~
pij~t)
which, under suitable regularity conditions that allow for the interchange of the limit and summation in this expression, yields the Kolmogorov forward differential equations,
P'ij = ~qkjPik(t) -- vjPij(t) k,j
o
9
[1 -- pjj(h)]Pij(t).
Pij(t + h) - Ply(t) = limfK"lz_~" ~'~Pkj(h) t'ik~'J h " h h~oO l. k.j 1 - pjj(h) ,.}, -
indicating that each event adds or subtracts an individual to the population. The Markov transition matrix for this problem is
--
Therefore
)ki q- ~i
P[N(t + s ) =jIN(t) = i] =
Pik(t)Pkj(h)
k*j
o
0
qi = ~ i / ( ) t i
nt- ~Li). T h e
entries in row 1 indicate that if the process is in state i = 0, then the only possible change is for an individual to be added to the population (necessarily through migration rather than reproduction). Entries in the last row indicate that the only possible change is for an individual to be subtracted from the population. All other states allow for either the addition or subtraction of an individual from the population, as indicated by nonzero entries in the off-diagonal positions. However, the process allows only for an increase or decrease of one individual with each transfer; hence the zero entries are everywhere but in the off-diagonal positions.
10.4.2. T h e K o l m o g o r o v Differential Equations
Recall that
Pij(s) = P[X(s + t ) = jlX(t) = i]
(Ross, 1996). They are called forward equations because the computation of the probability distribution at time t + h is conditioned on the state at time t through the Kolmogorov equation (see Kolmogorov, 1931). On reflection this transition equation makes sense. Thus, the summation term represents the addition to Pij(t) of probability mass from Pik(t), whereas the second term in the equation represents the loss of probability mass from Pij(t). By conditioning on h rather than t, we also can write the Kolmogorov backward equations,
P;j = E qikPkj (t) -- viPij(t)" k.~ i
Again, this equation makes sense; the summation term represents the addition to Pij(t) of probability mass from Pkj(t), whereas the second term again represents the loss of probability mass from Pij(t).
Example Consider a metapopulation of mice in a patchy environment, with local extinctions at a particular patch followed by recolonization from nearby patches. Let
10.5. Semi-Markov Processes
X(t) represent the presence of mice in the patch at time t, with X(t) = 1 if mice are present and X(t) = 0 if they are not. Let Ix be the extinction rate when the patch is occupied and )~be the colonization rate when the patch is unoccupied. Because there are only two states, we have transition probabilities P01 = Pl0 = 1 and P00 = P l l = 0. Furthermore, P01(t) = 1 - P00(t), so that the Kolmogorov forward equations for this system yield d
d~ P~176 = transfer of probability from Pl0(t) to Poo(t)
205
and
pij(t)-
Kj_lPi,j_l(t
) -
hjPij(t)
for j > 0, where h i -- qi.i+l and Ixi,i-1. It is straightforward to show that
Pii(t)
= e-Xit,
which is consistent with the fact that transition times are exponentially distributed. More generally, one can show that t
-
Pij(t) = Kj-le -~jt f o e~lSpi'J-l(s) ds
transfer from Poo(t) to P01(t)
= IxP01(t) - )~Poo(t)
for j > i (Ross, 1996). On assumption that )~j = j)~,
= -()~ + Ix)Poo(t) + Ix. By substituting back into this equation, it can be shown that IX
q_
P~176 = X + tx
~"
for j >-- i >-- 1. These equations provide a simple algorithm for the modeling of a pure birth process.
-(x+,)t
X + IX
with P01(t) = 1 - Poo(t) ~"
~"
---
-
-
-(x + ~)t
e
o
An analogous argument shows that P11(t) --
)k
h+~
IX
+-
)~+IX
e
Pij(t) = (i - 1 )1 e -xti(1 - e xt)j-1
-(x+~)t
The birth and death process models introduced above have been applied in population ecology, epidemiology, actuarial sciences, and evolutionary biology (e.g., see Bartlett, 1960; Bailey, 1964; Chiang, 1968), as well as in current conservation biology (see Chapter 11). Additional applications of the Kolmogorov forward and backward equations to population biology include the modeling of gene frequency dynamics (e.g., Wright, 1945; Kimura, 1957; Crow and Kimura, 1970).
and 10.5. SEMI-MARKOV
Pl0(t) = 1 - Poo(t) Ix K+IX
Ix e - ( x + , ) t . K+IX
Example For the general birth and death process described above, the Kolmogorov forward equations are
p;o(t)
= IxlPil(t)-
)~oPio(t)
and
p;j(t) = Kj_lPi.j_l(t) + Ixj+lPi.j+l(t) - (hj + Ixj)Pij(t) for j :/: 0. The first equation essentially says that the change in probability mass for Pio(t) is given by gains from Pil(t) (via death) minus losses to Pil(t) (via birth). The second equation asserts that for j :/: 0, the change in probability mass for Pij(t) is given by gains from Pi,j+l(t) (via death) and Pi,j_l(t) (via birth) minus losses from Pij(t) (via birth and death). In particular, the forward equations for a pure birth process reduce to
p;i( t) = - )~iPii(t)
PROCESSES
Thus far we have discussed processes satisfying the Markovian assumption that the future state of a process is influenced by its present state but not its past. In particular, we considered continuous Markov processes that are stationary over a discrete-state space and exhibit continuous random intervals of time between transitions. The transition probability structure Pij(s) for such a process is characterized by statistical independence of state transitions and the waiting times between those transitions. In this section we relax the Markovian assumption, but retain several other features of continuous Markov processes, including process stationarity, a discretestate space, and continuous random intervals between transitions. We also retain certain structural features that will allow us to recognize an "imbedded" Markov chain in the process. Thus, a semi-Markovian process is defined by the following characteristics: (1) at any given time the probability of transferring from state i to state j is Pij, and (2) the time until transition from i to j has a distribution that depends on both i and j.
206
Chapter 10 Stochastic Processes
From condition (2) a semi-Markov process fails to satisfy the Markovian assumption, because a prediction about the future state of the process is informed not only by the present state, but also by the length of time one has been there. Essentially, the stochastic prediction of transition times requires one to know the terminal state of the transition as well as its initial state. This adds considerable complexity to the process and stands in contrast to the continuous Markov process, for which the transition waiting times are assumed to be independent of the terminal system state. Let Fij(s) represent the distribution of time required for a transfer from state i to state j. By way of contrast, recall that transfer times for a continuous Markov process are distributed exponentially with rate parameter v i that applies to all transitions from state i, irrespective of the particular terminal state j to which the transfer is made. From condition (1) above, the semi-Markov process "imbeds" a Markov chain within it, in the sense that the stationary matrix _P = [Pij] defined in the semiMarkov process corresponds to a Markov chain. The latter process is called the imbedded Markov chain of the semi-Markov process, and it inherits its properties from patterns in the transfer probabilities in P. In particular, the semi-Markov process is said to be irreducible if the imbedded Markov chain is as well. If the process is irreducible, the expected value id,ii of the time sii between successive transitions into state i is finite.
10.5.1. Stationary Limiting Distributions Under certain conditions the limiting distribution of states for a semi-Markov process is stationary, with probabilities given in terms of the average transfer times. To see why, let Hi(s) be the average time required to transfer out of state i, based on the distributions Fij(s) of transfer times and the transfer probabilities Pij:
Hi(s) = ~_, PijFij(s). J Using the mean ~ii of the return time sii and the mean ~i
-~
f ~ S dHi(s)
of the distribution Hi(s), under rather mild conditions, the semi-Markov process can be shown to have a limiting distribution
pj = lim Pij(s) S ---~oo
= ~j/~jj that is independent of the initial state i (Ross, 1996). In essence, the value pj is the limiting proportion of
time the process spends in state j. Fortunately, these values can be identified without having to determine the mean return times txjj. If {'rri: i=1, ..., n} represents the stationary distribution of the imbedded Markov chain, i.e.,
"rrj = ~ "rriPij , i
then pj can be expressed as
~rj~j PJ = ~,i
"fribl'i"
On reflection this result makes sense. It asserts that the long-term proportion of time spent in a state increases with the stationary probability for the state from the imbedded chain and for the average amount of time the process resides in the state before making a transition. It is intuitive that large values for either of these factors will increase the proportional representation of j over the long term.
Example The movement patterns of a small mammal population are to be investigated by radio tracking. The study involves the periodic capture of individuals and fitting them with radio collars. At irregular intervals a transmitter fails, or the individual suffers mortality or leaves the study area, and another animal must be captured and fitted with a collar. The mean time required for replacement of an individual in the study is la,1 = 2 days. Experience thus far indicates that about twothirds of the individuals available for trapping are juveniles and adults are about one-third. About threefourths of all juveniles exit the study before becoming sexually active, either from mortality, migration out of the study area, or transmitter failure. The mean time to either maturation or death is t.1,2 = 10 days for juveniles. Of course, all adults ultimately exit the study through mortality, migration, or transmitter failure, with a mean time of hi,3 = 2 0 . This situation can be modeled as a semi-Markov process, with probabilities of transfer between states and with state-specific transition times that depend on both the initial and terminal states of the transition. Let the state indices 1, 2, and 3 represent individuals not in the study, and juveniles and adults that are in it, with the transfer from state I to state 2 or 3 representing capture, fitting with a radio collar, and release. Assuming equal trapping probabilities for juveniles and adults, the transfer probabilities from state I are P 1 2 = 2 1 and P 1 3 = 3, with a mean time of la,1 = 2 for trapping, collaring, and release. Because three-fourths of juveniles fail to enter the adult stage in the study, the trans-
10.6. Markov Decision Processes 1
3
fer probabilities for state 2 are P23 = 4 and P21 = 4, with a mean transfer time of ~2 = 10. Finally, the transfer from state 3 to state 1 is certain, so that P31 = 1 with a mean transition time of ~3 20. Under these circumstances the matrix for the imbedded Markov chain is =
P=
[0 3/4 1
1/3 0 0
2 / 3 ]] 14,
and it is easy to show that the corresponding stationary probabilities ~' = [-rr(1), "rr(2), w(3)] are "rr' = -rr'P =[1242-95] 25, 25,
"
It follows that the time spent in each state is in the proportions 'rrl[l,l:'rr2~2:Tr3~ 3 = 6:10:45.
Thus, one can expect to track juveniles about 16% of the time, to track adults about 74% of the time, and to lose about 10% of the study time capturing animals and replacing collars.
207
Let A i be the set of all possible actions available when the process is in state i. The available actions may well vary from state to state, i.e., it is not necessary that A i = Aj for i 4: j, though the set A = U i Ai of all available actions for the process is assumed to be finite. A policy for the process is defined by a mapping ~r that associates with a given state i at any given time t the action -rr(i, t). If ~r(i, t) = -rr(i) the policy is stationary, i.e., time independent. To apply a stationary policy, one need know only the process state and not the time t when it occurs; a particular state has the same action associated with it at every time in the time frame. To indicate the influence of decisions on the Markov transition probabilities, we represent by Pij[~r(i, t)] the probability of transfer from i at time t to j at time t + 1, assuming action ~r(i, t) is taken at time t:
Pij['rr(i, t)] = P~[X(t + 1) = jlX(t) = i], where the subscript ~r in the probability statement denotes a policy with action -rr(i, t) for state i at time t. If p,~(-rr) represents the probability of transfer in n time steps from i to j under policy -rr, then p~j('rr)
. . .in~ . 1
si1{ Piil , ['rr(i, O)]}{Pin_l,j['rr(in_l, 1"1-- 1)]}
n-2
10.6. MARKOV DECISION PROCESSES In this section we consider Markov processes for which the transition probabilities can be influenced by decisions at each point in time. To retain the Markovian assumption, we assume that at each decision point an action is taken based in the current state of the system, but not on previous states (or previous actions). The state space is assumed to be countable and therefore discrete. We also impose the condition that the range of decisions at each point in time is finite. In general, state-specific decisions are allowed to vary with time; thus, the corresponding Markov decision process is potentially a nonstationary process. However, we assume that the only sources of nonstationarity are statespecific actions that vary over time.
X H P,,//+l['rr(q,J )] j=l
by repeated application of the Chapman-Kolmogorov equation (10.4). Under a stationary policy the Markov decision process becomes a Markov chain that is defined by the stationary transition probabilities Pij['rr(i)]. Furthermore, the nth order transition probability matrix of the process is
pn
[P~j('n')] =
' r r !
where P11['rr (1)
...
PlN('rr(1)]
PTI" u 9
LpNiDr(N)
o
...
,
pNN(rr(N)]
10.6.1. Discrete-Time Markov Decision Processes We focus here on Markov decision processes with N states, over a discrete time frame T that is either finite or infinite in length. To simplify notation, we assume that an action is taken at each time in the time frame, with the action taken at time t influencing the probabilities of transition to a new state at time t + 1.
10.6.2. Objective Functionals An investigation of policies with Markov decision processes requires a measure of policy performance, by which different policies can be compared and contrasted and optimal policies can be identified. In what follows we describe an objective functional for measur-
208
Chapter 10 Stochastic Processes
ing policy performance, which aggregates utilities corresponding to time-specific actions and state transfers. Thus, let Rj[rr(i, t)] be the utility (e.g., returns net of costs) associated with the transfer from state i to state j w h e n action "rr(i, t) is taken. Then the average utility
Note that this expression is indexed by the initial state i. Thus, a policy -rr generates N such values, one for each of the possible states X(0) = i. In what follows we restrict our attention to processes with an infinite time horizon.
//
R[~r(i,t)] = ~ Pij[~r(i,t)]Rj[~r(i,t)l
10.6.3. Stationary Policies
j=l is an appropriate optimality index for discrete-time processes, and a corresponding objective functional is the expected sum of (possibly) discounted utilities, w h e n it exists:
IT
]
V~(i) = E ~ odR{'rr[X(t),tl}lX(O) = i . t=0
Additional structure in the values V~(i) can be recognized if the policy -rr is stationary. Let V'rr = [V~(1), ..., V=(N)] be the vector of values generated by a stationary policy ~r. Because the transition probabilities for a stationary process are time independent, average returns are as well:
(10.7)
N
R[~r(j,t)] = ~ Pjk['rr(j)lRk[~r(j)]
The s u m m a t i o n in this expression accumulates stochastic utilities over the time frame of the process, assuming the process begins in state i. The expectation is with respect to the stochastically determined values of process state, and the term oL _< 1 is a single-step discount factor that essentially devalues future utilities as time progresses. The notation Vrr(i) indicates that the value of the objective functional depends on both the initial state i of the process and the policy -rr that is used. We assume here that state-specific utilities are b o u n d e d for all policies, and therefore the value Vrr(i) exists whenever T < oo. If T = oo then Vrr(i) exists for all discount factors 0 < ot < 1. However, when ot = 1 the expectation in Eq. (10.7) can be finite or infinite, depending on the utilities and the pattern of transfer probabilities. If infinite, a different objective functional is required, based on the limit of timeaveraged utilities:
g~,(i) = lim (n + 1) -1E I ~_, n Rl'rr[X(t), t]} X(O) = i 1 . n--+oo t=0
(10.8)
Equation (10.8) can be shown to produce finite values of V=(i) for any stationary policy. For nonstationary policies the limit may be replaced by limit inferior. It is useful to consider an individual element in the expectation in Eq. (10.7) w h e n T = oo. A straightforward inductive argument shows that the expected utility for time t can be expressed as N
k=l = R['rr(j)]. The vector of average returns for a stationary policy is denoted here by R'~ = {R['rr(1)], R['rr(2)], ..., R[w(N)]}.
10.6.3.1. Finite Markov Decision Processes In matrix form, the objective functional in Eq. (10.9) is oo
Err = s oLtptarr. t=0
(10.10)
Because every stationary policy -rr has corresponding to it a stationary Markov matrix P~ and stationary vector Rrr of utilities, from Eq. (10.10) every policy also yields a vector of aggregate utilities Vrr. A simple alternative to Eq. (10.10) for determining Vrr can be derived by rewriting Eq. (10.10) as a recurrence relation: oo
Vrr = ~ ottptRrr t=O
= Rrr + otPrr[Rrr + oLPrrRrr + ""] = Rrr + oLPrrVrr. The vector Vrr therefore can be obtained as
E(R{,rr[X(t),tl}lX(O)=i ) = ~, ottp~j('rr)R['rr(j,t)],
V~ = (I - c~P:) - 1R~.
j=l and therefore the objective functional for finite processes may be written as
V~(i)= ~=o{~=loLtp~j(rr)R['rr(j,t)] }.
(10.9)
Of course, this computing formula requires the existence of ( / - otPrr)-1. The inverse clearly exists for discounted processes (i.e., 0 < oL < 1), because det(/-
N otPrr)= 1-Ill i=1
oLpii] =/h O.
10.6. Markov Decision Processes For essentially the same reason, the inverse for an undiscounted finite process also exists, though it is less obvious to demonstrate. 10.6.3.2. Infinite Markov Decision Processes As mentioned above, the limiting formula
V~,(i) = lim(n + 1)-IE
R['rr(xt, t)llx o = i
H----~oo
of Eq. (10.8) can be used as a measure of aggregate utility in cases in which Eq. (10.9) has no finite solution. It can be shown that the values V~(i) produced by Eq. (10.8) with a stationary policy -rr satisfy the recurrence relation
V~ + h~ = R~ + P~,h~,
(10.11)
where the vector h~, is defined by
209
using disturbances to manage the population in the most cost-effective way. For simplicity, population size is categorized as small, medium, and large populations, with stochastic transitions from year to year that depend on the type of disturbance. Three different actions can be taken, each with its own impact on the population and on its predators and other competitor species. Actions I and 2 can be used when the population is low, actions 1, 2, and 3 are available for midsized populations, and actions 2 and 3 are available when the population is high. The probability of transition from one population size to another depends on which action is taken at the time. At each point in time, management returns (net of costs) for an action depend on the population size at the time, as well as the action that is taken. The transition probabilities and returns are estimated to be -rr(1) = 1" {P11,P12,P13} {1/2, 1/4, 1/4} 9r(1) = 2: {P11,P12,P13} {1/4,1/8,5/8}
R[-rr(1)] = 8 R[-rr(1)] = 4
-rr(2) = 1" {P21,P22,P23} {1/16,3/4,3/16} -rr(2) = 2: {P21,P22,P23} {1/2, 0, 1/2} 'rr(2) -- 3" {P21,P22,Pa3} = {1/16, 7/8, 1/16}
R[-rr(2)] = 5 R[-rr(2)] = 12 R['rr(2)] = 9
"rr(3) = 2" {P31,P32,P33} {1/4, 1/2,1/4} 9r(3) = 3" {PBl,P32,PB3} {1/8, 3/4, 1/8}
R[~r(3)]= 6 R[~r(3)] = 4
- -
- -
z6 ( I - P,, + P*)h~ = ( I - P~)R~, =
with P* given by
=
n
_P* =
lim(n + 1 ) - 1 ~ _Pt. n~oo
t=O
=
Absent additional structure on the process, Eq. (10.11) represents N equations in the 2N unknowns in h~, and V~ and therefore cannot be solved. However, if the matrix P~, is ergodic, then the vector V~, can be shown to be of the form V~ = g~l, and the system of equations now involves N equations in the N + 1 unknowns h~(1), ..., h~(N), and g~. Setting one of the h~(i), say h~,(1), to zero reduces the system to N equations in N unknowns, which is solvable. The resulting h~,(2), ..., h~(N) then represent state-specific values relative to the value for state 1. From V~ = g~l the process gain g~ applies to every state and thus is independent of the initial state i. It can be shown that for an ergodic process the values h~ and g~ asymptotically satisfy
-
These transition probabilities define a total of 12 stationary Markov processes (two sets of transition probabilities for state 1, three sets for state 2, and two sets for state 3, each with state-specific returns). For example, the choice -rr(1) = 1, -rr(2) = 2, and -rr(3) = 2 in a stationary policy results in the Markov matrix [ i / /2 2 /4
Example Biologists are investigating the effect of disturbance on a population of small mammals, with a goal of
11/2 1/4 1/4
with average single-step returns R['rr(1)] = 8, R[-rr(2)] = 12, and R['rr (3)] = 6. State-specific values and system gain are found as solutions of the system of equations
__V~(n) = n[g.~l_] + h~, where V~,(n) is the vector of (asymptotic) cumulative utilities for policy -rr after n time steps. Cumulative returns thus are composed of a component for average long-term process gain and components specific to the initial state of the process. The values h~(i) may be thought of as utilities due to "transient" process behavior, whereas the gain g~ corresponds to "steady-state" utility.
1/4 0 1/2
3
g~, + h~(i) = R[~r(i)] + ~ Pij['rr(i)]h~(j), j=l
i = 1, 2, 3, and h~(3) = 0. It is easy to show that [g~, h~(1), h~(2)] = (160/19, 24/19, 80/19) solves this system of equations. Thus, the policy produces a system gain of 8.42 and transient values for states 1 and 2 of 1.26 and 4.2 (relative to state 3). Of course, a different policy would produce different state values and system gain. For this simple problem, one could determine the policy that produces
210
Chapter 10 Stochastic Processes
the largest system gain simply by enumerating the solutions for all 12 systems of equations. Obviously, such an approach becomes infeasible as the size of the process increases and the policy options multiply. We deal with optimization approaches for problems such as this in Part IV.
10.6.4. S e m i - M a r k o v D e c i s i o n Processes The results above can be generalized to allow for semi-Markov decision processes, involving sequential decision-making in which the times between decisions are random. A decision model for this situation assumes (1) the probability Pij(a) of transition between states i and j is Markovian, and is influenced by decision a, and (2) conditional on the terminal state j, the time s until transfer from i to j is random with probability density function fij(sla). An algorithm for implementing a semi-Markov process consists of choosing an action, determining the transition between states, identifying a random length of time before the transition, and repeating this sequence indefinitely. A policy ~r identifies the action "rr(i,t) = a to be used in the algorithm for every possible state at every possible decision time. As above, let Rj['rr(i,t)] represent the utility associated with transfer from state i to state j when action ~r(i,t) is taken. Assume also that there is a utility rate ri[~r(i,t)] (perhaps expressing delay costs) associated with the waiting time until transfer from i to j. Then a wait of s units of time followed by a transfer from i toj incurs a total utility of Rj['rr(i,t)] + s{r[~r(i,t)]}. Under these conditions the process is referred to as a semiMarkov decision process. Clearly, if the time between transitions is always unity, then the process is simply a Markov decision process. Note that if the policy is stationary, i.e., "rr(i,t) = "rr(i), the process is also. Then the transfer probabilities are Pij[~r(i,t)] = Pij[w(i)],
Rj['rr(i,t)] + s{r[~r(i,t)]} = Rj[~r(i)] + s{r['rr(i)]},
so that the average utility is N
Pij[~r(i)]Rj[w(i)]
+ s{r[~r(i)]}.
j=l
An appropriate objective functional is simply the expected value of the sum of (possibly) discounted utilities, when it exists: e
+ r[Tr(X n, tn)]
tn '
rs,,
e -~s ds
JO
}
-
-
V~(i) = E [ ~ e - ~ ( s ~
]
xo = i ,
1012
(10.13)
H
+(1-e~S')r{~r[X(n)]}/o~} X0= i]. If the process state space includes N transfer states and the time horizon is infinite, then Eq. (10.13) can be expressed explicitly in terms of transfer and waiting time probabilities:
u~(i) = E
e-~(s~
~. p~.(~r)
=0
•
j=l
RI~r(j)l + r [ ~ ( j ) l OL
•
and single-step utilities for the transfer from i to j are
R[w(i)] + s{r[w(i)]} = ~
where s n tn+ 1 -- t n is the waiting time for the nth state transfer and Xn is the state of the process after n transitions. This expression is analogous to Eq. (10.7) for Markov decision processes, with some notable exceptions. As before, the summation accumulates stochastic utilities over the time frame of the process; however, the number of terms in the summation now is random, because the waiting times between transitions are random variables. The expectation is with respect to the stochastically determined values of the states to which transitions are made, as with Markov decision processes; however, it also accounts for distribution of the waiting times between transitions. The sum of terms within parentheses accounts for both the utility R [ ~ ( X n , tn)] , associated with the decision at time t n, and the accumulated value (or cost) over the interval [tn_l, t n] before the nth transition occurs. Finally, the discount term e -st for continuous time has replaced oL-t for discrete time. Assuming that the process is stationary, Eq. (10.12) reduces to
pjk[~r(j)l
k=l
(1 - e~S)~.k[sl~r(j)l ds
xo = i ,
0
where the expectation now refers only to the waiting times s o, Sl, ..., Sn-1 between transitions. It also is possible to define time-averaged objective functionals for undiscounted processes with aggregate utilities that are infinite (Ross, 1970). The mathematics for this situation become rather complicated, and we leave further investigation to the interested reader.
10.7. B R O W N I A N M O T I O N Perhaps the best known continuous-state stochastic process is the Brownian motion or Wiener process. N a m e d after English botanist Robert Brown, who first discovered it while investigating particle movements
10.7. Brownian Motion in fluids, the process was given a concise definition by Norbert Weiner in 1918. It since has been used to describe behaviors of a great m a n y different phenomena, from q u a n t u m mechanics to m o v e m e n t s of stock prices. Brownian motion describes stochastic behaviors over a continuous time frame and continuous state space, on assumption that the process is normally distributed at any given time. Formally, a stochastic process {X(t): t -> 0} over continuous time is said to exhibit Brownian motion if (1) X(0) = 0, (2) {X(t): t - 0} has stationary i n d e p e n d e n t increments, and (3) for every t ~ 0, X(t) is normally distributed with m e a n ~t. From the assumption of stationary i n d e p e n d e n t increments, one can show that the variance of X ( t ) is var[X(t)] = 0.2t, where 0 .2 is linked to the underlying process and must be determined empirically. The probability density function for X(t) is 2
]" (10.14) 1 r 2x/ t W h e n ~ = 0 and 0. = 1, the distribution has the form ft(x) =
exp [ - ~l ( x - ~ tcr)
ft(x) =
1
exp -
(10.15)
and the corresponding process is called standard Brownian motion. Because a normal distribution can always be rescaled and translated so as to have any mean and variance, we assume in what follows below that the Brownian motion is standard.
211
tions of t), but nowhere differentiable. Basically, the random, i n d e p e n d e n t nature of the transitions over infinitesimally small time steps means that change is continuous but abrupt, so that the function X(t) cannot be differentiated. Based on the assumption of stationary i n d e p e n d e n t increments, it is possible to define a joint distribution for Brownian motion. Thus, the probability density function for X(t 1) = x I ~ "'" ~ X ( t n) = x n can be factored into ftl ..... t n ( X l , ...,Xn) = ftl(Xl)ft2_tl(X2
X ft,_t,,_l(Xn
-- Xn_l)
,
and stationary independent increments allow us to recognize the joint probability distribution in Eq. (10.16) as multivariate normal for all values t 1, ..., t n. Processes that meet this condition are said to be Gaussian. Because a multivariate normal is completely determined by its first two m o m e n t s (see Appendix E), one need only identify the covariance terms in the probability density function, Eq. (10.16), which can be shown to be c o v [ X ( t i ) , X(tj)] = min{ti, tj}. Thus, the probability density function of standard Brownian motion is Gaussian with E = 0 and tl tl tl
tl t2 t2
tl t2 t3
... ... ...
t1 t2 t3
.tn
tn
tn
...
tn.
X(tl, ..., tn) =
Example
Consider a population N ( t ) that fluctuates over time according to a combination of n o n r a n d o m and r a n d o m factors. N o n r a n d o m variation can be modeled by the continuous logistic equation, such that the population mean at each point in time is given by
(10.16)
-- X l ) " "
Equation (10.16) also allows us to compute conditional probabilities. For instance, it can be shown that the conditional distribution for X ( t ) given X(t 1) = A and t ~ t I is just the normal with mean E [ X ( t ) I X ( t 1) = A] = A t / t 1
E[N(t)] =
l+e
--Ft"
R a n d o m fluctuations about these average values can be modeled as Brownian motion with ~ = 0. Thus, a stochastic model by which to predict population size at time t -> 0 is described in terms of a normal distribution with logistic mean and variance var[N(t)] = 0.2t. The variance for the model increases linearly in t over the time frame. Though the probability distributions, Eqs. (10.14) and (10.15), have the familiar form of a normal distribution, inclusion of the continuous variable t in the distributions induces the very unusual property that the process is everywhere continuous (as might be expected, because its m e a n and variance are linear func-
and variance v a r [ X ( t ) l X ( t 1) = A] = t(t I - t ) / t 1.
Letting t/tl = e~, we thus have the conditional m e a n oLA, which increases from 0 to A as t increases from 0 to tl, and conditional variance o~(1 - oL)tl, which increases from 0 to a m a x i m u m of t 1/2 w h e n t = t 1/2, followed by a decrease back to 0 as t --> tl. These patterns make intuitive sense, in that the conditioning equation X(t 1) = A means that X(t) must converge to A as t approaches tl, which in turn means that the distribution variance must vanish as t approaches t 1. What is not so intuitive is the remarkable property of Brownian motion that the conditional variance of X ( t ) given X ( t 1) = A is i n d e p e n d e n t of A over 0 K t K t 1.
212
Chapter 10 Stochastic Processes
By extension, the conditional distribution for X(t) given X(t 1) -- A, X(t 2) = B, and t I < t < t2, is just the normal with n,ean E [ X ( t ) [ X ( t 1) = A, X(t 2) = B] = A + ~(t - t 1)
+
[B - A + ~(t I - t2) J t 2 -- t 1
(t-
t 1)
and variance
where Tx is the first time the process attains a value of x -> 0. Then {Z(t): t -> 0} is said to be absorbed, i.e., once having attained a value of x, the process remains at x forever. An example involves the absorbing state of zero for biological populations, in which stochastic p o p u l a t i o n change stops only w h e n the p o p u l a t i o n is extinct. It is easy to see that the first m o m e n t s for standard Brownian motion that is absorbed are
v a r [ X ( t ) l X ( t 1) = A, X(t 2) = B] = (t2 - t ) ( t - tl)" t2 - t 1
The conditional m e a n of the distribution therefore changes in a linear fashion from A w h e n t = t 1, to B w h e n t = t 2. O n the other hand, the conditional variance increases from 0 to a m a x i m u m of (t 2 - t l ) / 4 w h e n t = (t 2 4- t l ) / 2 , followed by a decrease back to 0 as t approaches t 2. As above, the conditional variance is i n d e p e n d e n t of the parameters A and B. Example
Biologists investigating the d y n a m i c s of a population of fruit flies record the n u m b e r of organisms at each of several points in time. Recognizing that the p o p u l a t i o n size is N(t 1) = A at the beginning of the observation period and is N(t 2) = B at its end, the investigators wish to determine w h e t h e r n o n r a n d o m factors have influenced population change over [tl, t2]. One w a y to investigate this issue is to compare the population size at several points in [tl, t 2] against the m e a n p o p u l a t i o n size predicted by Brownian motion. The equations s h o w n above for the m e a n and variance of constrained Brownian m o t i o n can be used to determine h o w well the recorded data fit a Brownian motion process. A reasonable fit suggests that changes in population size over [tl, t 2] are essentially r a n d o m , whereas a lack of fit suggests that population change is being influenced in some systematic w a y over [tl, t2].
10.7.1. Extensions of Brownian Motion A n u m b e r of stochastic processes that are applicable to biological populations can be derived from Brownian motion. Here we mention a few w e l l - k n o w n processes that result from simple process transformations or from restrictions on process values. 10.7.1.1. B r o w n i a n M o t i o n A b s o r b e d a t a Value
One potentially useful derived process assumes that Brownian motion is absorbed once the process attains a specified value: Z(t)={X~t)
if if
t- Tx,
0 x
E[Z(t)] =
if if
t < Tx t -> Tx
and {~ var[Z(t)] =
if if
t Tx.
10.7.1.2. B r o w n i a n M o t i o n R e f l e c t e d a t the O r i g i n
A n o t h e r variation on Brownian motion that is relevant to biology assumes that it can never be negative: Z(t) = IX(t)l
t -> 0.
Such behavior is said to be reflected at the origin. Reflected Brownian motion is especially applicable to processes such as population dynamics, in which process size m u s t remain nonnegative. It is not difficult to s h o w that the m e a n and variance for Z(t) is E[Z(t)] = V ' 2 t / ~ r
and var[Z(t)] = (1 - 2/~r)t. W h e n c o m p a r e d to m o m e n t s of the probability density function [Eq. (10.15)] for the standard Brownian motion, the nonnegativity restriction is seen to increase the process m e a n and decrease process variance. On reflection these results are intuitive; the nonnegative condition restricts the range to positive values, thereby reducing their spread and ensuring that their average m u s t be positive. 10.7.1.3. G e o m e t r i c B r o w n i a n M o t i o n
Yet another derived process is geometric Brownian motion, defined by Y(t) = e X(t). Y(t) is nonnegative, with m e a n ElY(t)] - e t/2
and variance var[Y(t)] = E[e 2X(t)] - [et/2] 2 = r 2t _ e t.
10.8. Other Stochastic Processes Again, these results are intuitive. The exponential transformation is monotone increasing and positive, so the mean and variance of the transformed process should reflect both the sign and the structure of the transformation.
213
by its mean, which is E[Z(t)] = 0, and its covariance structure, which for s ~ t is given by cov[Z(t), Z(s)] = s2(t/2 - s/6). Of course, process variances correspond to t = s: var[Z(t)] = t3/3.
Example
Geometric Brownian motion is especially useful for the modeling of percentage changes [i.e., Y ( n ) / Y ( n 1) rather than Y(n) - Y(n - 1)] that are held to be independent and identically distributed over time. An example involves the modeling of population trends over time. Consider a population that is represented by the exponential model
Example
Consider a population in which the intrinsic rate of growth (rather than the population itself) is assumed to be Brownian. Letting Z(t) represent the population and X(t) represent the population rate of growth, we have d Z(t) = X(t), dt
N(t + 1) = gt N(t),
with growth parameter gt - 1 + r t. We can describe population size in terms of a product of the growth terms:
or t
Z(t) = Z(O) + f
t
N(i) N(t) = N(O) I-I N ( i - 1) i=1 t-1 = X(0) I-I i=0
o
X(s) ds.
Because X(t) is Brownian, Z(t) is integrated Brownian, with E[Z(t)]
= e X(t) +
Z(0)
gi"
and variance as above.
On assumption that the values gt are independently and identically distributed and that the mean of r t is 0, the Central Limit Theorem ensures that the logarithm
10.8. OTHER STOCHASTIC PROCESSES
X(t) = ln[N(t)/N(0)] t-1
)
In this section we briefly mention some other stochastic processes that may arise in the modeling of animal populations.
t--1 = E In(g/) i=0
is approximately normally distributed with mean zero and variance to"2. With appropriate scaling, it follows that N ( t ) / N ( O ) = e X(t)
is geometric Brownian motion.
10.7.1.4. Integrated Brownian M o t i o n Yet another extension is integrated Brownian motion, as expressed by /. t Z(t) = ~ X(s) ds. d0
10.8.1. Branching Processes A useful class of stochastic processes with biological applications consists of branching processes (Harris, 1963; Jagers, 1975). To illustrate, suppose that a semelparous organism produces a random number Z of offspring and then dies (i.e., the generations do not overlap), as is the case with many species of insects, fish, and other taxa. Let {pj: j = 0, 1, ...} describe the probability distribution of Z for individuals in the population, assuming that all organisms reproduce according to the same distribution. Suppose also that offspring act independently of each other and produce their own offspring according to the same probability distribution. If there are, say, N(t) individuals in the population at time t, then N(t)
It can be shown that because {X(t): t - 0} is Gaussian, {Z(t): t -> 0} is as well. Thus, the process is specified
N(t + 1 ) -
~ i=1
Zi
214
Chapter 10 Stochastic Processes
describes the population transition from t to t + 1, where the time step corresponds to a single generation. This equation essentially aggregates the results of random, independent reproduction events across all individuals in the population, and realizations of such behavior over time describe a branching process. Because of the independence of reproduction events, it is easy to see that {N(t): t = 0, 1, ...} is a Markov process. If p~ and 0.2 represent the mean and variance of a random reproduction event, i.e., oo
I~ = ~, j(pj) j=O
and oo
0 .2=
~,(j-
~l,)2pj,
j=O
then the mean and variance of N(t) can be shown to be
E[N(t)]
=
p t
1,1,t -
1
and var[N(t)] =
0.2pt-1
p~- 1 to"2
if
p~ ~ 1
if
~ = 1.
Because each individual in the population produces individuals on average and then dies, it seems reasonable that the population should exhibit geometric growth in its mean for ~ > 1. One also might expect the variance to increase over time, either by tracking the growth of the population mean (if p~ :~ 1) or by increasing linearly with time when the population is stochastically stable (if ~ = 1). It is easy to see that the coefficient of variation for a growing population converges asymptotically to 0.(~2 _ t.i,)-1/2, which, if substantially exceeds unity, is approximately 0./~. Simple branching processes provide a ready model for species that reproduce only once in a lifetime, given that reproductive events are independent and reproduction is only stochastically predictable. In this situation one needs little more than an estimate of the mean and variance of individual reproductive success, to forecast population dynamics and other population attributes over the process time frame. Simple branching processes can also be adapted to organisms with other life histories (e.g., iteroparous) by defining reproduction in a manner that includes survival [i.e., the number of animals at time t + 1 "produced" by an animal at time t includes not only new individuals produced by reproduction but also the survival of the
animal itself (Caswell, 2001)] or by rescaling the time step to correspond to one generation. The original applications of branching processes are usually attributed to the French mathematician I. J. Bienayme (1845; also see Heyde and Seneta, 1972) and to F. Galton (1873) and H. W. Watson (see Watson and Galton, 1874), who used them to study extinction probabilities of family names. They were used in population genetics to study the probability of fixation of a mutant gene (Haldane, 1927; Fisher, 1930; Crow and Kimura, 1970), and they have been recommended for the study of extinction probabilities for animal populations in conservation biology (e.g., Caswell et al., 1999; Gosselin and Lebreton, 2000; Caswell, 2001). Multitype branching processes (Harris, 1963; Ney, 1964; Sevast'yanov, 1964; Pollard, 1966, 1973; Crump and Mode, 1968, 1969; Mode, 1971, 1985; Athreya and Ney, 1972; Jagers, 1975) relax the assumption of simple branching processes that all individuals are similar in their probabilities of survival and reproduction. Multitype branching processes thus can incorporate the more general age and stage structures presented for deterministic models in Chapter 8. For example, Pollard (1966, 1973) focused on stochastic analogs of the age-structured Leslie matrix (also see Mode, 1985), whereas Crump and Mode (1968, 1969) and Mode (1971) developed branching process analogs of agestructured models in continuous time. Note that the variation considered in branching process models as described above concerns the stochasticity of birth and death processes. Thus, an individual either survives until the next time step or it does not, and this process is a simple Bernoulli trial. Similarly, animals may produce 0,1, 2, ... offspring with probabilities described by a multinomial or Poisson distribution. This type of stochasticity typically is referred to as demographic stochasticity (e.g., Chesson, 1978; Shaffer, 1981). One also can envision environmental variation such that the underlying probabilities of death and of producing specific numbers of offspring vary with time and environmental conditions. This variation in the underlying probabilities of the birth and death processes often is called environmental stochasticity. Smith and Wilkinson (1969), Athreya and Karlin (1971a,b), and Keiding and Nielsen (1973) considered branching processes in random environments, thus incorporating both demographic and environmental stochasticity in stochastic process models. Mountford (1973) presented an ecological application, and Mode and Root (1988) applied a generalized branching process with both age-structure and environmental stochasticity to study bird populations. Lebreton (1982, 1990; also see Gosselin and Lebreton, 2000) considered
10.8. Other Stochastic Processes parameter estimation and demographic modeling of bird populations using a branching process model that included environmental variation and density dependence. Gosselin and Lebreton (2000) and Caswell (2001) noted the limited use of branching process models in ecology and conservation biology and provided excellent descriptions and examples of the approach. We suspect that readers of Caswell (2001) and Gosselin and Lebreton (2000) will devote increased attention to this class of models, and we thus expect to see increased use of branching process models to study animal populations over the next decade.
10.8.2. R e n e w a l Processes Renewal processes can be thought of as a generalization of the Poisson process. Recall that Poisson processes accumulate counts over a continuous time frame, with exponentially distributed interarrival times between Poisson events. Because interarrival times for a Poisson process are assumed to be independent, they constitute an exponential white noise process. Renewal processes generalize this situation, by allowing for independent and identically distributed interarrival times with nonexponential distributions. Using an earlier notation, we characterize a renewal process in terms of the interarrival time Y(i) between the i-lst and the ith occurrence in a process, assuming an arbitrary distribution F(Y) for interarrival time. Because process occurrences are independent, the process effectively "starts over" with each occurrence or "renewal." Let Z(k) represent the time until the kth renewal, i.e., k
Z(k) = ~ , Y(i) i=1
accumulates interarrival times for the first k renewals of the process. Letting b~ = E(Y)
it can be shown that average of the first k renewal times Z(k)/k converges to the mean renewal time tx for the process,
k--+oo
215
Furthermore, if N(t) is the number of renewals in the first t units of time, then the renewal rate N(t)/t converges to the inverse of ~, lim IN, t____))] t -*o"
1 -
~.
This same limit also applies to the mean m(t) = E[N(t)] of the number of renewals by time t: lim [mlt__~)] t-+oo
1 -- ~"
None of these results is particularly surprising. As the number of renewals increases, it is reasonable to expect that the finite average of renewal times will converge to the mean renewal time. It also is reasonable to expect the number of renewals per unit time, and the mean number of renewals per unit time, to converge to the reciprocal of the mean time per renewal. Though the expectation re(t) = E[N(t)] can be difficult to compute for certain underlying distributions of interarrival times, the renewal equation f t
m(t)
= F(t) + | d
m(t-
x)dF(x),
0
sometimes can be used to solve for re(t). It also can be useful in recognizing patterns of behavior in renewal processes. Example
Consider a process with alternating renewals between "on" and "off" conditions (e.g., feeding/nonfeeding behaviors), each with its own distribution of renewal times. The renewal equation for this process can be used to show that "on" and "off" conditions occur over the long term in the proportions of the distribution means:
E(X)
lim P(t) = t-+oo E(X) + E(Y)' where P(t) is the proportion of time spent in the "on" condition and E(X) and E ( Y ) a r e the mean renewal times for "on" and "off" renewals, respectively. The Euler-Lotka equation (Section 8.4), expressing population growth rate as a function of the life table birth and death parameters, can be derived as a renewal process (Sharpe and Lotka, 1911; Lotka, 1939; also see Caswell, 2001). Similarly, renewal equations have been applied to stage-based population projection models (Houllier and Lebreton, 1986), multisite projection models (Lebreton, 1996), and nonlinear agestructured models with density dependence (Tuljapurkar, 1987).
216
Chapter 10 Stochastic Processes
10.8.3. Martingales
for supermartingales. It is straightforward to show that
Martingales formalize the concept of a "fair game" over a discrete time frame. Specifically, a martingale is a stochastic process {Zt: t = 1, 2 .... } such that
E[Zt+I] >~ E[Zt]
E[Zt+I] 0, and
E[Zt+l ] Zl, Z2, ..., Z t] -- Zt"
(10.17)
Equation (10.17) indicates that the expected process value at time t + 1 is simply the actual process value at time t, irrespective of the process history. For example, if Z t represents a gambler's fortune at time t, then his expected fortune at time t + 1 after his next gamble is simply the current value of his fortune, no matter what has occurred previously. Because the stochastic behavior of a martingale is independent of its past behavior, martingales satisfy the Markovian independence assumption and therefore are special cases of a Markov process. It is easy to show that
E[Zt+I]--
EEZt]
from the martingale condition above, and therefore E[Zt] = E [ Z l l for all t > 0. A derived process of some interest involves the time until some value of a martingale is attained. Thus, a r a n d o m time N for the process {Zt: t = 1, 2, ...} is determined by the r a n d o m variables Z 1.... , Z,, in that knowledge of Z1, ..., Zn is sufficient to k n o w whether N = n. For example, let N = n if n is the first occurrence in which Z t exceeds some value Zmin. If N can only take finite values, then it is said to be a stopping time, and the process -
-
Zt =
{Z t ZN
if if
for submartingales and
t --< N t > N
defines a stopped process. This essentially says that the process continues to vary stochastically over time until a condition on the r a n d o m values is met, and then it retains the last process value from that time forward. It is not difficult to show that a stopped process is also a martingale. It also is possible to define submartingales and supermartingales in a natural way, by replacing condition (10.17) with
E~Zt+l ] Zl, Z2, ..., Zt] >~ Z t for submartingales and
E[Zt+l ] Zl, Z2, ..., Z t] 0} is stationary if for any given combination t I .... , t, of times the r a n d o m vectors [X(tl), ..., X(tn)] and [X(t I 4- s), ..., X(t n 4- s)] have identical distributions irrespective of the value s. A less stringent requirement, k n o w n as second-order (or weak) stationarity, requires only that process covariances be time invariant, i.e., cov[X(t), X(t + s)] must be independent of t. It follows that the first two moments of a second-order stationary process are temporally invariant, so that the covariance between X(t 1) and X(t 2) depends only on ]tI - t21. Gaussian processes can be used to illustrate the linkage between strong stationarity and second-order stationarity. Second-order stationarity manifests in the first and second moments, which parameterize a
10.8. Other Stochastic Processes Gaussian process because it is multivariate normal. Because the process is determined by its means and covariances, a second-order stationary Gaussian process is necessarily strongly stationary. Of course, most processes are not determined by their first two moments, so that weak stationarity does not guarantee strong stationarity. The advantage of stationarity in a process is that in order to predict process behaviors, one need know only the relative positions of process values with respect to time and not the actual times of their occurrence in the time frame. Thus, the same temporal sequencing of random variables anywhere in the time frame produces the same stochastic behaviors. In the following discussion we briefly mention two important classes of second-order time series models, the well-known autoregressive and moving average processes, that are applicable over discrete time frames.
10.8.4.1. Autoregressive Processes Let Z(O), Z(1), ... be a sequence of uncorrelated random variables with E[Z(t)] = 0 and
I var[Z(t)] =
0-2 1 -- q)2
0-2
if if
t = 0 t -> 1,
where q)2 < 1. Then the process {X(t): t = 0, 1, ...} defined by X(0) = Z(0) and X(t) = ~ p X ( t - 1) + Z(t)
(10.18)
for t >- 1, is called a first-order autoregressive process. An algorithm for implementing an autoregressive process updates the process state to X(t + 1) simply by multiplying the process state X(t) by q~ and adding a random term. It is straightforward to show that
t X(t) = ~_,
@t-iz(i)
i=0 and that
0-2q)s
cov[X(t), X(t + s)] = 1 -- q)2" Because E[X(t)] = 0 and the covariance is independent of process time t, it follows that {X(t) = 0, 1, ...} is second-order stationary. The covariance formula indicates that the statistical association between process values declines exponentially as the time s between values increases, with ~p controlling the rate of
217
decline. Of course, when s = 0 the covariance formula yields the process variance 0 "2/(1 - q~2). A straightforward generalization of autoregressive processes is obtained by allowing for lags of order greater than one, along with lag-specific weighting parameters. A general autoregressive process of order p is given by
P X(t) = ~ , q~iX(t-i) + Z(t).
i=1 Note that this expression reduces to Eq. (10.18) for the special case of p = 1 and q)i = q)" Autoregressive processes arise naturally in population dynamics through the consideration of density dependence, where vital rates (and hence population growth) for the period t to t + 1 are functions of abundance at time t (see brief discussion in Chapter 1). Thus, abundance at time t + 1 (Nt+ 1 o r log Nt+l), or population growth rate from t to t + 1 (Kt = Nt + 1/ N t or logK t) is modeled as a function of the abundances at time t and in previous periods ( N t, N t _ 1, ..., Nt_d). Royama (1977, 1981, 1992) presented general autoregressive models of population growth, describing them as "density-dependent" and "density-influenced" processes. Autoregressive models of population growth have been used extensively in the modeling of density dependence and as a basis for tests of density dependence (e.g., see Bulmer, 1975; Slade, 1977; Vickery and Nudds, 1984; Pollard et al., 1987; Wolda and Dennis, 1993, Dennis and Taper, 1994). However, it is difficult to obtain unbiased estimates of the parameters of autoregressive models using time series of population estimates. The source of the problem is simple: the sampling variances of abundance estimates reflect sampling and the uncertainty of the estimation process (see Part III of this book). Because abundance estimates/r t appear in the denominator of population growth rate estimates ~t = /Qt+l/fi4t, the sampling variance of/~t leads naturally to a negative sampling covariance between N t and ~t. Although this problem was identified some time ago (e.g., Kuno, 1971; Ito, 1972), it frequently has been ignored. In the simulation study of Shenk et al. (1998), it was concluded that sampling variation invalidated most of the tests for density dependence based on autoregressive models (Shenk et al., 1998). However, Viljugrein et al. (2001) used a Bayesian state-space modeling approach that accommodates sampling variation in an autoregressive population model, based on a time series of estimates of duck population size. The approach appears to work well and should prove useful in fitting such models in the future. In recent years, efforts have been made to fit autore-
218
Chapter 10 Stochastic Processes
gressive population models to time series data for animal populations for purposes other than the investigation of density dependence. One such use, described in Section 9.8, involves inferences about system characteristics based on general nonlinear autoregressive models. In particular, Section 9.8 contained a brief discussion of attractor reconstruction for the purpose of estimating Lyapunov exponents and system dimension. The numerical methods for attractor reconstruction use an autoregressive model of a system state variable (e.g., Takens, 1981; Cheng and Tong, 1992; Nychka et al., 1992), in this case, population size. An estimate of the dominant Lyapunov exponent in the reconstruction can be used to draw inferences about divergence or convergence of nearby trajectories in the attractor based on the behavior of the system (Turchin, 1993; Falck et al., 1995a,b). We note that by means of reconstruction of an attractor, autoregressive models can be used to draw inferences about the number of trophic interactions influencing population dynamics. For example, Stenseth et al. (1996) found evidence that most microtine populations can be characterized as two-dimensional systems and suggested that this dimensionality is consistent with density dependence and the simultaneous influence of rodent-specialist predators. Similar analyses with snowshoe hare (Lepus americanus) data from boreal forest areas of North America provided evidence of a three-dimensional system, indicating influences from density dependence, predation, and food plants (Stenseth et al., 1997). Analyses on Canadian lynx (Lynx canadensis) from the same region suggested two dimensions, indicating density dependence and the influence of prey populations (Stenseth et al., 1997). In both of these autoregressive modeling efforts, Stenseth et al. (1996, 1997) developed mathematical models of the relevant ecological interactions (e.g., density dependence, predation) and then rewrote parameters of the ecological models as functions of the coefficients of the autoregressive model. This work led to general inferences about system dynamics (e.g., about system dimension) from the more phenomenological (see Section 3.4.2) autoregressive modeling and more focused inferences based on a mechanistic, ecological reparameterization of these models. Finally, we note that Dennis et al. (1995, 1997) developed mechanistic models that included autoregressive parameters and fit these to time series data using methods of nonlinear time series analysis (e.g., Tong, 1990). The application of this work to flour beetles (Tribolium sp.) provides a nice example of the interplay between mathematical modeling and laboratory experimentation (Constantino et al., 1995, Dennis et al., 1997; also see Mertz, 1972).
Although the autoregressive modeling described above represents important efforts to investigate animal population dynamics, they still are hindered by reliance on time series of estimated, rather than true, abundance. The existence of sampling variances and covariances remains a problem that has not been dealt with in a completely satisfactory manner. The degree to which the conclusions from the cited analyses are influenced by sampling variation is unknown, but the potential problem is great. Efforts by Viljugrein et al. (2001) and others to develop methods to deal with this problem likely will lead to important contributions in population ecology.
10.8.4.2. Moving-Average Processes Let Z(O), Z(1), ... be a sequence of uncorrelated random variables with E[Z(t)] = ~ and var[Z(t)] = ~2, and consider the average X(t) =
Z(t) + Z(t - 1) + ... + Z(t - k) k + 1
(10.19)
for t --- k. An algorithm for implementing a moving average process simply updates the value X(t) to X(t + 1) by X(t + 1) = X(t) +
Z(t + 1 ) - Z ( t k+l
k)
It can be shown that E[X(t)] = p, and cov[X(t), X(t + s)] =
(k+l-s)~r 2 (k 4- 1) 2 0
I
if if
O<sk.
Because the covariance formula depends on the time s between process values but not on time t, the process is second-order stationary. Process values X(t) and X(t + s) that are less than k units of time apart share some of the same random variables in {Z(t): 0, 1, ...}, so the stochastic association between them is nonzero. However, this association weakens linearly as s increases [because fewer random variables are shared between X(t) and X(t + s)], until the association vanishes for process values farther apart than k units of time. As with autoregressive processes, it is possible to generalize moving-average processes by allowing for lag-specific weighting parameters. A general movingaverage process of order k is given by k
X(t) = ~
O i Z ( t - i).
i=0
Note that this expression reduces to Eq. (10.19) for the special case of 0 i = (k 4- 1) -1. A further generalization combines an autoregressive process and a moving-
10.8. Other Stochastic Processes average process into a mixed autoregressive-movingaverage (ARMA) process of the form p
219
and t-1
k
X(t) = ~ , ~ i X ( t -
i) + Z(t) + ~ , O i Z ( t -
i=1
Nt = No l-I ~'i, i).
i=0
i=1
A process of this form is often referred to as an ARMA process of order (p, k) (Box and Jenkins, 1976). Applications of ARMA models include the modeling of environmental processes and factors influenced by environmental variation. Some uses of such processes in stochastic demographic theory are briefly discussed below.
where N t is abundance at time t and )~t is the finite rate of population increase from t to t + 1. Environmental variation is assumed to be iid, so that kt is drawn from a stationary, nonnegative probability distribution. As noted by Lewontin and Cohen (1969), there are at least two ways to think about computing an average growth rate of the population. One way is to consider the growth of the mean population:
t l)
10.8.4.3. Demographic Stochasticity and Population Projection
With the exception of models for branching processes in random environments (Section 10.8.1), we have focused thus far in this chapter on demographic stochasticity and variation associated with the binomial nature of birth and death processes (a distinction was made between demographic and environmental stochasticity in Section 10.8.1). Because environmental stochasticity often is discussed in the context of stationary time series, in this section we highlight some efforts that have been made to incorporate environmental stochasticity into population projection models. The bulk of this work focuses on environmental stochasticity only, in the absence of demographic stochasticity. We restrict attention here to discrete-time models, although stochastic versions of continuous-time models have also been developed (e.g., see Goel and RichterDyn, 1974). Among the approaches to modeling environmental stochasticity, Caswell (2001) lists three as especially useful. The first approach simply uses independent and identically distributed (iid) sequences of random variables as environmental drivers, whereby the state of the environment at any time t is drawn from a specified distribution. The second approach considers a finite number of environmental states, with changes in state modeled as a Markov process (Section 10.3). Although many applications involve stationary Markov processes, it is possible to consider time-varying processes as well. The third approach uses autoregressive moving-average models as described above. In this case, the environment is modeled as a continuous state variable that is dependent on past states of the system. The simplest discrete-time model of a population in a stochastic environment may be that of Lewontin and Cohen (1969): X t +l --- ~.tX t
Because of the iid nature of the population rates of growth ki, the above expression can be rewritten as
tt-1 E(Nt) = N o ( I [
) E()~i)
= NoE(~ki) t
i=0 or
log E(n t) = t log E(Ki) + log N 0. Caswell (2001) showed with a numerical example that log E(K i) predicts growth of the mean population reasonably well. The other view of average growth rate is to use time-averaged growth of individual realizations of population size trajectories. Population size at time t can be written as: t-1
log N t = log N O + ~
log
~ki,
i=0
so that the time-averaged growth rate for such a trajectory is log N t - log N O lt-1 t = t-/~0 log
)k i.
The right-hand side of the above expression is simply the arithmetic mean of the logarithms of the realized )~i. Because the )ki a r e iid, as t becomes large, this quantity converges to E(log )ti): i.e., lim log(Nt) = E(log t~oo
)ki).
t
Thus, one measure of average growth, log E()~i), represents the growth of the mean population, whereas the other, E(log ~,i), represents the average rate at which
Chapter 10 Stochastic Processes
220
individual realizations will grow. By Jensen's inequality (Mood et al., 1974), E(log hi) -< log E(Ki); that is, the mean population size grows at a higher rate than most individual realizations. As noted by Lewontin and Cohen (1969), there are many cases where E ( N t) may grow infinitely large, yet each population may exhibit a very high probability of going extinct. Thus, the metric E(log h i) associated with individual realizations of population growth is more appropriate for population dynamics. Although the Lewontin-Cohen model development assumes that h i are iid, the inference that E(log h i) provides a good measure of population growth rate applies to more complicated stationary stochastic processes, such as Markov chains and ARMA models (e.g., Tuljapurkar, 1990; Caswell, 2001). The model of Lewontin and Cohen (1969) is relatively simple, yet some of their inferences about population dynamics in a stochastic environment also hold for more complicated models. For example, Cohen (1976, 1977a,b, 1979) and Tuljapurkar and Orzack (1980) considered population dynamics in terms of the product of a stochastic sequence of matrices. Denote as a i the projection matrix for environment i, representing one of k possible sets of environmental conditions. To obtain a realization of the population age-stage structure at some time t (denote as nt), we premultiply an initial vector n 0, by a random sequence of projection matrices: Fit "- [ a t - l A t - 2 " " 6 0 ] / / 0
9
If population size N t is simply the sum of the elements of the age-stage vector n t obtained as above, it can be shown that lim l~ t-,oo
t
-- E(log h s)
where h s is known as the stochastic growth rate and characterizes most realized population trajectories (see Cohen, 1976, 1977a,b; Tuljapurkar and Orzack, 1980; Tuljapurkar, 1990; Caswell, 2001). Computation of h s can be accomplished using results of a long simulation: lo---ghs = log N T - log N O T where log h s is a maximum likelihood estimator of log h s (Heyde and Cohen, 1985; Cohen, 1986). The stochastic growth rate can also be computed analytically in some cases (Cohen, 1977b; Tuljapurkar, 1990) and can be approximated as well (Tuljapurkar, 1982b, 1990).
As was the case for the Lewontin-Cohen model, it is possible to consider other measures (than log h s) of average growth rate for populations with random projection matrices. Cohen (1977a,b, 1979) and Tuljapurkar (1982a, 1990) showed how to compute the growth rate of mean population size (denote this growth rate as ~) lim log E ( N t) = log ~, t--,oo t from population projection matrices. In the case of iid environments, bL can be computed as the dominant eigenvalue of the average projection matrix (A) m
bL = h~a). Variances and confidence intervals for the estimation of average growth rates and population size are available (Heyde and Cohen, 1985; Cohen, 1986; Cohen et al., 1983; Caswell, 2001). In addition, stochastic analogs of deterministic life history quantities (see Chapter 8) such as reproductive value, stable age distribution, sensitivity, and elasticity have been proposed and investigated as well (e.g., see Tuljapurkar, 1990; Caswell, 2001). Bierzychudek (1982) and Cohen et al. (1983) presented two of the first examples of the use of these methods with stochastic matrix models of plant and animal populations, respectively, and Lee and Tuljapurkar (1994) considered the development of shortterm forecasts of human populations based on stochastic demographic models. The past decade has brought a number of examples [e.g., see Doak et al. (1994) and review in Caswell (2001)]. Much of stochastic demographic theory is based on the assumption of stationarity (see above, in this section) or else assumes only demographic stochasticity (see Section 10.8.1). However, short-term forecasts must be able to account for possible changes in vital rates. Lee and Tuljapurkar (1994) used time-series analyses to estimate trends in vital rates with an ARMA model and in turn used these vital rates to project changes in population size and stage structure.
10.9. D I S C U S S I O N From the preceding sections it should be clear that the field of stochastic processes has much to offer biological modeling and investigation. Indeed, stochastic processes can play a natural role in the modeling of animal populations, by providing a convenient way to introduce environmental variation and other stochastic elements into a population model. For example, one might think of a population trajectory as consisting of
10.9. Discussion two parts: (1) a deterministic component that essentially tracks the population mean through time, and (2) a stochastic component that allows for random variation about the mean, as modeled by a stochastic process. For example, a renewal process can be used to model the occurrence of random environmental events over continuous time, whereas Markov chains might be used in combination with dynamic population models to project random population change over discrete time. We illustrate the application of stochastic processes in later chapters, with special emphasis in Part IV on the use of Markov decision processes for dynamic decision-making under uncertainty. As potentially useful as they are, stochastic processes present a real challenge to most biologists, not least because of the mathematical complexities involved, but also because of the scope of the subject matter. We have presented only a few classes of stochastic processes and only a few of the issues that can be addressed for processes in each class. We have restricted attention here to certain well-known processes with obvious biological applications, omitting discussion about queuing processes, nonstationary time series processes, random walks, stochastic gaming, stochastic order relations, and a host of other processes with potential applicability. Nor have we dealt comprehensively with stochastic features and process derivations for the stochastic processes that are included here. For example, we have touched little if at all on such issues as delayed renewals, Markov renewals, stopping times, process approximations, process control (except for Markov processes), asymptotic process properties, and many other interesting and important features. The breadth and variability of the field of stochastic processes arise from the virtually limitless number of ways of defining linkages among random variables
221
across a time frame. Indeed, there are as many stochastic processes as there are relationships among timeindexed random variables. For example, a profusion of mathematical structures could be imposed on the moments of a joint distribution of process variables, with each corresponding to a stochastic process with its own properties and behaviors. Only relatively few of the possible processes that could be considered have been, no doubt in part because of formidable analytic complexities that can arise. As with mathematical models, it is important to limit this complexity to the extent practicable, by including only those stochastic features thought necessary to capture important stochastic associations. As in mathematical modeling, the application of stochastic processes in an investigation requires the identification of an appropriate stochastic structure and the estimation of means, variances, and other process parameters. Basically, model identification and estimation are required for stochastic processes, just as they are for the mathematical models of Chapters 7 and 8. Consider, for example, the sequential sampling of a biological system over some period of time. Assuming no statistical association among samples across time, sequential samples can be thought of as realizations of a white noise stochastic process, the distribution for which must be identified and parameterized via estimation of the distribution moments and other parameters. Repeated measures and other sampling approaches that induce a correlation structure across time periods also can be treated in terms of stationary processes, but with more complicated probability structures and greater numbers of parameters to estimate. We note in closing that there is an extraordinarily large, and often quite complex, literature on the subjects of process identification and estimation, which we leave to the interested reader to explore.
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CHAPTER
11 The Use of Models in Conservation and Management
11.1. DYNAMICS OF HARVESTED POPULATIONS 11.1.1. The Concept of Sustained Yield 11.1.2. Maximum Sustained Yield 11.1.3. Compensatory and Additive Mortality 11.1.4. Methods for Determining the Impacts of Harvest 11.2. CONSERVATION AND EXTINCTION OF POPULATIONS 11.2.1. Population Size and the Risk of Extinction 11.2.2. Extinction and Metapopulations 11.2.3. Models of Patch Dynamics 11.3. DISCUSSION
11.1. D Y N A M I C S O F
HARVESTED POPULATIONS 11.1.1. The Concept of Sustained Yield Management of many renewable resources is based on the idea that a portion of a resource stock (forests, stocks of fish, wildlife populations) can be removed without causing long-term resource depletion. This idea has become enshrined in the concept of a sustained yield that can be taken year after year without causing the resource stock to decline (Beverton and Holt, 1957; Clark, 1976; Errington, 1945; Caughley, 1977). Sustained yield can be expressed in discrete time by the generic harvest model
N(t + 1) = N(t) + f(N, H, t),
In this chapter we revisit some basic concepts and certain of the models covered earlier, as they apply to problems in the management and conservation of populations. In what follows we address two important areas of population management, the first of which involves managing harvest of stocks of animals. In particular, we discuss the sustainability of harvested populations and explore the linkage between harvest rate and population size under sustained yield. Closely related is the concept of compensatory mortality, which we describe in terms of the association between harvest rate and annual mortality. The second issue of concern involves the analysis of animal populations that are vulnerable to extinction, which we describe in terms of stochastic models that explicitly allow for the possibility of extinction.
where f(N, H, t) is the population annual growth increment and H is the annual harvest. On assumption that annual growth can be disaggregated into harvest and population growth, this model can be expressed as
N(t + 1) = N(t) + f(N, t) - H,
(11.1)
from which it is easy to see that population size is sustained by a level of harvest H* that just matches annual growth:
H* = fiN, t).
(11.2)
The growth function f(N, t) in Eq. (11.1) can be expressed as
f(N, t) = r(N, t)N
223
224
Chapter 11 Conservation and Management Models
with r(N, t) the per capita rate of growth, so that the maximum permissible harvest rate sustaining a population of size N is H*/N
= h*(t) = r(N, t).
This equation highlights the rather obvious fact that a population is sustained in equilibrium at size N whenever per capita harvest rate coincides with per capita growth. On further assumption that annual growth is dependent on population size but not on time, the growth function f ( N , t) can be written as f(N), with population dynamics described by N ( t + 1) -- N ( t ) + f i N ) - H,
(11.3)
with equilibrium harvest rate h* = f ( N ) / N
= r(N).
(11.4)
From Eq. (11.4) the sustainable yield h* is dependent on population size, in that the harvest level necessary to sustain a population varies with the size of the population. Conversely, population equilibrium depends on harvest rate, in that a given harvest rate h induces a particular equilibrium population size. If monotonic in N, the function r(N) in Eq. (11.4) possesses an inverse, which allows one to express the equilibrium population size for harvest rate h as N * = r-l(h). The point here is that a particular harvest rate induces an equilibrium in the absence of harvest population size, just as a targeted equilibrium population size requires a particular harvest rate to sustain it. Analogous forms apply for continuous models. Thus, a harvest model in continuous time is dN/dt
= f( N , t) - H
= Jr(N, t) - h(t)]N(t), with f ( N , t) = r(N, t ) N the instantaneous rate of growth of the population in the absence of harvest and h(t) the instantaneous per capita harvest rate at time t. Thus, the equilibrium condition d N / d t = 0 is satisfied by a harvest rate of h*(t) = r(N, t).
(11.5)
As above, the assumption that per capita growth is autonomous but dependent on population size leads to an equilibrium harvest rate of h * = r(N).
There are certain conceptual and practical difficulties with sustained yield as described above, especially as concerns discrete-time models. For example, many populations change over the interval [t, t + 1], and in consequence, the growth function f(N) also varies over the interval. However, the growth function describes change over the interval It, t + 1] in terms of a single population size for that interval. The corresponding biological notion is that growth (through reproduction or migration) occurs at a single point in time in the interval, an assumption that may or may not be appropriate for a particular biological situation. Another potential difficulty is that harvest frequently is assumed in discrete-time models to occur at a single occasion in the interval [t, t + 1]. Two points are noteworthy here. First, the timing of a discrete harvest event during [t, t + 1] can have consequences as to harvest yield and population response (Table 11.1). Second, harvest usually is seasonal, often occurring continuously over some part of the year. Thus, it often is necessary to model the occurrence of harvest during a part of the year, either by changing the definition of the time intervals [t, t + 1] to capture the seasonality of harvest or by replacing harvest rate h with an average harvest rate that accounts for variation over [t, t + 1]. A more fundamental problem is the potential for environmental and stochastic variation in the growth function. Assuming population change is of the form in Eq. (11.3), it is possible to determine a stationary harvest level h* that maintains the population at some specified size N. However, stationarity is sacrificed in the more general case of Eq. (11.1) with nonautonomous growth function f ( N , t). Under these conditions an equilibrium population size can be maintained only if one varies the harvest in accordance with Eq. (11.2), in which case the concept of a sustained yield that is stationary over time ceases to be applicable. Because it can be induced by environmental variation, biological interactions such as predation and competition, and other stochastic influences, temporal variation in the growth of populations is widespread. In this situation, the rather restrictive condition of an autonomous population growth can limit the usefulness of the concept of sustained yield for analysis of harvested populations.
11.1.2. M a x i m u m Sustained Yield Because sustained yield is defined by Eq. (11.5) in terms of population size, wherein different values of h* correspond to different population sizes, it often is useful to identify the maximum stationary harvest (and the corresponding population size) that is possible. Assuming autonomous growth as in Eq. (11.3), the
11.1. Dynamics of Harvested Populations TABLE 11.1
225
Illustration of Effect on Yield H of Differential Timing of Harvest a Population size
Month
Unharvested population
Harvest in month 1
Harvest in month 5
1
1000 b
1000 - 167 = 833
1000
2
976
813
976
3
953
794
953
4
930
775
930
5
908
757
908 - 152 = 756
6
887
739
738
12
768
640
640
138
750
625
625
13 c
750 + 750(0.6) = 1200
625 + 625(0.6) = 100
625 + 625(0.6) = 1000
a For a population with growth rate k = 1.20 and initial size N = 1000 in the spring just after reproduction. Annual birth rate is assumed to be b = 0.6, with birth occurring at the beginning of each year. Annual mortality rate is d = 0.25 in the absence of harvest, with mortality occurring throughout the year. C o l u m n 1 corresponds to m o n t h of the year, column 2 records population size in the absence of harvest, and column 3 corresponds to harvest in the first m o n t h at a rate of h = 0.167, producing a harvest of H = 167 individuals. Column 4 corresponds to harvest in the fifth m o n t h at the same harvest rate, producing a harvest of H = 152 individuals (from Caughley, 1977). Harvest occurs at the beginning of each m o n t h before nonhunting mortality. b After recruitment. c Before recruitment.
maximum sustained yield (MSY) can be determined simply by finding the zeros of the derivative of f(N) when it is differentiable: dH/dN
= f' (N) = O,
from which the optimal sustainable population size N*, and in turn the maximum sustained yield H* = f(N*), can be found. A typical application of MSY is described in terms of the model N ( t + 1) = N ( t ) + rN(t)[1 - N ( t ) / K ]
N = K(1 - h / r )
(11.6)
dH/dN
= r-
2r(N/K)
from Eq. (11.6), which gives N* = K/2
and a value for MSY of H* = rN*(1 - N* / K)
or
= (r/2)N* = rK/4.
h(t) = H ( t ) / N ( t )
(11.7) = r[1 - N ( t ) / K ] . This is the largest harvest rate that can be imposed on a population of size N without causing the population
(11.8)
that can sustain it and thus to a specific level of sustained harvest H = h N . It is clear from Eq. (11.8) that the population can be sustained in equilibrium for any value h that is less than the intrinsic growth rate r, which raises the question of which value corresponds to the largest sustainable harvest. The maximum sustained yield for this model is given by
=0
- H(t)
with logistic growth and postreproduction harvest. Population equilibrium is given by H ( t ) = rN(t)[1 - N ( t ) / K ]
to decline. From Eq. (11.7) a given harvest rate h(t) corresponds to a specific equilibrium population size
Then the optimal per capita harvest rate is h* = H * / N * = r/2.
226
Chapter 11 Conservation and Management Models
Thus, for this particular model, the optimal per capita harvest rate h* is one-half the intrinsic rate of growth and is not influenced by the carrying capacity K. On the other hand, the optimal equilibrium population size N* is one-half the carrying capacity K and is not influenced by the intrinsic rate of growth r. Finally, the m a x i m u m sustainable harvest H* is given by the product h'N*, or H* = rK/4. The m a x i m u m sustainable yield maintains the population in equilibrium at a size that is one-half the carrying capacity, where the population grows as rapidly as possible. These relationships are displayed in Fig. 11.1 for a population with intrinsic rate of growth r = 0.1 and carrying capacity K = 500 individuals.
H
"o
m
It is of course possible to incorporate more realism into the logistic model by including time lags, various forms of stochasticity, age structure, and other features (see Chapters 7 and 8). Nonetheless, the above model captures the biological ideas that frame the concept of MSY: 9 Populations have the potential to produce more offspring than the environment can sustain. 9 At low abundance, populations tend to grow rapidly, with the rate of growth slowing as population size increases. 9 The population asymptotically approaches a level at which it sustains itself.
*
a
8
.m
6 c-
b
'A"
0.100
0.075 (1)
(D
;=, o.o5o c-
0.025
0.000
N 250
~
500
population size
FIGURE 11.1 Yield relationships under logistic model with intrinsic rate of growth r = 0.1 and carrying capacity K = 500. (a) Harvest yield as a function of equilibrium abundance. Maximum sustainable harvest is H* = 12 for a population of size N* = 250. (b) Per capita harvest as a function of equilibrium abundance. Harvest rate for maximum sustainable harvest is h* = 0.05.
11.1. Dynamics of Harvested Populations 9 The population can be maintained below that level by removing the growth increment. For a logistic population, the MSY is achieved by maintaining population size at K / 2 , the point at which population growth is maximum. This result is of course specific to the logistic model, and other models can be expected to exhibit different equilibrium values. We note that an analogous treatment is possible for the continuous model dN/dt
= rN(t)[1 -
N(t)/K]
-
H(t),
with H ( t ) now representing instantaneous harvest. Equilibrium for this model is defined by d N / d t = 0, which requires that H(t) = rN(t)[1 -
N(t)/K]
227
and harvest rates. The additive mortality hypothesis was defined there by an approximately linear relationship between survival and harvest, with strict additivity producing an equivalent decrease in survival rate when harvest rate is increased. On the other hand, compensatory mortality was defined by changes in nonharvest sources of mortality that compensate for changes in harvest mortality, so that within limits, survival rate remains unchanged as harvest rates increase. Here we describe the compensatory and additive mortality hypotheses in terms of the relationship of harvest to mortality (rather than survival). Assume that harvest mortality occurs at the beginning of the year and is followed by nonhunting mortality. Then a simple linear model relating nonharvest mortality m ( t ) in the presence of harvest to mortality from harvest h(t) is given by
or
m(t) = m o + h(t)
= r[1 -
~h(t),
(11.9)
N(t)/K].
But this is the same equilibrium condition as for the discrete model, with the same formulas (properly interpreted in terms of instantaneous rather than discrete rates) for maximum sustainable yield and equilibrium population size.
where m0 is natural mortality in the absence of harvest, with - 1 -< ~ -< - ~ m 0 and oLclose to 1 (e.g., oL = 0.95) (Anderson and Burnham, 1976). The value f~ = - 1 represents complete compensation, with ~ = -oLm0 representing strict additivity. Thus, we have re(t) = m o -
11.1.3. Compensatory and Additive Mortality The concept of compensatory mortality was introduced previously in Chapters 3 and 8. Recall that under the compensatory mortality hypothesis (CMH), increasing harvest rates are compensated by densitydependent changes in nonharvest mortality factors. Thus, when harvest rates are reduced, densities are higher than they otherwise would have been, and natural mortality rates consequently increase. Conversely, as harvest rates increase, densities are lowered and natural mortality operates at a reduced rate. For complete compensation, there is no change in the annual mortality rate with changes in harvest rate, up to a threshold harvest rate, the maximum value of which is the mortality rate in the absence of hunting (Anderson and Burnham, 1976). Opposing the CMH is the additive mortality hypothesis (AMH), which presumably operates in the absence of density-dependent mechanisms that would affect nonharvest mortality. Thus, the AMH implies that as harvest mortality increases, total annual mortality increases proportionately, thereby producing a linear relationship between harvest rate and annual mortality. Recall that these relationships were described in Chapter 8 in terms of the relationship between survival
h(t)
(11.10)
under the CMH, which essentially says that within limits the sum of hunting and nonhunting mortality remains constant: m(t) + h(t)=
m o.
Note that Eq. (11.10) describes a linear relationship between nonhunting mortality and hunting mortality, with a slope of -1. Because the CMH operates through compensatory decreases in nonharvest mortality m(t), the amount of compensation cannot exceed m0, and this value provides an upper bound for the threshold C, beyond which additional harvest mortality becomes additive. On the other hand, Eq. (11.9) with oL = 1 yields m(t) = m o -
moh(t)
(11.11)
= m0[1 - h(t)] under the AMH. This expression is intuitively reasonable, because an animal must survive harvest mortality {with probability [1 - h(t)]} in order to have a chance of dying from nonharvest mortality. As with Eq. (11.10), this equation describes a linear relationship between hunting and nonhunting mortality. However, the slope of the relationship in Eq. (11.11) is - m 0, which is greater than -1. Thus, the reduction in nonhunting mortality attendant to increases in hunting mortality is less than that occurring beyond the threshold C under the CMH.
228
Chapter 11 Conservation and Management Models
The relationship between m ( t ) and h(t) is summarized under both the CMH and the AMH in Fig. 11.2a. Note that m ( t ) declines with increasing h(t) under both hypotheses, as a result of competition between risks associated with harvest mortality and nonharvest mortality (animals killed by harvest cannot be lost to other sources). However, this decline does not represent compensation, which is engendered by densitydependent mortality mechanisms (Anderson and Burnham, 1976; Nichols et al., 1984d). The region be-
m(t)
tween the two curves represents the range in potential compensation, from completely compensatory (lower curve) to completely additive (upper curve). By definition, total annual mortality is the sum of mortality from both harvest and nonharvest sources; that is, 1 -
S(t)
=
h(t)
m(t),
so that S(t) = 1 -
h(t) -
m(t)
= 1 -
h(t) -
[m 0 + f~h(t)].
Substitution of
m(t)
from Eq. (11.10) yields S(t) = 1 -
mo
+
mo
under the CMH [for h(t) < C], whereas substitution of m(t) from Eq. (11.11) yields S(t)
h(t) C
= S011 - h(t)]
(11.12)
under the AMH. Thus, the effect of the additive hypothesis is essentially to add a harvest component to nominal mortality m 0, thereby decreasing the survival rate as in Eq. (11.12) (Fig. 11.2b). On the other hand, the CMH leaves unchanged the survival rate over a range of values for harvest rate up to the compensation limit, with declines thereafter as harvest rate increases:
s(t) S(t)
=
1
So
0.0
(11.13) SO
0.2
0.4
0.6
0.8
1.0
h(t)
C
F I G U R E 11.2 Hypotheses of compensatory (CMH) and additive (AMH) mortality. (a) Relationship between natural mortality m and harvest rate h. Under both hypotheses, m declines from m 0 (natural mortality in absence of harvest, here taken as 0.5) as h increases, because of competition between these sources of mortality. Under the CMH, the decline is steeper and is sufficient to balance h. (b) Relationship between annual survival S and harvest mortality h. Under the AMH, each increment in h is additive to overall mortality, resulting in a linear decline in annual survival. Under the CMH, there is complete compensation up to the threshold C m i n this example C = 0.4; the maximum potential for compensation, Cmax = m0, is 0.5 in this example.
h(t) ~- C.
Thus, compensatory harvest has no effect on population dynamics if the harvest rate is sufficiently small, but reduces survival if the harvest rate is in excess of C. The relationships in Fig. l l.2a represent the results of a mechanism (density-dependent, compensatory mortality) by which a relationship between changes in harvest rates and survival rates (Fig. 11.2b) arises. The contrast between the phenomenological and mechanistic modeling of compensation can be clarified by a slight recasting of the definition of annual survival: S(t)
= 0t[1 - h(t)]
for 0 -< h(t) di, i = 1, 2, 3. Assuming only demographic stochas-
Poi(OO)=(dfii)Ni(~
3
Po(oO) = 11 Poi(OO) i=1
3 (dfii)Ni(O)
-"1 i=
In the special case in which demographic rates are identical for each population, probability of overall extinction is 3 ( d ) Ni(0)
P0(oo) = 111 i=
3 __
i=1 N(0) !
11.2. Conservation and Extinction of Populations that is, overall extinction is simply an exponential function of metapopulation abundance, and the metapopulation is essentially a single population with three biologically identical components. Individual probabilities of extinction are no longer independent if there is migration among populations. For instance, population 1 (with low growth rates) might decline to the threshold of extinction, but be "rescued" from extinction by immigration from populations 2 and 3. The situation is even more complicated if demographic rates are stochastic and nonindependent, i.e., there is a covariance structure among randomly varying parameters of the different populations. This might be expected if the separate populations share common environmental and habitat features, e.g., they all are subject to similar annual variation in climatic conditions that affect birth and growth rates. Though it is possible to incorporate these and other features into extinction models, the models quickly become analytically intractable. In practice, it is usually more straightforward to simulate metapopulation dynamics in terms of a system of interacting populations. By following the "fates" of a large number of simulated metapopulations with common initial conditions and parameters, one can determine how many populations persist and use that information to estimate extinction and persistence probabilities. This approach has the advantage of allowing for the inclusion of other sources of variation (environmental stochasticity, random catastrophic events, genetic effects) in addition to demographic stochasticity, thus providing a more comprehensive assessment of population viability. Its disadvantage is that the biological models and model parameters underlying the approach often must be identified in the absence of adequate field data.
11.2.3. Models of Patch Dynamics Here we consider two analytical and simulation approaches for modeling the dynamics of spatially structured populations: (1) patch-dynamic models, in which the population abundance is defined by the numbers of animals in discrete "patches" (habitats, areas, or other spatially defined regions), and abundance and other statistics are summarized for each patch; and (2) spatially explicit individual models, in which the spatial coordinates of individuals and their fates are simulated. A particularly simple model for patch dynamics considers only the presence (N > 0) or absence (N = 0) of animals in a system of patches, with probabilities of occupancy that are functions of patch-specific growth rates and the probabilities of migration among patches (Levins, 1969, 1970; Hanski 1992, 1994, 1997; Lande and Barrowclough, 1987; Lande, 1988). Our ap-
235
proach here extends this framework, to account for patch-specific abundance (also see Hastings and Wollin, 1989; Gyllenberg and Hanski, 1992; Gyllenberg et al., 1997). Let Ni(t) represent abundance in patch i at time t, with hi(t) the finite rate of population growth from birth and survival in patch i during time interval [t, t + 1] (i.e., excluding immigration into the patch or emigration from the patch). Let ~ri,j(t) represent the probability of movement from patch i to patch j during [t, t + 1]. Then the population dynamics for patch i are given by Ni(t + 1) = Ni(t)hi(t)'rri,i(t) 4- s Nk(t)Kk(t)'rrk,i(t) , k~i
(11.24)
where movement (if any) follows birth or mortality. For example, the dynamics of a system of three populations in a metapopulation are characterized by Nl(t + 1)= Nl(t)hl(t)'rrl,l(t) 4- N2(t))~2(t),rr2,1(t) 4- N3(t)h3(t)~rg,l(t) N2(t 4- 1)= Nl(t)hl(t)Trl,a(t) 4- N2(t)h2(t)'rr2,2(t) 4- N3(t)h3(t)~r3,a(t) N3(t 4- 1)= Nl(t)hl(t)'rrl,3(t) 4- N2(t)h2(t)'rr2,3(t) 4- N3(t)h3(t)~rg,3(t).
By specifying initial populations sizes Ni(O) and functional forms for Ki(t) and ~ri,j(t) (e.g., stationary patch-specifi~ migration rates), one can determine the trajectories of patch-specific population abundances as functions of time. For certain special cases, the population trajectories can be expressed analytically as function of time, but more typically one must use computer simulation. With simulation, the fates of simulated populations can be tracked over a selected time horizon (e.g., 100 yr), and the influence of the rate functions hi(t) and "rri,j(t) can be investigated via repeated simulation. Note that the above model incorporates patch-specific abundance, but no additional structure. The geographically structured projection matrix models of Section 8.6 can be used to develop detailed models of withinand between-patch dynamics for metapopulation systems (e.g., see Rogers, 1966, 1968, 1975, 1985, 1995; Schoen, 1988; Lebreton, 1996). Note also the close connection between the model in Eq. (11.24) and statistical models such as the multistate extensions of the Jolly-Seber model (e.g., Arnason, 1972, 1973; Hestbeck et al., 1991; Brownie et al., 1993; Schwarz et al., 1993a). In Chapters 17-19 we describe methods to estimate the demographic and movement parameters in Eq. (11.24).
11.2.3.1. Source-Sink Models A special case of Eq. (11.24) is the source-sink model described by Pulliam (1988). Suppose there are two "habitat types," one that is "suitable," in that hi(t) =
236
Chapter 11 Conservation and Management Models
)kI ~ 1 (e.g., habitat I provides adequate nest sites, food, and cover for its population component to increase) and one that is "unsuitable" [X2(t) -- X2 ~)s2).
i=1
and
Example m
-d = ~ , a i l m i=1
are the respective sample means of the counts and areas, and D = y/~ is an estimate of population density D = N / A . That this estimator is a particular case of the canonical form [Eq. (12.1)] can be seen by defining
Consider the example in Section 12.3.2 involving counts of muskrat houses, but now consider the sample plots to be variable in area (Table 12.2). An estimate of abundance for these data is now I
F=Y-A_ a 7.5
m
-
C = ~_j yi
i~100
i=1
= 536
and with variance
m
oL = ~
ai/A ,
i=1
V ~ ( / ~ r ) = M2 (1 - m / M ) ( s ~
= 200116.94 + (5.36)2(0.27)- 2(5.36)(1.56)~
/~/= C / o L .
= 1596.
The estimated variance of/~ is provided by V~(/~r ) = M2
+ f)2s2- 2L)Say)
m
so that
(1 - m / M ) ( S m
2 if- ~)2S2 _ Y a
2~)Say )_
(12.8)
From Eq. (12.5) the resulting approximate 95% confidence interval for N is (458, 614).
12.5. Complete Detectability of Individuals on Sample Units of Unequal Area TABLE 12.2 Example of Muskrat Houses with Complete Counts on Variable-Area Plots Plot
Houses counted
Plot area
1
15
2
2
8
1
3
6
1
4
8
1
5
7
2
6
3
1
7
3
1
8
3
1
9
9
2
10
13
2
y, a
7.5
1.4
2 sy, s2a
6.94
0.27
Say
1.56
12.5.2. W e i g h t e d Estimators
If var(yi) is not proportional to ai, estimators other than that shown in Eq. (12.7) are appropriate. It can be shown (Brewer, 1963; Royall, 1970) that if the residual variance of Yi about a i is proportional to var(ai), then the weighted estimator
249
aerial surveys, stratification often can be accomplished by delineating areas of similar habitat types (e.g., wetland types), which can be expected a priori to have similar densities of animals. Ideally, estimates of variance from pilot samples would be available to provide a basis for the optimal allocation of sampling units within strata. In practice, relative densities may be all that are available, but these are adequate if there is a linear relationship between the mean and variance of counts. Absent any other information, allocation should be proportional to the size of the strata. If one fails to stratify when it is appropriate, heterogeneous densities induce heterogeneity in the linear relationship between Yi and a i, resulting in suboptimal estimation of D and N. One solution to this problem is the separate estimation of a subpopulation size for each stratum. Even if densities are similar among strata, there still may be nonhomogeneous variances among strata (e.g., because of differing stratum areas), in which case an estimator based on stratification of the counts, but estimating a common value of D for all strata, may be more efficient. On assumption that a ratio (vs. weighted) estimator is optimal (there are analogous choices for weighted estimators) (see Cochran, 1977), this suggests two choices for estimators in stratified designs. The separate ratio estimator (Cochran, 1977) is
&=
I --
l~s = ~_j Yi a i i=1 ai
--
~,~ wiaiy i
~_j wi a2 A,
i=1
i=1
/
with w i = 1 / v a r ( a i ) , is best linear unbiased. On assumption that var (a i) -- ai,2 the best linear unbiased estimator of population size becomes
]~)iai
i=1
with estimated variance
I
i=1
._ ( ~~ / i ~ l ~ ) a i = l
m m
= ~
v~(/~s ) = ~ M2 (1 - m i / M i) (S2y + D" 2i s2 ia - 2Disiay),
Nw = Dw A
=
(12.9)
I
mi
where Siy , Sia, and Siay a r e the estimated variances and covariance of counts and areas for stratum i. An alternative is the combined ratio estimator
l(lc- 19cA
a "m..~
--(i=~1 Mi~]i/i~= 1 Mi-ai) a (Cochran, 1977), with estimated variance with estimated variance V~r(~w ) = A21 - m/M m
~_~ 1 a~ (Yi -- E)wai )2/(m - 1). i=1
I
var(/~c) = ~ M/2 (1 - m i / M i) (S2y + ~2s2 a _ 2~cSiay). i=1
12.5.3. Estimators B a s e d o n Stratified D e s i g n s
Under conditions described in Section 5.3.2, stratification can reduce variance. In sample counts such as
mi
The separate ratio estimator has lower variance if density D is not constant across strata and is appropriate when there is sufficient replication within strata. However, if stratum samples are "small" and there are many
250
Chapter 12 Estimating Abundance Based on Counts
strata, there may be substantial bias in estimator (12.9) (Cochran, 1977). Because of the potential for bias, the separate ratio estimator should be used only if there is good empirical evidence for between-stratum variation in D (Cochran, 1977).
abundance from the total sample is obtained as in Eq. (5.23), = )~/~,
(12.10)
where f =1/~
12.6. P A R T I A L D E T E C T A B I L I T Y OF I N D I V I D U A L S O N SAMPLE UNITS As noted earlier, incomplete counts on sampling units can result in estimates that are biased low, in that E(/Q) < N. To the extent that detectability 13 varies over time, space, or among individuals, comparative inferences also can be compromised. It therefore is important to estimate 13so as to obtain unbiased estimates of N and also to test for homogeneous detectability so as to ensure comparability across population cohorts. Two general approaches are described below. In the first, detectability is estimated on a subset of sample units that appear in the sample, and this estimate is effectively applied to all sample units (double sampling; see Section 5.4.3). In the second approach, detectability is estimated on all sample units selected in the survey.
12.6.1. Estimation of Detectability Based on a Subset of Sample Units 12.6.1.1. General Approach The adjustment of counts by the detectability 13ordinarily requires collection of auxiliary data in addition to the counts. Sometimes these data can be collected simultaneously with the count data, but often they must be collected via independent or interleaving sampling. We consider here the use of separate but complementary surveys, each survey recording counts on sampling units according to a double sampling scenario as discussed in Section 5.4.3. Thus, an extensive survey records counts x i on m' sampling units, with counts Yi recorded by independent observers on a subset of m units. The Yi counts on the subset of m units are assumed to be obtained with detection probability of 1, and data from this subsample are to be used to estimate 13 for the extensive count data. The bivariate values (x i, Yi), i = 1, ..., m are used to establish a linear relationship between counts from the two surveys, which then can be used to adjust counts for the units sampled in the extensive survey. On assumption that the relationship between the two survey counts has zero intercept, an estimator of
m
m
= i=1 ~ Y i / ~i =1 Xi estimates the reciprocal r = 1/13 of detectability and m'
2=M~xi
.
m r
i=1
The variance of this estimate is v,.d~(/Qr) =
M 2
(1 - m / M ) (s} + f 2 s 2 - 2fSxy). m' x
(12.11)
When an assumption of a zero intercept between the two counts is not warranted, a regression estimator should be used, based on the model E(y) = f3o + [31x rather than E(y) = f3x as above. The parameters of this model can be estimated using ordinary or weighted least-squares methods (Draper and Smith, 1966) as appropriate. In either case, var(/Q) must be estimated using a variant of Eq. (12.2), where in addition to the variance due to incomplete sampling (described in the previous section), a component due to the estimation of [3 ([30 and [~1 for the regression model) is required (see Thompson, 1992). Although the double-sampling approach frequently is described as above for the situation where detection probability is one for the subset of m sample units, this restriction is not necessary. In many cases, complete counts on even a subset of sample units is not possible, and the data needed to estimate detection probability are expensive to collect. In such cases it is reasonable to estimate detection probability (and thus the actual number of organisms) on the sample units in the double sample, and then apply this estimate to all surveyed units. Survey design in double-sampling includes specification of both the number of sample units selected for extensive survey and the number of units selected for intensive survey and detectability estimation. Optimal survey design in such cases is an area of active research, and some initial recommendations have been provided by Thompson (1992) and Pollock et al. (2002). 12.6.1.2. A i r - G r o u n d Comparisons in Aerial Surveys Aerial surveys from fixed-wing aircraft offer an important example of survey-based correction for detect-
12.6. Partial Detectability of Individuals on Sample Units ability. Two sampling features typically are associated with aerial surveys. First, narrow, rectangular sampling units are usually more practical than square or circular sampling units (although the latter are feasible with helicopters). Second, the area of each sampling unit typically is variable. The width of these units is usually fixed, as a function of the flight altitude and an observation angle determined by window or strut marks (Rudran et al., 1996). Typically, observations are obtained from either side of the aircraft, with a "blind spot" directly below the aircraft, although in practice the observations are often aggregated into a single plot (e.g., Conroy et al., 1988). The length of each unit (and thus the area) usually is variable. The use of aerial surveys in wildlife biology has been reviewed by Pollock and Kendall (1987). A recommended procedure for delineating aerial survey units (Seber, 1982) is to (1) stratify the study area into more or less homogeneous strata (e.g., forest or wetland types), (2) establish a baseline in the direction of least environmental change (e.g., parallel to a coastline), and (3) select sampling units at random, perpendicular to this baseline (Fig. 12.1). In practice, the rectangular units (sometime called "transects") will be flown in sequence, with a random starting point. In some circumstances, an aerial survey may use sampling units that are either square or circular and of either fixed or variable area. For example, counts of breeding American black ducks (Anas rubripes) typically are based on square, fixed-area quadrats. Aerial surveys of breeding ducks in North American
/ boundary
/_,
,: "-.../
Aerial quadrat Baseline FIGURE 12.1 Illustration of an aerial survey design. Baseline (dashed line) is oriented perpendicular to environmental gradient (e.g., upland to wetland to pelagic habitats). Aerial quadrats are selected randomly along baseline, with central lines perpendicular to baseline and with endpoints determined by the study area boundaries (e.g., marsh habitat for ducks). Quadrat width is determined by aircraft elevation and angles of detection (see text).
251
prairies (Pospahala et al., 1974; Smith, 1995) employ a double-sampling scheme, in which ground counts of a subsample of sampling units (i.e., transects) are used to correct for visibility bias, on assumption that the ground counts provide nearly unbiased estimates of abundance on the sampled units. In this scenario, m' aerial units are flown, and from this sample, a subsample of m units is selected from which accurate ground counts are recorded. The bivariate values (x i, Yi), i = 1, ..., m are used to establish a linear relationship between the ground and aerial counts, which then is used to adjust aerial counts for the units sampled only by air as in Eqs. (12.10) and (12.11). Stratified (separate) estimates of [3 are warranted when surveys combine data from different habitats or species having differing detectabilities (e.g., Pospahala et al., 1974; Smith, 1995). For instance, Srnith et al. (1995b) found a nearly threefold difference in visibility rates between forested and nonforested habitats in aerial surveys of wintering mallards. Under these circumstances, combining count statistics from different habitats without habitat-specific correction leads to biased and misleading estimates of abundance. Even the use of the counts as indices to trend might be misleading, if the relative numbers in the different habitats change from year to year, as they do for wintering ducks (Smith et al., 1995b). The double-sampling approach assumes that ground surveys are complete (detection probability of one), but this assumption may not be always be appropriate (e.g., Jarman et al., 1996; Short and Hone, 1988). Additionally, it must be assumed that the complete and incomplete counts are of the same, closed population and are independent of one another. Obviously this assumption is violated if immigration or mortality on the sampling units occurs between the times at which complete and incomplete surveys are recorded or if the sampling units are imperfectly matched, as could easily happen with air-ground comparisons. At the same time, simultaneous surveys often are infeasible for logistic reasons and, even if feasible, would be difficult to conduct without violating the independence assumption (e.g., different observers "cueing" on each other; disturbance of animals by observers). Despite these difficulties, double-sampling remains a valuable, if imperfect, means of dealing with incomplete detectability.
Example We use an example of an aerial survey of moose (Alces alces) abundance described by Thompson (1992) to illustrate visibility adjustment via double-sampling. Sample counts of moose were taken on m' = 20 aerial survey plots from a study area of M = 100 plots of equal
252
Chapter 12 Estimating Abundance Based on Counts
area, and 240 moose were counted. For a subsample of m = 5 of these plots, 70 moose were counted on the ground, whereas 56 moose had been seen from the air. The resulting estimate of detection is ~ = 1/~ = 56/70 = 0.80. Based on the aerial survey alone the estimated count for the total population is
or objects are seen by one method but not the other, and which are seen by both; and (4) the population is closed between the two samples. The idea here is to consider as a marked sample the organisms or objects observed in the ground survey and use the proportion of these that are detected in the aerial survey to estimate detectability and hence abundance (see Section 14.1). The resulting estimator for the number of organisms or objects present in the subset of sample units surveyed by air and ground is /~/= (n I + 1)(n 2 + 1 ) _ 1, m+l
m'
x=M~_jx i m'
l=1
- 12~(240 ) = 1200, resulting in an estimate = 1.25(1200) = 1500
where n I is the number of objects seen by the aerial observer, n 2 is the number of objects seen by the ground observer, and m is the number of objects seen by both observers (see Section 14.1). It can be shown that lZ(/~) -- (nl + 1)(n2 + 1)(nl - m ) ( n 2 - m) (m + 1)2(m + 2)
The assumption that all individuals are counted in the sample ground plots frequently is unrealistic. In such cases, detection probability can be estimated from intensive efforts on a subset of sample units using any of a number of approaches (e.g., see Section 12.6.2 and Chapters 13 and 14), and double-sampling still can be used. For example, an aerial survey approach described by Magnusson et al. (1978) allows both aerial and ground counts to be incomplete and uses capture-recapture models (Chapter 14) to estimate abundance. The approach was developed for sessile organisms or objects associated with animal presence and activity (e.g., nests). Thus, some sample units are surveyed both by aerial survey and by a ground crew. In order to match detections from the two surveys, the organisms or objects typically are mapped by personnel conducting both the aerial survey and the ground survey. Following completion of the two counts on the sample units, the maps are compared and numbers of organisms or objects detected by aerial survey only, by ground survey only, and by both surveys are recorded. The assumptions of this approach are (1) that the sightings by aerial and ground observers are independent; (2) the detection probabilities are homogeneous, i.e., the same detection probabilities apply for all organisms or objects; (3) one can determine which organisms
(12.13)
is an essentially unbiased estimate of the variance (see Section 14.1). The detection rate for the survey is thus estimated as ~ 1 --
of abundance after adjustment for detection.
12.6.1.3. Incomplete Ground Counts
(12.12)
nl//~.
This estimate of detection probability then can be applied to all the sample units surveyed from the air using the double-sampling estimator of Eq. (12.10). The estimator in Eq. (12.12) is the well-known LincolnPetersen estimator and is discussed more fully in Section 14.1.
Example This approach to estimating detectability was used with aerial and ground surveys to estimate the abundance of osprey (Pandion haliaetus) nests (Henny and Anderson, 1979; cited in Pollock and Kendall, 1987). A total of n I = 51 nests were seen from the air, n 2 = 63 from the ground, and m = 41 from both locations. In this particular example, the entire area of the study was surveyed by air and ground, so a double sampling approach was not needed. Application of Eqs. (12.12) and (12.13) provides estimates of N = 78.24 and v ~ (/~) - 9.67. The estimated detection rate for the aerial method was ~ = 51/78.24 = 0.65. Unfortunately this method is unlikely to be useful for mobile populations, because of the difficulty of determining which animals are seen by either or both methods (Pollock and Kendall, 1987). Though the method may prove useful for fixed objects (such as nests, roosting sites, biologically relevant terrain features), we do not recommend its application to mobile populations.
12.6. Partial Detectability of Individuals on Sample Units
12.6.2. Estimation of Detectability Based on the Set of All Sample Units Here we again consider a survey with the objective of estimating the total number of animals in some large area of interest. However, instead of using a doublesampling approach, the detection probability is estimated on all sampling units that are surveyed. We consider both simple random sampling and sampling proportional to size of the sample unit. In both cases, we assume that the survey method includes a means of obtaining counts and estimating the associated detection probability. Thus, survey efforts yield an estimate N i of abundance and its conditional sampling variance va"'r(Ni[Ni), for each sample unit i. Additional details on estimation under such survey designs can be found in Skalski and Robson (1992), Thompson (1992), and Skalski (1994)
12.6.2.1. Simple Random Sampling Assume that there are M sample units in the area of interest, from which m are randomly selected with equal probability, yielding c~ = m/M. An unbiased estimate of the total abundance for the area of interest (denote as N T) is given by m NT -- M E m i=1
w h e r e / ~ i is based on counts adjusted as necessary for
detectability. The estimated variance is
253
12.6.2.2. Sampling Proportional to Size Because the variance of the total population estimate under simple random sampling is a function of the variation in abundance among sample units, it may be inflated by variation in the size of the sample units. Thus, if sample units are of unequal size, it may be reasonable to consider sampling the different units with probability proportional to their size. If units are sampled without replacement, then total abundance can be estimated using the Horwitz and Thompson (1952) estimator m /~i
=/El / where Pi is the selection probability for sample unit i; i.e., the probability that unit i appears in the sample of m units from the total of M possible units. Note that when selection probabilities are equal for all units (Pi = P), the above expression equals the previously presented estimator under simple random sampling. In many ecological surveys, sampling costs are fixed and depend on the sizes of the selected sample units. In that situation, sample size m is a random variable, so that the variance for N T is estimated as (see Skalski, 1994) m m m v,.d,r(/~/T) = ~ (1 - Pi)fil2 i=1 p2 + 2 ~i= 1 j>i ~ m
(Pij-
Pi Pj)l~il~j ~gii ~9/~j.
~-
+ E var(NilNi) va"r(/~/T) =
M2[ ( 1 - m/M)d2im
+
E[~r(l~i[Ni)]]M
(see Skalski, 1994), where m "2 __ E i = l
sNi
"
N
~
(l~i _ ~])2
(m - 1)
"
E rn 1~i i=1 ! m
~
and m
E(~r(1Cqi[Ni) =
Ei=l
V~r(l~i[Ni ) m
The first term of the sum in brackets reflects spatial variation in abundance among the different sample units and hence variation associated with selection of the m sample units. If the entire area is surveyed, m = M and this term vanishes. The second term of the sum is the average sampling variance or measurement error associated with the fact that [3k
and
aj_k:kO
The terms OXis/OOj involve the factors used to scale x i to xis. For example, scaling x by x s = x / r leads to OXis/ OOj = - x i / r 2. The terms Opj,(Xis)/OXis are given by
Opj'(Xis) OXis
q
l(Xis)
(simple and Hermite polynomials) (Fourier series)
(Buckland et al., 1993). This system of k + m equations in the parameters 0i, i = 1, ..., k + m, can be solved using numerical optimization procedures to provide maximum likelihood estimates of the parameters. Variances and covariances of the estimates follow from the Fisher information matrix, which is obtained from the Hessian matrix evaluated at the maximum likelihood estimates (see Appendix F). Note that changing the key function oL(x) involves the specification of OoL(x)/ 00j and cOXis/OOj, whereas specifying new adjustment factors requires redefining pj(Xs). The normalizing factor ~ and 013/O0j can be evaluated by numerical integration.
13.2.5.2. M a x i m u m Likelihood Estimation for Grouped Distance Measurements
In [/=I~1{f3 f(xi)}][3
O{ln[L(O)]} " 00/ = ~2
so that
J-P~'~rsin(j-rrx s)
1"/
ln[L(O)] =_
273
It generally is preferable to record distances as continuous measurements, so that the above procedure can be used to estimate density. However, it sometimes is either inefficient or impossible to record distances accurately, and in some instances there is a tendency for measurements to be clustered at certain values irrespective of the care with which they are recorded. In such cases one is required to estimate density with data that are grouped into a limited number of distance categories. Fortunately the robust estimation methods described above are readily adapted to this situation. We assume here that the range of potential distance values is partitioned into a fixed number of distance categories by "cutpoints" {c0, c1, ..., c k} that define k distance categories, where category i includes distances between ci_ 1 and c i, with co = 0 and c k = w. The distance of an observed individual lies in one (and only one)
274
Chapter 13 Estimating Abundance with Distance-Based Methods
of these categories. Thus, grouped survey data consist of the numbers {n1, n2, ..., nk} of individuals with distances in each of the categories. Given the assumptions for transect estimation as listed earlier, the counts for a total of n observed individuals are distributed according to a multinomial distribution, with the multinomial probabilities dependent on the distribution f(x) of observed individuals. Recall that the probability density function for a multinomial distribution is
(
f(nl~r) =
)k
n
H "]T~/i~'
H1, ...r Hk
i=1
where "rrI + "'" + "rrk = 1 and n I + ... + nk = n. In this case, ~ri is the probability that the perpendicular distance of an observed individual is in the ith category. As with ungrouped data, the key to estimating density with grouped data is to estimate the distribution f(x), so that f(0) can be used i n / ) = nf(O)/(2L). This estimation is facilitated by recognizing that the probability '1ii corresponding to the distance between ci-~ and ci is simply the area under the curve f(x): "rri =
k
n i ln(-rri) + C,
i=1
where C = log[n!/II~=l(ni!)] is a constant given the data. Differentiation with respect to the model parameters yields 0{In[L(_0)]}
k ni O~i
c90j
.= gri OOj'
a0j
3f3
]
-~ L ooj aofr~
.
(13.17)
Numerical integration can be used to determine
Pi =
fci ci-1
f(x)f3 dx
and
3P i = OOj ,J
fc, Ci-1
and
0__~ = ~ 3Pi. OOj i - 1 OOj These forms can then be used in Eq. (13.17) to determine MLEs for the values 0j. Note that the same implications and requirements hold for changing the key function and series adjustments as for ungrouped data (Buckland et al., 1993). A computer is necessary to compute the iterative maximum likelihood computations and numerical integrations and to calculate the parameter estimates and the estimated variances and covariances. The program DISTANCE performs these calculations, computes likelihood ratio tests for model comparisons (e.g., to test effects of adding adjustment terms), computes AIC for model comparison and selection, and tests the resulting model for goodness of the fit to the distance data.
[f(x)[~] dx,
The statistical properties of the estimator/) are inherited from f(0) and n, the two components of /) that are subject to random variation. The estimator is sensitive to statistical behaviors of both components and in particular to the behavior of f(0). Of special concern is the variance o f / ) . On condition that f(0) is asymptotically unbiased {i.e., if f(0) converges to E[f(0)] as n increases}, the asymptotic sampling variance o f / ) is var(/))
and the values of 0j for which these expressions vanish are the MLEs. From Eqs. (13.15) and (13.16) the probabilities 7ii are parameterized by the parameters of the key function and the series adjustment that define f(x). A reparameterization by Pi = "rri~ allows us to write 3"rri= 1 [OPi
i=1
13.2.6. Estimating the Variance o f / )
The log-likelihood for grouped data is In[L(_0)] = ~
k
=EPi
(13.16)
f(x) dx.
ci-1
which in turn can be used to produce ~ and 0[3/00j by
= D 2 { [ c v ( F / ) ] 2 q-
cv[f(0)]2]}
(13.18)
(Burnham et al., 1980). An estimated variance is obtained by using estimates of the coefficients of variation: V~(/~)
=
/~)2{[C"V(H)]2 +
C"v[f(0)]2]}.
(13.19)
If var(n) = aE(n) (as is the case with the Poisson and certain other distributions of distance), it can be shown (Burnham et al., 1980) that the variance of/5 is of the form var(/)) = (1/L)[D.f(O)/2][a
+ b/f(O)2],
(13.20)
suggesting that a combined estimate of density based on replicate transect lines of varying length should weight the replicate estimates by transect line length (Burnham et al., 1980). Note the relationship between Eq. (13.18) and the canonical variance estimator, Eq.
13.2. Line Transect Sampling (12.2), both of which emphasize variation in the count statistic n and variation due to the estimation of detectability. There are several ways of estimating the variance of a density estimator (Burnham et al., 1980). For example, if multiple transects are run, one could use the empirical estimator k va'~(b) = ~ Li(]~ i - D ) 2 / [ L ( k i=1
k LiDi/L
13.2.7. Density Estimation with Clusters
i=1
with L - ~ i Li (see Appendix F). In essence, data from transect i are used to develop an estimate of D i, and transect-specific estimates then are treated as estimate replicates. The resulting estimator of variance has the advantage that no assumptions about the distribution of D are required, but it has the disadvantage that minimum data requirements must be met for each replicate transect. Because no distribution assumptions are necessary, it is the estimator of choice when data requirements can be met. Unfortunately, they can be met only infrequently. In the event that minimum data requirements for each transect cannot be met, a second approach involves the use of a "jackknife" estimator (see Appendix F). Here the data from all transects but one are pooled, and an estimate of density is derived. This is repeated for all transects, leaving each transect out and computing a corresponding estimate of density. This results in k such estimates, designated by D_i, i - 1, ..., k, where the negative subscript is used to indicate that transect i is omitted from the computations. These values then are used to define the jackknife "pseudovalues," defined by Di--
combine them according to Eq. (13.19). An estimator of var[f(0)] can be derived from the procedure for estimating f(x). An estimator for var (n) can be obtained from k ff~r(n) = L ~ , Ci[rli/C i - n / C ] 2 / ( k 1) i=1 if replicate lines are available (Burnham et al., 1980). If not, one can either assume some spatial distribution for individuals in the study area, from which is derived a value for w'r (n), or one can simply assume an expression for var(n) as a function of n.
1)1,
where ]~ = ~
275
[LE) - (L - L i ) D _ i ] / L i ,
i = 1,..., k, which in turn are used to calculate
Animals often are detected in clusters, such as coveys, flocks, and schools. In this situation, interest may focus on the density D s of clusters, the total density D of individuals, the average cluster size E(s), or any combination of the above. Clearly, these three parameters are related to one another. Statistical estimation depends on assumptions about the relationship between detectability g(x) and cluster size s, with the possibility that observed cluster size depends on the distance from the transect. 13.2.7.1. O b s e r v e d Cluster Size Is Independent
of Distance Under this situation, estimation of density and its variance is straightforward, with the estimator of overall density simply the product of estimated cluster density [cf. Eq. (13.9)] and estimated mean cluster size, i.e., /~ =/~s g
(13.21)
= [nf(O)/2r]g,
where ~ = ~ 7=1 si/n, si is the observed size of the ith cluster, and n is the number of observed clusters. A large-sample estimate of variance is provided by va'~r(/~) =/~2([cv(n)] 2 + {cv[f(O)]} 2 + [CV(S)]2),
k /~)jackknife-- ~ LiDi/L i=1
(13.22)
where cv(n) and cv[f(O)] from Eq. (13.18) are applied to the observed clusters, and cv(g) = N/~v~(g)/g with H
~i=l(Si -- ~)2 var(g) = n(n - 1)
and k V~(/~)jackknif e) -- ~ Li(D i - L)jackknife)2/C(ki=1
1).
(Burnham et al., 1980; Buckland et al., 1993). Yet a third approach to the estimation of variance for the estimator of density is to estimate the components of variance in Eq. (13.18) separately and then
(Buckland et al., 1993). 13.2.7.2. O b s e r v e d Cluster Size Is Dependent
on Distance This situation typically arises when cluster size influences the detection probability g(x), which naturally
276
Chapter 13 Estimating Abundance with Distance-Based Methods
complicates estimation. If not adequately addressed, this influence can result in positively biased estimates of density, be~.duse of the tendency to overrepresent large clusters and underrepresent small clusters in the sample. There are several alternatives to account for the nonindependence of group size and distance. One approach involves the estimation of the detection function g(x) using robust methods that do not depend on cluster size. A method for this approach uses the observed clusters to estimate E(s), though including clusters only within some maximum distance x0 over which detection is close to 1, so that detection is not an issue. Another is to use regression methods to estimate E(slx). Other approaches that avoid the influence of cluster size are (1) to treat individuals as the observations (thus avoiding the issue of estimating cluster size) but violating the assumption of independent detections, or (2) poststratify by cluster size, fit detection models for each stratum, and compute a weighted average of the stratum counts rlis i. In each of the above, once E(s) is estimated, it is used along with the unconditional estimate of D s to estimate D as in Eq. (13.21). An alternative approach, described by Drummer and McDonald (1987) and Drummer et al., (1990), uses a data transformation and bivariate parametric detection models to estimate detection, average group size, and density, corrected for size bias. Drummer (1991) documented the use of computer program SIZETRAN for implementation of these procedures. Yet another approach is a regression of si or ln(si) on d(x i) to estimate E(s) where ~(x i) ~ 1, i.e., where detectability is certain and size bias thus should not occur. Buckland et al. (1993) particularly discourage replacing the observed clusters by the individual objects, although they concede that this procedure may be useful for "loosely aggregated clusters." If this approach is used, it is most effective if distances to each individual can be measured. Of the methods described above, the regression approaches seem to offer the greatest robustness and efficiency (Buckland et al., 1993). 13.2.7.3. Full L i k e l i h o o d E s t i m a t i o n
The likelihood approaches described above are based on a conditional likelihood argument, in which parametric models are applied to the distance portion of the data x, but not to the observed sample counts n or the cluster sizes s. Parametric models are avoided by using empirical variance estimates for n and by computing confidence intervals o n / ~ under assumptions of log normality. Likewise, E(s) and var(s) are obtained in a least-squares regression framework, thus
avoiding the specification of a probability model for the number and sizes of the clusters (Buckland et al., 1993). In contrast, the full likelihood approach requires that probability modeling be extended to the sample counts and cluster sizes. The full likelihood for cluster data that include both distances and cluster sizes is given in terms of the joint probability density function P(n,
X1, ...,
Xn, $1,
...,
Sn),
where {X1, ..., X n} are the distances and {$1, ..., Sn} are the cluster sizes associated with n observations (clusters). This probability can be expressed in terms of conditional probabilities, as P(n)P(x I ..... Xnln)P(Sl, ..., shin,
X 1, ...,
Xn),
whereby the estimation of density is represented as a series of separate likelihoods. Buckland et al. (1993) note the difficulties of developing such an approach but point to several advantages, including (1) improved estimator efficiency, (2) availability of a welldeveloped likelihood theory for computing profile likelihoods (Section 4.2.3) and model comparison by AIC (Section 4.5), and (3) the possible extension of Bayesian approaches (Section 4.5) to distance estimation. Presently there is no general, full likelihood approach for distance estimation, and the remainder of this chapter is confined to the conditional approach described above.
13.2.8. M o d e l Selection and Evaluation The approach of combining key functions with series adjustment functions can result in a large number of potential models. On the one hand, this provides users with a great deal of flexibility in fitting detection functions to sample data. On the other hand, there is the problem of how to choose an appropriate model from among the large number of possible models that may be constructed. As indicated earlier, a detection model should meet estimation criteria such as model robustness, pooling robustness, shape criterion, and estimator efficiency. For a given data set, these criteria can be achieved with a combination of methods such as data screening, including the use of histograms to identify general patterns of detection and obvious outliers. This step may be helpful in identifying one or more key functions with which to start the analysis. For a given key function, the issue becomes how many terms to include in the adjustment series. The alternatives form a hierarchy, with simpler models (fewer adjustment terms) forming nested subsets within more complex models. Likelihood ratio and
13.2. Line Transect Sampling similar procedures thus are appropriate for model comparisons. However, frequently more than one key function, or type of adjustment series, may be plausible, so that the models do not form a nested hierarchy as required for likelihood ratio testing. For example, consider a model with normal key function plus the lead term of a cosine series and an alternative model consisting of a hazard function and no adjustment. Both models contain two parameters, and they do not form a nested hierarchy and cannot be compared by likelihood ratio. Akaike's Information Criterion (AIC) (Akaike, 1973; Burnham and Anderson, 1998) provides an alternative method for model selection that views model selection as an optimization rather than a hypothesis-testing procedure (see Section 4.4). The computing formula AIC = - 2 In(L) + 2q includes ln(L), the natural logarithm of the maximum of the likelihood function, and the number q of model parameters. Essentially this expression represents the tradeoff between bias reduction through improved model fit [achieved by minimizing the deviance - 2 ln(L)] and a penalty for increased variance as additional parameters are added (the 2q term) (see Section 4.4). For a given data set (AIC comparisons among data sets are meaningless), the procedure is to compute AIC for each candidate model and select the model providing the lowest AIC statistic, recognizing that models with AIC values less than two units apart are essentially equivalent. We note that for the special case where nested models differ by one parameter, model selection based on AIC is equivalent to a likelihood ratio test with X2 = 2.0 (oL = 0.157) (Buckland et al., 1993). AIC thus can be used for ranking models that are either nested or non-nested. Occasionally, the AICs for more than one model are essentially tied (i.e., differ by ~2). In these cases, the models all are seen as acceptable competitors and should be further evaluated based on other criteria, such as prior biological knowledge. Alternatively, model-averaged estimates (Burnham and Anderson, 1998) can be computed. Once the estimated detection function and the corresponding densities are produced, goodness of fit statistics and graphical analysis of residuals are useful in determining model adequacy. Goodness of fit can be tested by a Pearson chi-square statistic (see Section 4.3.3), provided the n distances are first split into, say, k groups with sample sizes n 1, ..., n k. A model fitted to the (original) data then can be used to estimate the cumulative probability "rri under the probability density function between the "cutpoints" ci_ 1 and c i. Finally, these estimated probabilities can be used to compute a test statistic as
277
k (n i _ n~ri)2, X2-- E i~-1 tllTi which follows a chi-square distribution with k - q - 1 degrees of freedom under the null hypothesis that the candidate model appropriately represents the data.
13.2.9. Interval Estimation Variance estimates f o r / ) are obtained from application of Eq. (13.19) or (13.22), with the estimate v'~[f(0)] obtained from the conditional maximum likelihood methods described in Section 13.2.5. As noted in Section 13.2.6, empirical estimates of var(n) can be used in lieu of likelihood approaches. However, empirical estimates are not available if lines are not replicated, and one then is forced to rely on a distribution-based relationship such as var (n) = n for the Poisson distribution, possibly adjusted by a constant (Burnham et al., 1980). An approximate (1 - 2c~)100% confidence interval may be computed by invoking asymptotic normality of D as /~ _ z~X/v~r(/~) where z~/2 is the upper a point of the standard normal distribution. However, Buckland et al. (1993) note that the distribution of /~ is skewed and suggest that a confidence interval based on assumed log normality o f / ) provides superior coverage. This interval is computed as ( ~ / c , [) . c)
where C = exp [ G V ' v ~ (ln/~i ] and v~r(ln D ) = ln[1 + v~(D)//~2]. The above approach is used in program DISTANCE (Buckland et al., 1993) to calculate confidence intervals, except that the normal deviate is replaced by a t statistic with degrees of freedom computed by a Satterthwaite (1946) adjustment. Example
Burnham et al. (1980) describe an experiment in which a known number of wooden stakes were placed in a sagebrush meadow, with a density of 37.5 stakes/ ha. Teams of students walked transect lines and recorded perpendicular distances from the lines to the stakes that were detected. Here we report the results for
Chapter 13 Estimating Abundance with Distance-Based Methods
278
one transect line, from which 68 stakes were detected. Program TRANSECT (Burnham et al., 1980) was used to compute estimates based on Fourier series (equivalent to the uniform key function with a cosine adjustment term), and a model with two adjustment terms was selected, providing an estimate of density /~ = 39.3 s t a k e s / h a (~~ = 0.15). These same data were reanalyzed with p r o g r a m DISTANCE, using (1) the uniform key function with 0, 1, 2, and 3 cosine adjustment terms and (2) the half-normal key function with 0, 1, and 2 adjustment terms (Table 13.2). The seven models formed by these combinations of key functions and adjustment series were ranked by descending AIC, and the top two models were indistinguishable based on AIC (AAIC < 2). The second ranked model is based on fewer parameters, with a resulting higher precision in the density estimate (~'v = 0.13 vs. 0.16); both models evidenced adequate fit (P > 0.20). The second-ranked model yielded an estimated density of 33.08 stakes/ ha with a log-based 95% confidence interval of (25.38, 43.12).
13.3. P O I N T
SAMPLING
In the previous section the sampling units were line transects of fixed length and (possibly) indefinite width. However, in some applications the sampling unit is a point (or "point transect") with observation distances recorded in terms of radial distance from the point. We have already seen some examples of this approach, in the point-to-object methods considered in Section 13.1. Point sampling often is used in surveys of singing birds, whereby observers stop at predetermined stations and attempt to identify all birds in the vicinity, sometimes visually but often by detecting their songs. Point sampling also occurs in the context of cue
TABLE 13.2 Key function
Uniform Uniform Uniform Half normal Half normal Half normal Uniform
O~ r3~
FIGURE 13.3 Example of point sampling and measurements. Open circles represent detected individuals. For detected individuals, r is the observer-to-individual distance.
counting and trapping webs (Buckland et al., 1993) (see Section 13.5).
13.3.1. S a m p l i n g S c h e m e Data Structure
and
The sampling units in point sampling are k replicate points at each of which individuals are detected and the radial distances r i to each individual are measured (Fig. 13.3). Field ornithologists using point sampling ("point counts") have tended to emphasize sampling over an area of fixed radius w about the point, within which detection is assumed to be perfect, or at least uniform. Though sometimes justified, this assumption, which is analogous to perfect detectability near the transect line, is unnecessarily restrictive. We advocate recording distances to all objects detected in point sampling, along with the use of robust methods to estimate empirically detection functions and density. A modification of point count sampling for birds,
Example of Line Transect Estimation Using Laake's Wooden Stake Data a
Adjustment
Number of adjustment terms
Goodness of fit AIC
~AIC
X2
df
Cosine Cosine Cosine Hermitepolynomial -Hermitepolynomial m
2 1 3 2 0 1 0
382.14 384.11 384.14 384.16 385.78 387.73 409.24
0 1.97 2.00 2.02 3.64 5.59 27.1
8.87 13.42 4.65 8.88 16.21 16.13 39.37
9 10 7 8 10 9 11
P
0.45 0.20 0.70 0.35 0.09 0.06 N (Chapman, 1951). The variance for N can be estimated as va,--~(/qr) = (nl + 1)(n2 + 1)(nl - m 2 ) ( n 2 - m2) (m 2 + 1)2(m2 + 2)
v~r (/(1) =
(Seber, 1970a). Confidence intervals for Lincoln-Petersen estimates of population size can be constructed in various ways. One approach (Seber, 1982) is to rely on the asymptotic normality of N and construct the approximate 95% confidence interval for N as/Q + 1.96V'v~(/~). Skalski
(n.2_
m.2)(n.1 _ m.2)(/~1 + /~2)
11.111.2ra
n.1 -- nal + nbl, n.2 -- na2 + 11,b2, and m.2 -ma2 4- rob2. In general, var(/(1) < var(K). This makes intuitive sense, because/( requires separate estimation of capture probabilities for the two populations, so that v~(K) includes variance components for both capture probability estimates. On the other hand, /~1 ass u m e s that capture probabilities are identical for the two populations, so that the resulting variance is smaller. The use of /(1 requires some method for testing
where
(14.4)
- mb2)(11b1 -- mb2)ma211a111a2
14.1. Two-Sample Lincoln-Petersen Estimator whether the capture probabilities for two populations actually are equal. Skalski et al. (1983) and Skalski and Robson (1992) recommended the use of a 2 • 3 contingency table to test for homogeneous capture probabilities. A contingency table for capture history data from populations a and b is given as follows: Capture history
Population a
Population b
Sum
1 1
Xal I -- ma2
Xbl I = mb2
X.11 -- m . 2
10
Xal o = Ylal -- ma2
Xbl 0 = Ylbl ~ mb2
X.10 = 1"/.1 -- m . 2
01
Xa01 -- Yla2 m ma 2
Xb01 -- Ylb2 ~ mb 2
X.01 = r/.2 _ m . 2
ra
rb
r
Sum
The dot notation in this contingency table denotes summation over the two populations, for example r = r a + r b and x.11 = Xal I 4- Xbl 1. Expected cell frequencies under the null hypothesis of equal capture probabilities for the two populations are given by Expected frequency Capture history
Population a
Population b
Sum
11
Na PiP2
Nb PlP2
(Na + Nb) Pl P2
10
Na Plq2
NbPlq2
(Na + Nb) Pl q2
01
Na ql P2
Nb ql P2
(Na + Nb) ql P2
Na(Pl + P 2 - Pl P2)
Nb (Pl + P 2 - PlP2)
(Na + Nb)
Sum
X (Pl + P2 - PIP2)
Under the null hypothesis of equal capture probabilities for the two populations, the test statistic associated with this contingency table follows a chi-square distribution with two degrees of freedom. The test can be extended readily to the situation of more than two sampling periods and more than two populations, by considering expected numbers of animals for each population exhibiting all possible capture histories (Skalski et al., 1983; Skalski and Robson, 1992). 14.1.3. V i o l a t i o n of M o d e l A s s u m p t i o n s 14.1.3.1. C l o s u r e
The closure assumption can be violated in several ways, including mortality during sampling (i.e., during capture in sample 1). One way to deal with sampling mortality is simply not to include the number of sampling deaths (denote this number as d) in the n 1 statistic. Then the Chapman (1951) estimator/~ in Eq. (14.3) estimates the population size after the sampling deaths, whereas/9 + d is an estimate of the presampling population size.
293
The closure assumption also can be violated by deaths between sampling occasions. To see the effect of mortality on the Lincoln-Petersen estimator, define q0 as the probability that an animal alive at the time of the first sampling occasion is still alive and present in the population at the time of the second sampling occasion. Here we assume that r applies to all individuals in the population, whether captured or not. Because population size differs between the two sampling periods, it is necessary to designate by N 1 the population size at sampling occasion 1. Though E(n 1) = N i p 1, the expected values of the other two summary statistics are influenced by q0, with E(n 2) = NlCpp 2 and E(m 2) = NlPlCpp 2. Substituting these expectations into the standard Lincoln-Petersen estimator, Eq. (14.2), we obtain E(Iq) ~ E ( n l ) E ( n 2 ) / E ( m 2 ) = (NlPl)(Nlq~p2)/NlPlCpP2 -- N1"
Thus, the Lincoln-Petersen estimator provides an estimate of the population size at the time of the first sampling period (Robson, 1969; Seber, 1982). N o w consider mortality associated with handling or marking, which is imposed only on members of n 1. In this case, the expected values of the summary statistics can be written as E(n 1) = N i p 1, E(m 2) = NlPlq~p2, and E(n 2) = Nl[PlCpp 2 4- (1 - Pl)P2]. Substituting these expectations into Eq. (14.2), we obtain E(lxl) ~- E ( n l ) E ( n 2 ) / E ( m 2 ) = ( N l P l ) N l [ P l @ p 2 4- (1 - p l ) P 2 ] / N l P l @ P 2
= NI[Pl + (1 - pl)/q~]. Thus, the Lincoln-Petersen estimator is positively biased in the presence of deaths associated with handling or marking. Intuitively, the estimator/31 = m 2 / n 2 is too small because some of the animals marked in sample 1 die and therefore are not available to be caught in sample 2. A negative bias in ~31 then leads to positive bias in/9. The closure assumption also can be violated because of immigration of new animals between the two sampling periods. If we denote the number of animals entering the sampled population between the first and second periods as B, the expectations of the summary statistics become E(n 1) = N i p 1, E(n 2) = (N 1 + B)p2, E(m2) = NlPlP2. Substitution of these expectations into Eq. (14.2) yields E(lCq) ~, E ( n l ) E ( n 2 ) / E ( m 2) = ( N l P l ) ( N 1 4- B ) p 2 / N l P l P 2
=NI+B.
294
Chapter 14 Mark-Recapture Methods for Closed Populations
Thus, the Lincoln-Petersen estimator provides an estimate of the population size at the time of the second sample (see Seber, 1982). Finally, consider the case where both mortality/emigration and immigration occur between the two samples. The expected values for the summary statistics are now E(n 1) = Nip 1, E(n 2) = (Nlq~ 4- B)p2, and E(m 2) = Nlq~plP2. Substitution of these expectations into Eq. (14.2) yields
E(I~I) ~ E(nl)E(n2)/E(m 2) = (Nlpl)(Nlq~ + B)p2/NlPl~P2
(14.5)
same probability of being caught. The assumption of equal catchability can be violated in two ways. First, members of the sampled population can be heterogeneous with respect to capture probability, such that some animals have a higher probability of being caught than other animals. Consider the Lincoln-Petersen estimator in Eq. (14.2) as an example of the canonical estimator, where n I is the count statistic and ]91 = m2/n2 is the estimate of the corresponding sampling probability. Animals with higher capture probability than average have a greater chance of being caught in both samples, so that
= N 1 4- B/q~.
E(]~I)
Because q~ < 1, the expectation in Eq. (14.5) is larger than N 1 + B, and the Lincoln-Petersen estimator is positively biased for population size at either sampling time (also see Robson and Regier, 1968). A special case of a population that is open to both gains and losses considers the animals in the population at time j to represent a subset of animals in a superpopulation of size N ~ with animals in the superpopulation moving freely in and out of the sampled area. Assume that the animals in the sampled area at either time represent a random sample from the superpopulation with probability ,rj, i.e., E(NjlN ~ ,rj) = N%j. If the superpopulation is closed and the capture probability pj is redefined to be conditional on being in the sampled area at time j, Lincoln-Petersen estimation produces an estimator of the capture probability ,rjpj for animals in the superpopulation, and the Lincoln-Petersen estimator for population size now estimates the number of animals in the superpopulation (see Kendall, 1999). Time specificity can exist in "rj or in pj (or both), and the Lincoln-Petersen parameterization and estimators are still appropriate for the superpopulation. This result is consistent with the more general situation expressed in Eq. (14.5), as seen by writing the expected values for the quantities in Eq. (14.5) in terms of the random movement model:
Pl
-
T1)T2,
b,
4-
where b is a bias factor (b > 0). Because the estimated capture probability is too large, the population size estimate is too small, that is
E(l(4) ,~ E(nl)/E(~I) = N[pl/(Pl
4-
b)].
A second form of unequal capture probability is known as trap response, referring to a tendency for animals caught in the first sample to have a different capture probability in the second sample compared to animals not caught in the first sample. Denote the capture probabilities for sample period 2 as Pc for captured, previously uncaught animals and Pr for recaptured animals that were captured in the first sample. The expected values of the summary statistics under this scenario can be written as E(nl)
=
Xpl ,
E(n2) = N[plp r 4- (1 - Pl)Pc], and E(m2)
=
XplPr.
Substitution of these values into Eq. (14.2) yields
E(l(4) ~ E(nl)E(n2)/E(m2) -- ( N F 1 ) N [ F I F r 4-
E(N1) = N~ E(B) = N~
=
-- X p l
4-
(1
-
pl)Pc]/XplPr
N(1 - Pl)Pc/PF-
Substitution of these expectations into Eq. (14.5) yields the approximate expected value of the LincolnPetersen estimator as E(N)-~N ~
The approximate expectation in this expression equals N if Pc = Pr, that is, if there is no trap response. When Pr > Pc (trap-happy response), the Lincoln-Petersen estimator is negatively biased. When Pr < Pc (trap-shy response), the estimator is positively biased.
14.1.3.2. Equal Capture Probability
14.1.3.3. Tag Loss
Capture probabilities for the Lincoln-Petersen estimator need not be the same for the two samples, but within each sample, all animals are assumed to have the
The third assumption underlying the LincolnPetersen estimator is that marks are neither lost nor overlooked. Consider the situation where a mark is
E(q~) = "r2.
14.1. Two-Sample Lincoln-Petersen Estimator lost between the first and second samples with probability 1 - 0, where 0 < 0 < 1. Then expected values of the summary statistics can be written as E(n 1) = Npl, E(n 2) = Np2, and E(m 2) = NplOp2. Substitution of these expectations into Eq. (14.2) yields
E(1Q) ~ E(nl)E(n2)/E(m2) = (Xpl)(Xp2)/XplOp2
= N/O. Because 0 < 0 < 1, tag loss (or failure to recognize tags) produces positive bias in the Lincoln-Petersen estimator. If the probability of tag loss can be estimated [e.g., via a double-tagging study as described in Seber (1982)], an improved estimate of population size is given as the product/Q0.
14.1.4. Study Design A design for a two-sample capture-recapture study should produce precise and unbiased estimates of abundance when the underlying model assumptions are met. Of particular concern is the closure assumption, which is influenced by the time period separating the two capture occasions. Deaths, recruitment, and movement in and out of the population are much more likely to occur over long time periods. Thus, there should be only a short time period separating the two sampling occasions for most populations. To avoid trap mortality at the first sampling occasion, also a violation of the closure assumption, traps should include sufficient bait to keep animals alive while they are in the traps. In areas experiencing high temperatures, trap covers should be used to shield traps from direct sunlight, and traps should not be set during the hot periods of the day. If trap mortality does occur on the first sampling occasion, then the animals experiencing mortality should be removed from the initial computations and not be included in the n I statistic. The number of trap deaths can be added to the estimated population size subsequently, to obtain an estimate of the pretrapping population size. The variance of the adjusted population estimate is unchanged by the addition of a known number of trap deaths. Also important is the assumption of equal capture probabilities, recognizing that in reality this assumption is seldom if ever met exactly. If capture probabilities are likely to vary with visible characteristics of captured animals ( e.g., age, sex, weight), then samples can be stratified and stratum-specific estimates computed. The distribution of sampling devices (e.g., traps, nets) relative to the distribution of animals can be an important influence on the heterogeneity of capture probabilities, and one should avoid leaving some ani-
295
mals with small probabilities of capture and other animals with high probabilities. Even or uniform trap spacing is often desirable, with multiple traps per average home range size of the studied species. Of course, there sometimes are not enough traps to allocate multiple traps per home range over the entire area of interest. In such cases, division of the sampled area into quadrats, with random allocation of traps to quadrats at each of the two sampling occasions, should help equalize the underlying capture probabilities. As noted above, a special case of unequal capture probability involves a behavioral trap response. In this situation, animals caught in the first period have either a lower (trap-shy) or higher (trap-happy) probability of being caught in the second period compared to unmarked animals. Prebaiting before the first sampling period can reduce a trap-happy response, and minimization of handling time in the first period may reduce trap-shyness. We note that although the Lincoln-Petersen estimator is held to require equal capture probabilities of all animals within each sample, certain kinds of heterogeneity are allowed. If the animals exhibit heterogeneous Capture probabilities, yet the capture probabilities for an individual in the two sampling periods are completely independent (so animals with a relatively high capture probability in the first period do not necessarily have a high capture probability, again in the second period), the Lincoln-Petersen estimator still provides an unbiased estimate of population size (e.g., Seber, 1982). This observation has led to designs involving different capture methods for the two sampling occasions. If the initial marked animals are obtained as a random sample, then the second sample can be highly selective and still yield an unbiased estimate of abundance (Robson, 1969; Seber, 1982). For example, sometimes it is possible to use hunting or fishing as a way of obtaining the recapture sample for the LincolnPetersen estimator. In addition to closure and homogeneous capture rates, the assumption of no tag loss is required for Lincoln-Petersen estimation. This assumption is likely to be met for short-term studies for which the LincolnPetersen estimator typically is used. Tag losses can be investigated by the double-tagging of individuals with two standard tags or with a single standard tag and a more durable "permanent" tag. In this way, the loss of standard tags can be recognized, estimated, and accounted for in estimating population size (e.g., see Seber, 1982). While attempting to meet underlying model assumptions, study designs also should focus on obtaining precise abundance estimates. Precision increases with increasing capture probabilities, so design efforts
296
Chapter 14 Mark-Recapture Methods for Closed Populations
should be directed at catching a large proportion of animals in the sampled area. There are many ways of influencing capture probability, depending on the capture methods used. Robson and Regier [1964; reprinted in Seber (1982)] provided plots of sample sizes (n 1, n 2) needed to achieve Lincoln-Petersen abundance estimates with specified levels of accuracy for different population sizes. Robson and Regier (1964) presented an approach to optimal allocation of effort to the first and second samples as a function of the relative costs of the two types of sampling. Because two-sample studies are typically of short duration, it often is possible to conduct a pilot study to obtain an idea of capture probability and population size. Information about these parameters then can be used to design a more comprehensive study with the desired precision.
of animals exhibiting each possible capture history o~. For example, Xl01 denotes the number of animals caught at the first and third sampling occasions of a three-sample study. The counts x~ can be collapsed further into summary statistics for estimating parameters in specific capture--recapture models. There are assumptions underlying capture-recapture models for closed populations: (1) the population is closed to additions (via birth and immigration) and losses (via death and emigration) during the course of the study, (2) marks are neither lost nor overlooked by the investigator, and (3) capture probabilities are appropriately modeled. The first two assumptions are identical to the assumptions for Lincoln-Petersen estimation. The third assumption generalizes the Lincoln-Petersen assumption of equal capture probability.
14.1.5. Example
14.2.2. Modeling Approach
Skalski et al. (1983) reported results from a study of Nuttall's cottontail rabbits (Sylvilagus nuttallii) in central Oregon, in which 87 cottontails were captured and then released after their tails and hind legs were marked with picric acid dye. A follow-up sample yielded 14 animals counted on a drive count, and 7 of these were marked. Numbers of animals exhibiting each possible capture history were X l l -- 7, X01 = 8 0 , and x01 = 7. Thus, the summary statistics were n I = 8 7 , n 2 = 14, and m 2 = 7. The Chapman estimator, Eq. (14.3), for these data is/q = 164, with estimated variance from Eq. (14.4) of v~(/~ r) = 1283.33 and standard error SE(/~/) = V ' v ~ ( / ~ = 35.82.
We consider here a number of models that make different assumptions about the sources of variation in the capture probabilities, the primary parameters needed to model capture-recapture data for closed populations. To illustrate, consider the sampling of a closed population on three occasions, with unique marking of individuals so that individual capture histories can be recorded. For this situation there are 23 = 8 possible capture histories {i,j,k}, with the binary indices i, j, and k indicating capture outcome for the three sampling occasions:
14.2. K-SAMPLE CAPTURE-RECAPTURE MODELS 14.2.1. Sampling Scheme and Data Structure Here we consider capture-recapture models for sampling situations with K > 2 sampling occasions. An example might involve the trapping of a small mammal population for five consecutive nights. At each sampling occasion, previously uncaptured animals are marked with individually identifiable tags, and the identification codes of previously marked animals are recorded. Individual marks or some other scheme permitting reconstruction of the individual capture histories is required, so that the complete capture history of each animal encountered can be known unambiguously. The data from a K-sample capture-recapture study can be organized in an X matrix as shown in Eq. (14.1) and summarized in statistics x~ denoting the number
{1, 1, {1, 0, {0, 1, {0, 0, {1, 1, {0, 1, {1, 0, {0, 0,
1}, capture 0}, capture 0}, capture 1}, capture 0}, capture 1}, capture 1}, capture 0}, capture
all three times first time only second time only third time only first two times only last two times only first and third times only at no time
Let Xijk be the number of individuals with capture history {i,j,k}, where ~i,j,k Xijk = N. If probabilities for these capture histories are the same for all individuals in the population, the appropriate statistical model is a multinomial distribution
P(xijkIN, ,rijk) =
N! x~ I I ~ij~, IIi,j,k Xijk! i,j,k
with eight cell probabilities, "rrijk, where the subscripts representing sampling period take a value of I (indicating capture) or 0 (indicating noncapture). Thus the fully parameterized model includes eight parameters: the population size N and seven of the eight probabilities for capture histories [the eighth probability is given by 1 - (sum of the other seven)].
14.2. K-Sample Capture--Recapture Models Additional assumptions about the capture history probabilities can lead to model simplification. With the assumption of independence of capture events (i.e., no trap response) the probabilities associated with the different capture histories can be expressed as functions of time-specific capture probabilities. For example, the probability of catching an animal on all three occasions can be written as 11"111 = PlP2P3, where Pl, P2, and P3 are the probabilities of capture on occasions 1, 2, and 3, respectively (Table 14.1). This results in a model with four independent parameters (N, Pl, P2, P3), down from the original eight. A further assumption about equiprobable capture across periods (Pl = P2 = P3 = P) leads to a model with only two parameters (N and p). Reductions in model complexity also are possible under an assumption of differences between capture probabilities for marked and unmarked individuals. Under this scenario the probability structure of the model is written in terms of the probabilities Pc for first capture and Pr for recapture (Table 14.1). This assumption allows the number of model parameters to be reduced to three (N, Pc, Pr)" With the additional assumption of independence of capture events (Pr = Pc = P) the model again reduces to one containing only two parameters (N and p). In general, a fully parameterized model for K sampling periods requires 2 K parameters (population size N and 2 K - 1 of the probabilities corresponding to 2K possible capture histories). The corresponding model with an additional assumption of independent captures, allowing for temporal variation in capture probabilities (pj), requires only K + 1 parameters. On the
TABLE 14.1 Possible Capture Histories and Associated Probabilities a Capture
Probability
history
Mo b
Mb a
Mt c
111
p3
PlP2P3
Pcp2r
110
P2( 1 -- P)
PlP2( 1 -- P3)
PcPr( 1 --Pr)
101
P 2(1 -- P)
Pl( 1 -- P2)P3
PcPr( 1 -- Pr)
100
p(1 -- p)2
p1(1 -- p2)(1 -- P3)
Pc( 1 -- PF)2
011
p2(1 -- p)
(1 -- Pl)P2P3
(1 -- Pc)PcPr
010
p(1 -- p)2
(1 -- p1)P2(1 -- P3)
(1 -- pc)pc(1 -- Pr)
001
p(1 -- p)2
(1 -- pl)(1 -- P2)P3
(1 -- pc)2pc
000
(1 -- p)3
(1 -- pl)(1 -- p2)(1 -- P3)
(1 -- pc )3
other hand, the assumption of equiprobable captures across time, allowing for different probabilities of marked and unmarked individuals, results in a model requiring only three parameters. Finally, the addition of an assumption of both independent and equiprobable captures always requires just two parameters. For example, a model for four sampling periods requires either 16, 5, 3, or 2 parameters, depending on the assumptions of the model, whereas a model for five sampiing periods requires either 32, 6, 3, or 2 parameters. Thus, the impact of additional simplifying assumptions is exponentially greater as the number of sampling periods increases. The broadest possible class of models allows for separate probabilities Pij for each individual i and each capture period j. The models below allow for behavioral responses to trapping, differences in capture probabilities over time, and even heterogeneity in capture probabilities among individuals. For example, one can model capture probabilities so that capture events are independent (no trapping response) and equiprobable across trapping periods (no temporal variation in trapping probabilities), but specific to each individual in the population. The assumption of distinct capture probabilities for each individual is referred to as heterogeneity of capture probability. This source of variation is distinct from time-specific variation in capture probabilities, referred to as temporal variation. It also is distinct from a response to trapping, for which the probabilities of capture are the same for all marked individuals and the same for all unmarked individuals, but differ between the two groups. The latter effect is referred to as behavioral response. These three potential sources of variation in capture probability represent key elements in the modeling and estimation of closed populations (Pollock, 1974; Otis et al., 1978; White et al., 1982). Statistical modeling of closed populations based on multiple-recapture data is essentially an exercise in the comparison of models incorporating the various combinations of these three assumptions. Each combination of assumptions results in a distinct parameterization of the capture probabilities, and the challenge is to sift through the associated models to find one that best represents the sample data while minimizing model complexity. Conceptually, eight models can be defined:
M0 a U n d e r d i f f e r e n t m o d e l s in a t h r e e - s a m p l e c a p t u r e - r e c a p t u r e s t u d y of a c l o s e d p o p u l a t i o n . b p = c a p t u r e probability. Cpj = c a p t u r e p r o b a b i l i t y for s a m p l i n g p e r i o d j. a Pc = c a p t u r e p r o b a b i l i t y for u n m a r k e d a n i m a l s ; Pr = c a p t u r e p r o b a b i l i t y for m a r k e d ( r e c a p t u r e d ) a n i m a l s .
297
Mb Mt
Neither behavioral nor temporal variation nor capture heterogeneity (model parameters: N, p). Behavioral response only (model parameters: N, Pc, Pr)" Temporal variation only (model parameters: N, pj, j = 1.... ,K).
298
Chapter 14 Mark-Recapture Methods for Closed Populations
Individual capture heterogeneity only (model parameters: N, Pi, i = 1, ..., N). Mtb Behavioral and temporal variation only (model parameters: N, Pcj, Pry, j = 1, ..., K). Mbh Behavioral response and capture heterogeneity only (model parameters: N, Pci, Pri, i = 1, ..., N). Mth Temporal variation and capture heterogeneity only (model parameters: N, Pij, i = 1, ..., N, j = 1 ..... K). Mtbh Behavioral response, temporal variation, and capture heterogeneity (model parameters: N, P cij, Prij, i = 1.... , N, j = 1, ..., K).
Mh
The models M 0, Mb, and M t all possess MLEs, but additional assumptions or alternative approaches are required for estimation with models Mh, Mbh, Mth, Mtb, and Mtb h. MLEs for model Mtb can be obtained by assuming a relationship between the time-specific initial capture probabilities (Pcj) and recapture probabilities (Pry) (see Otis et al., 1978; Rexstad and Burnham, 1991). Estimates for models Mh, Mbh, and Mth can be obtained by assuming that capture probabilities for individuals are random samples of size N from an underlying distribution of probabilities (Pollock, 1974; Burnham and Overton, 1978, 1979; Chao, 1987) or by using an approach based on the concept of sample coverage (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). If time effects on capture probabilities are known (as when temporal variation is associated with relative trapping effort) it is even possible to obtain coverage-based estimates under model Mtb h (Lee and Chao, 1994). Pledger (2000) has used a finite mixture approach to obtain estimates under all four heterogeneity models (also see Norris and Pollock, 1996). Thus, with adequate data and appropriate parametric restrictions the parameters of all eight models can be estimated, and the models can be tested for goodness of fit. In addition, M b, M h, and M t can be compared to M 0, and Mbh can be compared to M h as an aid in model selection. Operationally, the idea is to do as follows:
6. Compare different models to identify the "best" model based on between-model tests, goodness-offit, and parameter parsimony. For the models for which MLEs can be determined, maximization of the likelihood function can be thought of as a two-step process. Let p represent the vector of capture probabilities defining a capture-recapture model, with N again denoting population size. The likelihood function can be expressed as L(N, p]{x~}), where the set {x~}denotes the number of animals exhibiting each capture history. Maximization of L(N, pl{x~}) is accomplished in program CAPTURE (Otis et al., 1978; Rexstad and Burnham, 1991) using the following general approach: 9 Consider N to be fixed, and choose the value ]3 maximizing L(N, p]{x~}) conditional on N. Let ~ denote the (conditional) KILE of p. In all cases for which the MLE can be described in closed form, a mathematical expression can be derived for fi in terms of N. 9 Incorporate ~ into the likelihood function, and choose N maximizing L(N, fi(N)]{xJ). Because fi is a function of N, the likelihood function now involves only the single variable N. With a single exception,/~ must be determined numerically, and because only the single parameter N is involved, this is a relatively easy numerical problem. m
14.2.3. E s t i m a t i n g P o p u l a t i o n S i z e
14.2.3.1. Constant Capture Probability~Model M 0 The simplest K-sample model assumes no variation in capture probability among animals or sampling occasions, i.e., Pij = P for all i and j (Darroch, 1958; Otis et al., 1978). Model M 0 contains only the two parameters, p and N. It is straightforward to show that the joint probability distribution for the data under model M 0 can be written as =
1. "Model" the capture probabilities by incorporating capture heterogeneity, behavioral response, a n d / o r temporal variation into the parametric structure. 2. Identify the probability density function that incorporates this parametric structure. 3. Incorporate the capture-recapture data {x~} in the probability density function, thereby identifying the likelihood function. 4. Maximize the likelihood function by choosing the appropriate estimates for the parameter values. 5. Calculate standard errors and confidence intervals for the estimates of population size and other parameters.
P({x~~
N!
p n.(1 - p)KN-n.,
P) [1-[0xo~!](N_ MK+I)!
(14.6)
where K /'/ -- ~ Y/j j=l is the total number of captures, and MK+ 1 is the total number of unmarked individuals caught during the study. The MLE ]~ for the capture probability under model M 0 can be derived by differentiation of the likelihood function in Eq. (14.6). In this case,/3 is just the number
14.2. K-SampleCapture-Recapture Models of captures divided by the number of opportunities for capture, or p = n/KN.
be missed (not caught) on each sampling occasion of the study in order to be missed for the entire study. 14.2.3.3. Behavioral Response--Model M b
Substituting this expression into the likelihood function and maximizing with respect to N yields the MLE /~ for population size. 14.2.3.2. Temporal VariationmModel M t
This model has a long history (Schnabel, 1938; Darroch, 1958) and is usefully viewed as the K-sample analog of the model underlying the Lincoln-Petersen estimator. Under model Mt, each animal has the same capture probability on any given sampling occasion (Pij = Pj for all i), but capture probabilities can vary from one occasion to the next. The model has K + 1 parameters, N and Pl, ..., PK (Table 14.1). The joint probability distribution for the data under model M t can be written as N~
P({x~o}]N, p)
299
= [l-I~o x 0 0 [ ] ( N - MK+I)[
(14.7)
The behavioral response model (Pollock, 1974; Otis et al., 1978) incorporates change in capture probability as a result of previous capture. Thus, captured animals not previously captured exhibit capture probability Pc, whereas marked (recaptured) animals exhibit capture probability Pr" The response may be either trap-happy (increased probability of capture after initial capture, P r > Pc) or trap-shy (decreased probability of capture after initial capture, Pr < Pc)" The model includes only the three parameters N, Pc, and Pr (Table 14.1). To describe the likelihood function for this model, let mj be the total number of marked animals caught on sampling occasion j, with Mj the number of marked animals in the population at the time of sampling occasion j. The probability distribution for model M b can be described in terms of the total number of recaptures K m =~mj j=2
K
• l-I pTJ(1 - pj)N-nj, j=l
where p is the vector of capture probabilities, Pl, ..., PK. Thus, the statistics needed for estimation are simply the number of animals caught on each sampling occasion (nj) and the total number of individuals captured m
during the study, the total number of marked individuals at the completion of the study (MK+I), and the sum (over all occasions) of the number of marked animals available for capture at each capture occasion K
M = 7_, Mj. j=2
(MK+I).
The MLEs/~j are determined by differentiating the likelihood function, Eq. (14.7). It is easy to show that /~j is just the number of animals captured in each period divided by the number in the population:
The corresponding probability density function is N! cMK+1 P({x~o}lN, Pc, Pr) = [l-I00 x~o!](N - MK+I)[ p X (1 --
~j = n j / N for j = 1, ..., K. The MLE of N is determined by substituting these expressions into the likelihood function and maximizing with respect to N. We note that in the special case of K = 2, the estimator/~/is simply the Lincoln-Petersen estimator [Eq. (14.2)]. Darroch (1958) showed that N could be estimated under model M t by solving the equation
pc) KN-MK+I-M
• prm'(1
-
(14.9)
pr)M.-m..
Under model M b, the MLE of the probability of first capture Pc is determined from Eq. (14.9) as the total number of first captures over the course of the experiment divided by the number of first capture opportunities: Pc = MK+I/(KN -- M.).
1
MK+I -- ~ N
1 --
(14.8)
for N. The left side of Eq. (14.8) estimates the probability that an animal is not caught during the study. The right side of Eq. (14.8) is the product of estimates of not being caught on each sampling occasion of the study (i.e., products of 1 - i0j). Thus, an animal must
The MLE for Pr is the total number of recaptures divided by the total number of potential recaptures: Pr = m . / M . Substituting these expressions into the likelihood function and maximizing with respect to N produces the MLE of N.
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Chapter 14 Mark-Recapture Methods for Closed Populations
Under likelihood Eq. (14.9), the estimation of N depends only on initial captures, and recaptures are used only~ for estimation of Pr" Because of the dependence of N only on first captures, estimation of population size under the behavioral response model is equivalent to estimation under a removal model (e.g., Zippin, 1956, 1958), in which animals are removed from the population on initial capture (e.g., as in snap-trap surveys of small mammals).
14.2.3.4. Heterogeneity among Individuals--Model
Mh
Under model Mh, there is no temporal variation in capture probabilities and no behavioral response associated with initial capture. However, every individual animal in the population is permitted to have its own capture probability independent of that of every other individual, i.e., Pij = Pi for all j. The model is thus parameterized with N capture probabilities Pl, ..., PN as well as population size N, for a total of N + 1 parameters. The large number of model parameters led Burnham and Overton (1978) to consider alternatives to maximum likelihood estimation for this model. Their approach was to conceptualize the vector of capture probabilities {Pi} as a random sample of size N from some probability distribution F(p) defined on the interval [0,1] (Burnham and Overton, 1978, 1979; Otis et al., 1978). The corresponding statistical model can be described in terms of the number fj of animals caught on exactly j occasions: N!
P(fl .... , &IF)
= [I_[K=I fj!](N
K
-
MK+I)I'ITN-MK+Ij=lI-I Try, 9
where "rrj
= fo
)!j!pJ(1 - p)K-j dF(p).
(14.10)
where k denotes the "order" of the jackknife estimator and the ajk are constants generated by the jackknife procedure (see Appendix F). Each order k of the jackknife generates a different set of constants ajk and thus a different estimator /qk (see Burnham and Overton, 1978). Burnham and Overton (1979) recommended a series of statistical tests for selecting the appropriate order jackknife estimator for any data set. If the appropriate order jackknife lies between the values k and k - 1, an interpolation algorithm is then used to compute an estimate of N lying between/Qk and/~k- 1 (Burnham and Overton, 1979). Usually, k is chosen to be no greater than 5. Although the Burnham and Overton (1978, 1979) jackknife estimator is the most commonly used approach in animal abundance estimation under model M h, other estimators also have been proposed. For example, Pollock and Otto (1983) proposed a momentbased bias-corrected estimator, Smith and van Belle (1984) used a bootstrap estimator, and Chao (1987, 1988, 1989) introduced a moment-based estimator for use with sparse data. In what follows we describe in somewhat greater detail two additional approaches to estimation under M h. Chao et al. (1992), Chao and Lee (1992), and Lee and Chao (1994) have proposed estimators based on the idea of sample coverage C, defined as the sum of the individual capture probabilities for animals that are caught as a proportion of the total of individual capture probabilities for all N animals in the population. If all individuals in the population have the same constant or time-specific capture probabilities (as in models M 0 or Mt), then the sample coverage effectively estimates the probability that an animal is caught during the study. Thus, an estimate of the sample coverage can be used to estimate population size as = MK+I/C
The cell probability "rrj in Eq. (14.10) can be viewed as the average probability that an individual is caught exactly j times. Burnham and Overton (1978) considered estimation in the case where F(p) is the class of beta distributions, but this approach was found not to be satisfactory. Instead, they used an estimation approach based on the generalized jackknife statistic (Quenouille, 1949, 1956; Gray and Shucany, 1972), in which MK+ 1 is viewed as a naive estimator of N, and bias reduction is accomplished using a linear function of the capture frequencies fi. This approach leads to estimators of the form K
~I k = ~ ajkfj, j=l
(14.11)
(see Darroch and Ratcliff, 1980; Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). Note that Eq. (14.11) is an example of the canonical estimator (Eq. 12.1). Estimators for C can be constructed from capture frequency data (Good, 1953; Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). For example, a widely used estimator is K
= 1 - fl/j
14.12
and bias-corrected versions are available (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). The estimator in Eq. (14.11) is negatively biased when there is heterogeneity in capture probability among the members of the population, with the magnitude of the bias a function of the coefficient of variation
301
14.2. K-Sample Capture-Recapture Models of the capture probabilities. This coefficient of variation can be estimated as a function of the capture frequencies fj and used in turn to estimate population size in the face of heterogeneity (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994) (also see model Mth below). A disadvantage of the jackknife and sample coverage estimators is that they are not maximum likelihood estimators and thus are not easily evaluated using familiar likelihood-based approaches. For example, likelihood ratio tests between models and model selection criteria such as Akaike's Information Criterion (AIC) are not available for testing sources of variation or selecting parsimonious models. Thus, nonparametric maximum likelihood estimators recently proposed by Norris and Pollock (1995, 1996; also see Agresti, 1994) are promising. This approach considers the joint estimation of N and a generating distribution F for the capture probabilities. The generating distribution is based on a finite mixture model in which the population is viewed as being composed of some finite (hopefully small) number of groups of animals having similar capture probabilities. The number of groups, the proportions of animals in each group, and the capture probabilities for the different groups are unknown and must be estimated. The approach to estimation involves cycling through each integer n between MK+I and some predetermined upper bound on population size, and, for each n, using the EM algorithm (Dempster et al., 1977) to obtain the MLE of F. The nonparametric MLE is then the (n, F) pair that yields the largest value of the likelihood function. This approach is computationally intensive, but is very general, yields MLEs, and thus has considerable promise. Pledger (2000) recently considered a somewhat different approach that utitlizes finite mixture models to deal with heterogeneous capture probabilities. Instead of estimating the number of groups in the mixture distribution directly within a single model, she proposed using multiple models defined by specific numbers of groups. Simulations and work with actual data sets indicate that two-group distributions frequently provide parsimonious models and estimators with good properties (Pledger, 2000). Pledger (2000) has derived estimators for all eight models of Otis et al. (1978). This estimation and modeling approach holds great promise, and we anticipate its becoming a standard approach for dealing with models for closed populations. Many of the competitor estimators under model Mh, including the jackknife estimator of Burnham and Overton (1978, 1979), the moment-based bias-corrected estimator of Pollock and Otto (1983), the bootstrap estimator of Smith and van Belle (1984), the moment-
based estimator of Chao (1987, 1988, 1989), the sample coverage estimators of Chao et al., (1992), Chao and Lee (1992), and Lee and Chao (1994), and the nonparametric MLE of Norris and Pollock (1996), have been investigated via simulation studies. These investigations have been documented in Otis et al. (1978), Norris and Pollock (1996), and the papers cited above. Some of these estimators perform especially well in specific sampling situations and poorly in other situations. On the other hand, the jackknife estimator of Burnham and Overton (1978, 1979), the oldest widely used estimator for this model, performs reasonably well in a variety of situations based on various simulation results. If the investigator can identify covariates (e.g., a size variable) associated with variation in capture probability among individuals, it is possible to use this additional information in estimation under a special case of model M h. Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) all considered modeling capture probability as a linear-logistic function of individual covariates, e.g., as e f30+ f31xi
Pi = 1 +
e ~O+f31xi'
where 130 and ~1 are parameters to be estimated and x i is the covariate value for individual i. The unconditional approach of Pollock et al. (1984) that includes N in the likelihood requires the grouping of covariates into a finite number of discrete classes and can be implemented using the program LINLOGN (Hines et al., 1984). The conditional approaches (conditional on MK+ 1 animals being captured) of Huggins (1989, 1991) and Alho (1990) do not include N in the likelihood and permit the estimation of individual capture probabilities Pi using continuous covariates. Estimation of abundance following the conditional approach is based on the estimator MK+I
19= i~1 1 . p';' where ~3~is the estimated probability that individual i was caught at least once during the study: K
/~* = 1 - ]-I(1 -19i) j=l
= 1 - (1 -/~i) K. The above abundance estimator is of the general form described by Horwitz and Thompson (1952) and, in the case of equal capture probabilities for all individuals, Pi -- P, is identical to the canonical estimator of Eq. (12.1). The conditional approach of Huggins (1989,
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Chapter 14 Mark-Recapture Methods for Closed Populations
1991) and Alho (1990) is implemented in program MARK (White and Burnham, 1999). In situations in which the variation in capture probability among individuals is closely associated with easily measured covariates, the models of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) should be useful in estimating population size. As with the finite mixture models of Norris and Pollock (1995, 1996) and Pledger (2000), these covariate models have the advantage of yielding MLEs and permitting likelihood-based inference and model selection.
14.2.3.5. Behavioral Response and Individual
Heterogeneity--Model Mbh Thus far we have considered temporal variation, behavioral response, and heterogeneity singly. However, it is also useful to consider sampling situations in which capture probabilities incorporate multiple sources of variation. For example, model Mbh includes both behavioral response and heterogeneity among individual animals. Thus, every animal in the population is assumed to have a specific pair of capture probabilities: Pci, the capture probability if individual i has not been captured previously, and Pri, the capture probability if individual i has been caught at least once. We assume that the pairs (Pci, Pri) are a random sample from a bivariate distribution F(pc, Pr)" Under the most general formulation, this model includes capture probabilities Pci and Pri for each of the N animals, along with population size N, for a total of 2N+1 parameters. Assuming independence of initial and subsequent capture probabilities, the probability density function can be factored as F(pc, Pr) = FI(Pc)F2(Pr). In this instance, all of the information needed to estimate N is provided by initial captures (as was the case for model Mb). To estimate parameters for this model, define uj, j = 1, ..., K, as the number of unmarked animals caught on sampling occasion j. If F l(p) is the unknown distribution of initial capture probabilities (the subscript c is dropped for this development), we can write the probability distribution of the unmarked captures as P ( u 1, ...,
N~
UKIF 1) -~
~I-I~.= 1 uj!~(N - M K+I) !
ta )( / j=l
where
j=l
~rj = El(1 - p)j-lp-] 1
= f (1 - p)j-lp dF l(p). 0
Estimation under this model can be accomplished by first transforming the K parameters -rrj into a new set of parameters pj via the relationship -try = p j j-1 1-Is=1(1 -- Ps), where pj is the average capture probability of individuals that have not been captured prior to the jth sampling occasion. Otis et al. (1978) based estimation on the assumptions that Pl ~ P2 ~ "" ~ PKand (Pl -- P2) > (P2 -- p3) > "" > ( F K - 1 -- FK)" The first assumption captures the idea that individuals with the high initial capture probabilities tend to be caught in the first sample, animals with slightly lower capture probabilities tend to be caught next, and so on until primarily animals with relatively low initial capture probabilities remain uncaught in the later samples. The second assumption is that differences between the average capture probabilities of animals caught in adjacent samples tend to be largest in the initial sampling periods and decline over time. Estimation involves sequential testing for differences among the pj. The first test is for equality of all the ~j. If this hypothesis is not rejected, one concludes that heterogeneity is not important and model Mb is appropriate for the data. If the hypothesis is rejected, then one next allows Pl to differ and tests for equality of the remaining pjs, P2 = P3 . . . . . FK" Sequential testing continues until it is concluded that the final K - r capture probabilities do not differ significantly, where r is the number of initial pjs that are modeled separately (r -~ K). Estimation of N is based on the resulting model. Several other estimators for model Mbh were considered by Pollock and Otto (1983). One of these is K-1
1~ = ~
u i + KUK,
(14.13)
j=l
based on the generalized jackknife statistic of Gray and Shucany (1972). Estimator (14.13) has performed well in simulation studies, especially with relatively small numbers of sampling occasions (e.g., K = 5). Lee and Chao (1994) presented an estimator for model Mbh based on sample coverage, and simulation results (Lee and Chao, 1994) indicated that it performed better than the generalized removal estimator of Otis et al. (1978) in terms of root mean squared error, but not as well as the jackknife estimator of Pollock and Otto (1983). Norris and Pollock (1995, 1996) and Pledger (2000) developed MLEs for model Mbh using the finite mixture model approach outlined above for model Mh, which simulations suggest is competitive with the other estimators referenced here (Norris and Pollock, 1996). As with the finite mixture MLE for model M h, the estimator has the advantage of placing model Mbh in the likelihood framework that is so useful for model evaluation. If capture probabilities can be modeled
14.2. K-Sample Capture-Recapture Models using individual covariates, then the logistic models of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) can be used with Mbh (see previous discussion under model Mh).
14.2.3.6. Temporal Variation and Individual HeterogeneitymModel Mth This model permits variation in capture probabilities Pij both over time, j = 1.... , K, and for individual animals, i = 1, ..., N. The likelihood under the model was described by Otis et al. (1978), but associated estimators were not developed until later (Chao et al. 1992; Lee and Chao, 1994; Pledger, 2000). Otis et al. (1978) viewed the set of capture histories {xij} as mutually independent random variables, with Pij described by Pij -- Pi ej, where 0 -< piej G 1. They viewed Pi as a random sample from some probability distribution F(p) and described the probability distribution of the observed sample {xij} as
303
model, Chao et al. (1992) provided guidelines for which estimators work best, depending on the magnitude of the sample coverage and the coefficient of variation of the capture probabilities. The preferred estimator of Chao et al. (1992) is implemented in program CAPTURE (Rexstad and Burnham, 1991). Pledger (2000) also has developed estimators under Mth , using the finite mixture model approach. If capture probability can be modeled using individual covariates, then the logistic modeling approach of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) can be implemented for this special case of Mth. In particular, capture probability Pij for individual i at time j can be modeled as e ~oj4- f3lXi
Pij = 1 +
e ~~
Again, the advantage of the mixture model and covariate approaches is that they permit likelihood-based inference and model selection.
P[xij ] = P[xij J MK+I]PEMK+I] ,
14.2.3.7. Temporal Variation and Behavioral ResponsemModel Mtb
with
P[{xq} J MK+I] =
h e;J
j=l
1 1
L i=1
fo
K pyi[I-I(1
-- pej)l-xij] d E ( p ) ,
j=l
This model assumes a behavioral response to capture and also permits temporal variation in both initial capture and recapture probabilities. The model contains 2K parameters: population size N, a vector Pc = {Pc1.... , PcK} of initial capture probabilities, and a vector Pr = {Pr2, "", PrK} of recapture probabilities. The corresponding joint probability distribution for the data can be written in several ways (Otis et al., 1978), including m
where Yi is the number of times animal i is captured and P[MK+ 1] is the probability distribution of the number of animals caught in the study, depending on the parameters N, el, ..., eK, and the distribution F(p). Chao et al. (1992) utilized coverage estimators for this model with the general form 1Q
-
-
MK+I
4
+
f1~]2
4'
n/f
11
N!
I~,T
'-tlXod l l~, Pc, Vrl = [l-]~ Xo~]](N -
MK+I)!
(14.14)
Xihpc~(l_Pcj)N_Mj+l](14.1B)j=l where ,~2 is an estimate of the coefficient of variation of the individual capture probabilities. The latter quantity can be estimated (Chao et al., 1992) as r~J(1 -
x
.~2 = max
K ~,k=l k(k--1)fk (IVIK~+I~ ~R-~ ..... \c ! 2~j=1 ~k=j+l njnk
O) M j - m j
[
j=2 1, 0 .
(14.15)
Chao et al. (1992) presented three estimators for C [including Eq. (14.12)] for use in Eqs. (14.14) and (14.15). In particular, they found that the estimator
4 = 1 -h-2f2/(K1) K Ek=l kfk performed well in simulation studies. Based on their simulation work with model Mth as the underlying
where Uj and mj are the numbers of unmarked and marked animals, respectively, that are caught at time j, and My is the number of marked animals present in the population at the time of sample j. Because the probability distribution, Eq. (14.16), contains only 2K-1 statistics, the 2K parameters of the model are not identifiable. On the other hand, estimation of N is possible if a relationship is specified between Pcj and Prj" Otis et al. (1978) considered the multiplicative relationship PFj = 0pcj, j = 2, ..., K, but concluded that a constant relationship between initial capture and re-
Chapter 14 Mark-Recapture Methods for Closed Populations
304
capture probabilities is not realistic (Otis et al., 1978). Rexstad and Burnham (1991) considered the relation.~1/o ; ..., ship Prj = Vcj , J = 2, K, where 0 -< Pcj I +
ak= Uk+-------~ 1
ej+lUj, \ ek+l / j-1 Uj+I
where
fo = 1 ~ - MK+ 1 Ck = 1 --
/'/k+l/ek+l
and
ul/el
and
2:maxlI,lllUluael,e2, u2
] 10}
for k = 1.... , K - 1, with M k + l the number of distinct animals captured in the first k samples; Simulations by Lee and Chao (1994) suggested that N ( K - 1) from Eq. (14.17) is the most appropriate estimator for population size when the coefficient of variation of the capture probability distribution is greater than 0.4 (i.e., in the face of substantial heterogeneity). Pledger (2000) considered estimation for model Mtbh using linear-logistic modeling in conjunction with her
c exp{19611nlI+vi2111'2} The lower bound of this confidence interval cannot be smaller than M K + 1, but the upper bound frequently is larger than upper bounds computed with the information matrix under the assumption of normality. Another approach to interval estimation makes direct use of the likelihood function, and the resulting intervals are frequently termed "profile likelihood intervals" [for general applications see Hudson (1971) and Venzon and Moolgavkar (1988); for capture-recapture see Otis et al. (1978) and Rexstad and Burnham (1991)]. The profile likelihood approach is based on
14.2. K-SampleCapture-Recapture Models lnL(0_), where 0 is a vector of parameters consisting of N and the capture probability parameters p (see Section 4.2.3 for general discussion). It reduces lnL(0) to a function of a single parameter (N) by treating the capture probability parameters as nuisance parameters and maximizing over them. The profile likelihood confidence interval then consists of all values of N for which the log-likelihood function evaluated at N is no more than 1.92 units from the maximum value of the log-likelihood function (the log-likelihood function evaluated at the MLEs, including N). The value 1.92 comes from the 0.95 quantile of the chi-square distribution, based on the generalized likelihood ratio test (Venzon and Moolgavkar, 1988; Rexstad and Burnham, 1991). Thus, profile confidence intervals include values of N that correspond to values of the likelihood function that are "close" to its maximum (Otis et al., 1978). D
14.2.5. Testing Model Assumptions A discussion of model assumptions is more involved with K-sample models than with the twosample Lincoln-Petersen estimator, because K-sample studies permit tests of underlying assumptions. Here we address both the testing of assumptions and the assessment of estimator performance when assumptions are violated. We focus on population closure and the absence of tag losses during the investigation. The third assumption (Section 14.2.1) of proper modeling of variation in capture probabilities is dealt with in Section 14.2.6 on model selection. 14.2.5.1. C l o s u r e
All of the models described in this section were developed under the assumption that the sampled population does not change during the course of sampling. We first consider tests of the closure assumption and then discuss consequences of its violation. 14.2.5.1.1. Tests for C l o s u r e
The most commonly used closure test (Otis et al., 1978) uses the null hypothesis H0: Pij = Pi, j = 1 . . . . , K, for all animals captured two or more times. The alternative hypothesis is that some capture probabilities were zero prior to initial capture or subsequent to final capture, because the animals arrived after the study began or departed before the study was completed. Thus, the alternative hypothesis is Ha: Pil -- Pi2 ..... Pir = 0 a n d / o r Pis -- Pi,s+l . . . . . PiK = 0, with r and s the first and last times of capture, respectively. Under H a we would expect the time between first and last capture to be less than under H 0. Otis et al. (1978)
305
developed a closure test based on the observed times between first and last capture for all animals caught at least twice, which is computed by program CAPTURE (Rexstad and Burnham, 1991). The test is sensitive to behavioral and temporal variation in capture probabilities (e.g., low capture probabilities at the beginning or end of a study can confound assessment). In addition, the test is not suitable for detecting situations in which animals emigrate temporarily during the middle of the study. Pollock et al. (1974) considered the testing of four hypotheses about time-specific variation in capture probabilities that are relevant to the closure assumption: (1) no mortality and no recruitment (complete population closure), (2) mortality but no recruitment, (3) recruitment but no mortality, and (4) both recruitment and mortality. Burnham (1997) considered the probability distributions under hypotheses 2, 3, and 4 above, and Stanley and Burnham (1999) have used these results to develop an overall test for population closure using time-specific capture-recapture data. The resulting chi-square test essentially tests the null hypothesis of complete closure (hypothesis 1 above, which corresponds to model M t) against the alternative hypothesis of a completely open population with both mortality and recruitment (hypothesis 4 above, which is the Jolly-Seber model to be described in Chapter 17). The overall test statistic of Stanley and Burnham (1999) can be decomposed into components that provide information about the nature of the closure violations. Under the first decomposition, one component represents a test of the null hypothesis of no recruitment (hypothesis 2 above) versus the alternative of the Jolly-Seber model (hypothesis 4). Another component tests null hypothesis M t (hypothesis 1) against the alternative of mortality but no recruitment (hypothesis 2). The chi-square test statistics for these two tests are independent, and their sum (also distributed as chisquare under the null hypothesis of closure) provides a test of null hypothesis 1 (M t) against alternative hypothesis 4 (Jolly-Seber model). Under the second decomposition of the test statistic (Stanley and Burnham, 1999), one component provides a test of the null hypothesis of no mortality (hypothesis 3) versus the alternative hypothesis of the Jolly-Seber model (hypothesis 4). The other component tests null hypothesis M t against the alternative of recruitment but no mortality (hypothesis 3). The chi-square test statistics for these two components also can be summed to obtain the overall closure test of Stanley and Burnham (1999). Thus the two decompositions have the same overall null (M t) and alternative (Jolly-Seber) hypotheses but involve different intermediate hypotheses.
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Chapter 14 Mark-Recapture Methods for Closed Populations
Stanley and Burnham (1999) provided information about the power of these test components to the alternatives of permanent and temporary emigration and immigration. Thus, behavioral response in the absence of migration can lead to false indications of closure violations, but some violations are still detectable even in the presence of trap response. Stanley and Burnham (1999) recommend that their closure tests be used in conjunction with, rather than instead of, the test of Otis et al. (1978). The null model of the Otis et al. (1978) test permits heterogeneity of capture probabilities, but is sensitive to time and behavioral variation. On the other hand, the null model of the Stanley and Burnham (1999) test permits temporal variation, but not heterogeneity or behavioral response. The closure tests of Stanley and Burnham (1999) are implemented in software CLOSTEST written for that purpose, whereas the test of Otis et al. (1978) is implemented in CAPTURE (Rexstad and Burnham, 1991).
14.2.5.1.2. Consequences of Closure Violation The consequences of violations of the closure assumption for estimates based on closed population models were reviewed by Kendall (1999). An interesting form of closure violation considers animals in a population to be a subset of animals in a superpopulation of size N ~ Members of the superpopulation move freely in and out of the sampled area, and animals in the sampled area at time j are essentially random samples with probability Cj from the superpopulation. Under these conditions the expected size of the population in the sampled area is a function of the size of the superpopulation and the probability ~'j: E(NjlN ~ = N~ To illustrate, consider model Mt, with pj now reflecting the conditional (on being in the sampled area at time j) capture probability. On assumption that the superpopulation is closed during the study, the M t estimator for capture probability now estimates the product r the capture probability for an animal in the superpopulation. Thus, the population size estimator under M t estimates the number of animals in the superpopulation. Time specificity can exist in r or p (or both), and the M t parameterization is still appropriate. If neither 9 nor p varies over time, then estimation for the superpopulation should be based on M 0. A different scenario for the violation of closure allows for the entire population to be available for capture at the first sampling occasion, but permanent emigration by some individuals can occur before the study is completed (emigration only). Alternatively, some animals could enter the population during the study period (immigration only). Under these scenarios, the K-sample model estimators are biased, and the partially open models of Darroch (1959; also see Jolly,
1965; Burnham, 1997) can be used for estimation (see Chapter 18). Yet another scenario corresponds to a migration stopover site, with animals entering the population during the study and then (potentially) emigrating before the study is completed. Again, estimates obtained under closed models are biased in this situation, and models for open populations should be used for estimation. One approach utilizes the idea of a superpopulation (Crosbie and Manly, 1985; also see Schwarz and Arnason, 1996), which permits direct estimation of the number of animals that were members of the population at some time between the first and last sampling occasions (see Chapter 18).
14.2.5.2. Tag Loss As with the Lincoln-Petersen two2sample estimator, tag loss induces a positive bias in N because capture probability following initial capture tends to be underestimated. In certain cases, it may be possible to recognize recaptures as animals that have been caught before (e.g., in small mammal studies, animals losing ear tags can be identified by torn ears), even though individual identification is not possible. It may be possible to reconstruct capture histories fairly reasonably in such cases. Otherwise, the behavioral response models M b and Mbh , which do not rely on recapture information, can be used to provide unbiased estimates if other assumptions hold true. Tag loss can be investigated with double-tagging studies, in which some animals are marked with two tags, either of the same or different types. Recaptures of double-tagged animals with only one tag provide evidence of tag loss, and numbers of recaptures with one and two tags provide the data needed to estimate tag loss (e.g., see Seber, 1982).
14.2.6. M o d e l Selection One strategy to guard against the failure to incorporate important sources of variation in a model would be to select the most general of available models. However, the sample data allow one to estimate a few parameters with greater statistical precision (at a potential cost in bias) or a greater number of parameters with less precision (but potentially less bias). One therefore faces a tradeoff between greater complexity, with the advantages to accuracy and realism it confers, against greater precision with the potential for informative inference that it confers. A useful approach is to select parsimonious models that achieve an acceptable tradeoff between bias and precision (Otis et al., 1978; Burnham and Anderson, 1992, 1998; Lebreton et al., 1992). In this sense the "appropriate model" can be viewed
14.2. K-Sample Capture-Recapture Models as "the simplest model that fits the data" (Otis et al., 1978). If all of the above models and their estimators were based on likelihood theory, we could use likelihood ratio tests and optimization criteria such as Akaike's Information Criterion (AIC) and its relatives (e.g., see Anderson et al., 1994; Burnham and Anderson, 1998) as tools to aid in model selection. If the finite mixture models of Pledger (2000) prove to be as useful as we suspect, then it may soon be possible to use AIC in model selection for the full set of closed models. However, the models of Otis et al., (1978) that include heterogeneity of capture probabilities (models Mh, Mbh , Mth , and Mtbh) do not fit easily into the standard likelihood framework, and model selection strategies must rely on other approaches than maximum likelihood. Here, we follow the approach of Otis et al. (1978) and Rexstad and Burnham (1991) for model selection, based on model goodness-of-fit tests and between-model tests.
14.2.6.1. Goodness of Fit The multinomial distributions in capture-recapture modeling can be used as a basis for assessment of model goodness of fit (see Section 4.3). For example, assume that one wants to test the fit of model M t to data from a capture-recapture study and that maximum likelihood estimation yields the estimates/~ = 200, ]~1 -- 0 . 2 5 , ]92 = 0 . 4 0 , and ]93 -- 0 . 3 0 for a study with three sampling occasions. The expected number of animals exhibiting the capture histories can be estimated with these values [e.g., E(Xll 1) = /~]911921~3 -- 6; see Table 14.1]. The difference between the observed numbers of animals with each capture history and the numbers expected under model M t then provides information about the likelihood that the data were actually generated by this underlying model. Program CAPTURE (Rexstad and Burnham, 1991) computes goodness-offit tests for models Mb, Mt, Mh, and Mtb. The computation of these statistics is described for all models except Mtb by Otis et al. (1978).
14.2.6.2. Between-Model Tests When MLEs can be computed for two nested models (i.e., one model is a special case of a second, more general model), then a likelihood ratio test can be used for comparative testing (Section 4.3.4). The null hypothesis of such a test is represented by the more restrictive model, and the alternative hypothesis is the more general model. The test is conditional on the more general model fitting the data and essentially addresses the question of whether the more restrictive model is adequate to represent the data (see Section 4.3.4). Program CAPTURE computes tests to compare models
307
M 0 vs. M b and M 0 vs. M t based on MLEs, although they are not computed as standard likelihood ratio tests (Otis et al., 1978). Program CAPTURE also computes tests to compare models M 0 vs. M h and M h vs. Mbh , though they are not based on MLEs.
14.2.6.3. Use of Discriminant Analysis for Model Selection In the absence of an optimization criterion such as AIC, it seems reasonable to base model selection on an examination of the results of the described goodness-of-fit and between-model tests. Otis et al. (1978) developed such a model selection procedure, which is included in program CAPTURE. The procedure utilizes data that were simulated under all eight general models for closed populations, with various test statistics and associated probability levels computed for each simulated data set. Discriminant function analysis (e.g., Cooley and Lohnes, 1971) then was used to develop a model classification function based on the test statistics and probabilities. The procedure subjects actual data sets to the various tests of program CAPTURE, with test results used as input data for the classification function to compute a score that is treated as a model selection criterion (McDonald et al., 1981). In a simulation study assessing the performance of their model selection algorithm, Otis et al. (1978) found that the algorithm performs well when capture probabilities are high, but performance declines rapidly as capture probability declines. Menkens and Anderson (1988) also assessed the performance of the CAPTURE model selection algorithm via simulation and noted that when the population and sample sizes are small, the underlying model generating the data is selected relatively infrequently. They concluded that when sample sizes are not large, it may be wise to pool data from multiple periods into two periods and use the Lincoln-Petersen estimator to estimate population size (Menkens and Anderson, 1988). Stanley and Burnham (1998) investigated possible improvements to the model selection procedure of program CAPTURE. Although they followed the same general approach as in the CAPTURE procedure, their methods differed in some important details. For example, they used not only linear discriminant function analysis but also multinomial logistic regression to develop the classification function. They also used a different vector of predictor variables, specifically the probabilities corresponding to between-model and other tests, as well as coefficients of variation of some of the capture-recapture statistics. Finally, they based their classification function not on the ability to select the underlying generating model but instead on the
Chapter 14 Mark-Recapture Methods for Closed Populations
308
root mean squared error of the resulting estimators. The resulting classifiers performed marginally better than that of plogram CAPTURE. In addition to exploring model selection, Stanley and Burnham (1998) investigated a model-averaging approach to estimation (Buckland et al., 1997), in which they estimated population size as 1~I = ~ , W kl~ k,
k where/~k is the abundance estimate from model k, and w k is the predicted probability associated with model k based on the multinomial logistic regression classifier. The associated variance estimator is va"~(/~) = [ ~
WkV'v~r(l~lk[~k)+~12,
k
where
By incorporating estimators and probability significance levels for multiple models, this estimator incorporates model uncertainty. Stanley and Burnham (1998) recommended considering implementation of the above model-averaging procedure in CAPTURE. The mixture models of Pledger (2000) place all eight basic closed-population models and several variants in the likelihood framework. One of the most important advantages of the likelihood framework is the ability to use AIC as a model selection criterion. Use of AIC also permits model averaging and allows for the incorporation of model uncertainty in variance estimates (Buckland et al., 1997; Burnham and Anderson, 1998; Stanley and Burnham, 1998). We thus expect the Pledger (2000) model set to become widely used in closed population estimation. 14.2.6.4. D i a g n o s t i c S t a t i s t i c s f o r Capture-Recapture Models
As mentioned above, model overparameterization leads to declining precision in all model estimators, with extreme overparameterization leading to parameter estimates containing so little information that they are essentially useless. For this reason it is important to select a model that includes the fewest parameters necessary to fit the data (see Burnham and Anderson, 1992, 1998). Overfitting of a model should be evident on investigation of the goodness-of-fit and model comparison tests, along with other diagnostic statistics. An overfitted model typically has quite wide confidence intervals for the model parameters, corresponding to a lack of precision in parameter estimates. The goodness-of-fit
statistic for an overfitted model indicates a good fit between model and data, but typically one or more reduced models also indicate a good fit. This suggests that a reduced model is adequate to represent the data, i.e., that the full model includes more parameters than necessary. Finally, the test statistics comparing an overfitted and a reduced model typically indicate that a reduced model compares favorably to the overfitted alternative, again suggesting that the reduced model does about as well as the overfitted model in representing the data. Of course, underparameterization of a model also carries risks. A model that fails to account for key sources of parameter variation may result in very precise but very biased results. For example, if model M 0 with constant capture probability is incorrectly used when capture probabilities are in fact highly heterogeneous (i.e., model M h is the "true" model), population size is precisely estimated but the estimate can be severely biased downward. Again, such a situation should be evident in the standard diagnostic statistics. Goodness-of-fit statistics typically indicate a poor fit for the model, and model comparisons indicate that a more fully parameterized model compares favorably to the reduced model, i.e., the more fully parameterized model does a better job in representing the data. Beyond the issues of overfitting and underfitting, certain patterns in the data are useful as diagnostics of particular models. Thus, the expected number of captures for model M0 is the same for all sampling occasions [E(nj) = Np], so the actual number of captures should be similar and show no trends over capture periods. The number of captures of unmarked animals should decline through the study according to E(uj) = Np(1 - p ) J - 1, whereas the captures of marked animals should increase according to E(mj) = Np[1 (1 - p ) J - 1]. These patterns are illustrated in Table 14.2, which shows results of a simulation in which 120 animals were subjected to a capture probability of p = 0.30 for each of seven sampling periods. Note that
TABLE 14.2
Summary of Simulated Capture Histories under M o d e l M0 a
Population data Occasion (j)
Measure 1
2
3
4
5
6
7
Animals caught
32
40
35
42
23
41
31
Newly caught
32
30
17
12
8
8
6
0
10
18
30
15
33
25
36
39
24
12
2
0
0
(nj) (uj) Recaptures (mj) Frequencies (fj)
a For a population consisting of 120 individuals, with capture probability p = 0.3.
14.2. K-Sample Capture-Recapture Models TABLE 14.3
Summary of Simulated Capture Histories under Model Mh a
Population data
Measure
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies 0~)
1 38 38 0 34
2 31 20 11 21
3 32 11 20 23
4 27 7 25 10
5 31 5 22 5
6 32 6 25 2
7 34 8 24 0
the n u m b e r nj of captures s h o w s no a p p a r e n t trend t h r o u g h time, w h e r e a s the n u m b e r uj of first captures decreases a n d the n u m b e r mj of recaptures increases as the s t u d y proceeds. U n d e r m o d e l M h the n u m b e r of captures again should be relatively constant over time. Animals with higher capture probabilities tend to be captured early in the study, so that the captures of u n m a r k e d animals should decline m o r e rapidly than u n d e r m o d e l M 0. A high p r o p o r t i o n of captured animals exhibits capture frequencies that are very low (e.g., fl) or very high (e.g., fK-1, fK) relative to expectations u n d e r m o d e l M 0. These general patterns are illustrated in Table 14.3, w i t h a simulated p o p u l a t i o n consisting of 60 animals w i t h p = 0.15 and 60 animals with p = 0.40. U n d e r m o d e l Mb, the n u m b e r s of u n m a r k e d animals in samples should decline over the study, as u n d e r M 0. U n d e r a t r a p - h a p p y response, m a r k e d animals have higher capture probabilities than u n m a r k e d individuals. Thus, the total n u m b e r of captures should increase with time according to E(nj) = N p r - N(1 - pc) j-1 (Pr - Pc), as increasing n u m b e r s of m a r k e d animals are exposed to traps and are recaptured (Table 14.4). U n d e r a trap-shy response, m a r k e d animals have lower capture probabilities than u n m a r k e d animals, so that the
Summary of Simulated Capture Histories under Model Mb a
Population data
Summary of Simulated Capture Histories under Model Mb a
Population data
a For a population consisting of 120 individuals. Sixty individuals have capture probability p = 0.15 and 60 individuals have p = 0.40.
TABLE 14.4
TABLE 14.5
309
Measure
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (~)
1 42 42 0
51
2 41 32 9 38
3 33 18 15 20
2 48 30 18 19
3 54 17 37 25
4 67 12 55 33
5 57 8 49 19
6 63 8 55 4
7 62 6 56 1
a For a population consisting of 120 individuals. Animals show trap-happy response with initial capture probability Pc = 0.30 and recapture probability Pr = 0.60.
6 29 5 24 0
7 22 1 21 0
a
total n u m b e r of captures should decline w i t h time as m a r k e d animals accumulate in the p o p u l a t i o n (Table 14.5). There typically are relatively more animals captured only once (fl) u n d e r a trap-shy response than u n d e r a t r a p - h a p p y response, given similar initial capture probabilities Pc. The p r i m a r y diagnostic for m o d e l M t is t e m p o r a l variation in the n u m b e r nj of animals caught per trapping occasion, reflecting t e m p o r a l variability in capture probabilities according to E(nj) = Npj. It often is possible to tell from a quick look at a data set w h e t h e r substantial t e m p o r a l variation is present, simply by examining the n u m b e r s caught. It s h o u l d be clear that different patterns of t e m p o r a l variation p r o d u c e different patterns in the capture histories and associated statistics. Table 14.6 s h o w s results of a simulation based on capture probabilities that increase until the m i d d l e s a m p l i n g occasion and then decline, w i t h the n u m b e r of captures reflecting this pattern. The patterns expected u n d e r the models with multiple sources of variation are more complicated and difficult to recognize. For m o d e l M b h , a t r a p - h a p p y response in the presence of heterogeneity still s h o u l d p r o d u c e an increase in the n u m b e r of captures (nj) t h r o u g h time, as animals in the p o p u l a t i o n become
TABLE 14.6
Measure
1 32 32 0 12
5 20 5 15 1
For a population consisting of 120 individuals. Animals show trap-shy response with initial capture probability Pc = 0.40 and recapture probability Pr = 0.20.
Summary of Simulated Capture Histories under Model Mt a
Population data
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (fj)
4 37 15 22 8
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (~)
Measure
1 22
22 0 35
2 37 29 8 41
3 35 18 17 19
4 54 20 34 10
5 23 8 15 3
6 37 8 29 0
7 21 3 18 0
a F o r a population consisting of 120 individuals with capture probabilities Pt = 0.20, 0.25, 0.30, 0.35, 0.30, 0.25, and 0.20.
Chapter 14 Mark-Recapture Methods for Closed Populations
310 TABLE 14.7
Summary of Simulated Capture Histories under Model Mbh a
Population data
Measure
Occasion (j)
1
2
3
4
5
6
7
Animals caught
28
34
39
47
39
41
40
Newly caught
28
20
17
16
9
5
2
0
14
22
31
30
36
38
28
19
17
19
10
3
1
Recaptures
(nj) (uj) (mj)
Frequencies (fj)
a For a population consisting of 120 individuals showing a traphappy response. Sixty individuals have initial and recapture probabilities of 0.15 and 0.20, respectively, and 60 individuals have initial and recapture probabilities of 0.40 and 0.53.
marked and thus have increased capture probabilities. Patterns in the data are more difficult to predict under a trap-shy response, but numbers of captures should decrease through time, or at least not increase. Tables 14.7 and 14.8 show results of simulations under model M b h with trap-happy and trap-shy response, respectively. Temporal variation in capture probabilities can be a dominant feature producing patterns in capture history data. Any of the multiple-factor models containing time as one of the sources of variation in capture probability (Mth , Mtb , and Mtb h) c a n produce temporal variation in the number nj of animals caught each sampling occasion. However, general patterns are difficult to predict, because they depend on the magnitude and specific pattern of temporal variation. 14.2.7. E s t i m a t o r R o b u s t n e s s
Even with testing, model diagnostics, and model selection algorithms, selection of an appropriate model is not guaranteed. It thus is important to determine
TABLE 14.8
Summary of Simulated Capture Histories under Model Mbh a
Population data Occasion (j)
Measure 1
2
3
4
5
6
7
(nj) Newly caught (uj) Recaptures (mj)
33
31
35
24
34
30
25
33
24
17
10
6
6
0
0
7
18
14
28
24
25
Frequencies (~)
38
41
22
4
2
0
0
Animals caught
a For a population consisting of 120 individuals showing trap-shy response. Sixty individuals have initial and recapture probabilities of 0.20 and 0.15, respectively, and 60 individuals have initial and recapture probabilities of 0.40 and 0.30.
how well a selected estimator performs when the underlying model on which it is based is not appropriate for the data. Estimator robustness has been investigated primarily by computer simulation, whereby data are generated under a particular model with specified parameters, and the estimates from different capture-recapture models are compared against the known parameter values. In a few instances it has been possible to examine estimator performance with field data for a population of known size (e.g., Greenwood et al., 1985; Manning et al., 1995). The results of these investigations indicate that the MLE for model M 0 is generally not robust to variation in capture probability. Heterogeneity of capture probability among individuals produces negative bias, as does a trap-happy behavioral response, whereas a trap-shy response yields positive bias (Otis et al., 1978). The estimator based on M 0 is somewhat robust to temporal variation in capture probability (Otis et al., 1978). Performance of the estimator for model M t is similar in many respects to that of model M 0. Heterogeneity and a trap-happy response produce negative bias in estimates of population size, whereas a trap-shy response produces positive bias (Otis et al., 1978). The magnitude of bias depends on the degree of heterogeneity a n d / o r the magnitude of the behavioral response. With model M b, valid estimates can be obtained only when K
~ (K + 1 - 2 j ) ( n j - mj) > 0 j=l
(Seber and Whale, 1970). This condition essentially reflects a requirement for "depletion" of the unmarked population through the marking of previously unmarked animals. Temporal variation in capture probabilities can produce pattern in the captures of unmarked animals that is unrelated to change in the number of unmarked animals available for capture, resulting in large biases in estimates of population size (Otis et al., 1978). Heterogeneity of capture probability results in a negative bias in abundance estimates, with the magnitude of the bias strongly dependent on the number of individuals with low capture probabilities (e.g., p < 0.10). As mentioned above, several different estimators have been proposed for model M h. The jackknife estimator of Burnham and Overton (1978, 1979) has been the most frequently used, and it consistently performs well with respect to model robustness. However, simulation studies by Burnham and Overton (1979) and Otis et al. (1978) indicate that the jackknife estimator can exhibit negative bias when some members of the
14.2. K-SampleCapture-Recapture Models population are essentially untrappable. Simulation results indicate relative robustness of the jackknife estimator to temporal variation and to behavioral response under some scenarios (e.g., Otis et al., 1978) although not all (e.g., Chao, 1989). Simulation studies based on small populations with low (p ~ 0.10) and heterogeneous capture probabilities led Rosenberg et al. (1995) to favor first-order and second-order jackknife estimators, but to question the use of higher order jackknife estimators even in cases where they are selected by the algorithm in program CAPTURE. The jackknife estimator performed the best among all estimators tested by Greenwood et al. (1985) on known populations of striped skunks (Mephitis mephitis). The jackknife estimator and the moment-based estimator of Chao (1989) outperformed other estimators for graytailed vole (Microtus canicaudus) population sizes of 60 and 90 animals (Manning et al., 1995). In contrast to the jackknife estimator, the momentbased estimator of Chao (1987, 1988, 1989) has performed well in simulations of heterogeneous populations with sparse data, the situation for which it was developed. The coverage estimators of Chao et al. (1992) and Lee and Chao (1994) also have been found to perform well, especially when sample coverage is relatively high (e.g., >50%). Finally, the nonparametric MLE of Norris and Pollock (1996) did not perform as well in simulation studies as some of the other estimators (including jackknife and coverage). Because the generalized removal estimator for model Mbh requires a substantial drop in numbers of new animals captured over the course of the study (Otis et al. 1978), certain patterns of temporal variation can cause the estimator to perform poorly. In simulations to evaluate the various estimators proposed for model Mbh, the jackknife estimator of Pollock and Otto (1983) performed fairly well [also see simulation results of Lee and Chao (1994) and Norris and Pollock (1996)], as did the coverage estimator of Lee and Chao [1994; also see Norris and Pollock (1996) simulation results] and the MLE of Norris and Pollock (1996). In a simulation study of their coverage estimator for model Mth , Chao et al. (1992) and Lee and Chao (1994) found that the estimator for model M t performed well when heterogeneity was relatively small (coefficient of variation of capture probability distribution ~ 0.4), but the estimator for model Mth performed better in the presence of substantial heterogeneity (Chao et al. 1992; Lee and Chao, 1994). The only simulation work of which we are aware on estimators for models Mtb and Mtbh involves special cases of these models where the pattern of temporal variation in capture probability is known (Lee and Chao, 1994).
311
14.2.8. Study Design The design of studies to estimate population size using K-sample capture-recapture should involve two general considerations. First, the study should be designed in such a way as to minimize violation of underlying model assumptions to the degree possible. Second, study design should focus on producing precise estimates. A key assumption of the capture-recapture approach in this chapter is that populations are closed to gains and losses over the course of a study. It thus is important to design studies with short duration, because shorter studies reduce the possibility of death, recruitment, and movement in and out of the population. Closed models frequently are used with daily sampling (e.g., small mammal trapping; mist-netting of songbirds) over 5- to 10-day study periods. Study timing also is relevant, because it is useful to avoid sampling during migration and during periods of recruitment or high mortality. Trap mortality is a violation of the closure assumption and should be reduced to the extent possible. When trap mortality does occur, there are at least two ways to deal with it (see Flickinger and Nichols, 1990). In studies with relatively small numbers of trap deaths, the capture histories of animals that die prior to the final day of capture can be removed from the data set, with estimation based on the reduced data set. Trap deaths can be added to the resulting population estimate in order to estimate the pretrapping population size. The variance of the adjusted estimate is the same as that for the estimate obtained from the capture-recapture model, because the number of trap deaths is known and does not add additional variance to the estimate. If trap deaths substantially reduce the number of recaptures, it may be necessary to use one of the removal models M b or Mbh. AS indicated above, the estimators for these models are based on initial captures only, so that trap deaths do not reduce the data used for estimation of pretrapping population size. However, elimination of trap mortality is preferable to the use of removal models, which restrict the analysis in not allowing for temporal variation in capture probabilities. In addition, the restriction with removal models to initial captures clearly reduces the data available for estimation, because recapture data are not used to estimate N, resulting in reduced precision. The assumption of no tag loss typically is not a problem with closed population estimators because of the short duration of such studies. It should be noted that all the models described above were developed for use with individual marks. The use of a single
312
Chapter 14 Mark-Recapture Methods for Closed Populations
"batch mark" yields data that cannot be analyzed fully, though the relevant statistics for some of the models described above can be obtained from batch-marking studies. However, batch marking does not permit adequate testing of model assumptions and precludes the use of many of the models. On the other hand, occasion-specific batch marks sometimes are applied so that at each capture, the previous capture history can be ascertained (e.g., see White et al., 1982). Finally, it should be noted that animals of some species are individually recognizable, so that observations and reobservations can be used with closed-population capture-recapture models in the absence of physical captures [see example with camera-trapping of Indian tigers, Panthera tigris (Karanth, 1995; Karanth and Nichols, 1998)]. High capture probabilities are nearly always desirable, regardless of whether one focuses on model selection, estimator precision, or bias reduction. However, the addition of trapping occasions to increase capture probability reflects a tradeoff between competing objectives. Large numbers of trapping occasions usually increase estimator precision and increase the performance of the model selection algorithm. On the other hand, multiple trapping occasions over an extended time also increase the probability that the closure assumption will be violated and increase the probability that time will be an important source of variation in capture probabilities. Design considerations for estimator precision include those that influence both sample size and the sources of variation in capture probabilities. Because estimator precision is a function of the number of model parameters, one should eliminate nuisance parameters associated with capture probability to the extent possible. Time variation is likely to be most easily influenced by the investigator, via standardization of trapping procedures. For example, the investigator should expend the same effort at each sampling occasion and use the same bait, number of traps, and daily trapping schedule throughout a trapping study. Of course, some factors cannot be dealt with adequately via standardization. For example, weather has the potential to influence capture probabilities. In situations where a study includes a single day of anomalous weather (e.g., hard rain; very cold temperatures reducing animal activity) among "normal" weather days, it sometimes is worthwhile to extend the study an extra day. The investigator then can examine both the full data set and the reduced data set (omitting the day of anomalous weather). It may be that the cost in estimator precision of using a model with time variation for the full data set may exceed the cost of discarding the data associated with the single day of bad weather. If
a likelihood-based model (e.g., M t, M b) appears to be appropriate for the data, then it will be possible to build models that are tailored to specific data sets. For example, it would be possible to develop a special version of M t in which capture probabilities were constant for days 1, 2, 3, and 5 of trapping, but different for day 4. Sometimes nontarget animal species can disturb traps and produce temporal variation in capture probabilities. For example, in small mammal studies, traps can be disturbed and "tripped" by both predators (raccoons, Procyon lotor) and large herbivores (white-tailed deer, Odocoileus virginianus). As is the case with anomalous weather patterns, it may be reasonable to either discard data from days on which substantial disturbance occurs (e.g., see Nichols et al., 1984a) or develop special models that have separate parameters for anomalous days. Behavioral response can be a troublesome source of variation in many sampling situations. The use of bait, an important component of many trapping studies, can induce a trap-happy response, yet in most trapping studies it would be foolish to recommend that bait not be used in order to eliminate the response. Prebaiting (placing baits at traps or trap stations several days before the traps are actually set) is useful not only for increasing capture probabilities, but also for reducing trap-happy responses. On the other hand, one can reduce problems with trap-shy responses by minimizing handling time. Trap deaths can be viewed as an extreme trap response, which can be reduced by minimizing the time animals spend in traps, reducing trap temperatures through the use of trap covers, leaving traps open during hot periods of the day, and other common-sense precautions. Some degree of heterogeneity of capture probabilities among individuals is likely to characterize all populations. In some cases, variation in capture probabilities is associated with visible characteristics of captured animals (e.g., age, sex, weight). Heterogeneity of this kind can be accommodated at the analysis stage, either by stratification or by covariate modeling of capture probabilities (Pollock et al. 1984; Huggins, 1989, 1991; Alho, 1990). The source of heterogeneity most likely to be associated with study design involves spatial distribution of capture devices a n d / o r animals. Investigators should try to avoid a situation in which some animals in the sampled area have very small probabilities of appearing in the captured sample, whereas other animals have high probabilities of appearance. When possible, one should include multiple traps per animal home range [e.g., Otis et al. (1978) recommend four traps per home range], though this may not be possible because of inadequate numbers of traps relative to the size of the sampled area. In
14.2. K-Sample Capture-Recapture Models these situations, one can divide the sampled area into quadrats smaller than the average home range size (e.g., four quadrats per home range) and then randomly select quadrats for trap placement at each sampling occasion. Capture probabilities for such a design are likely to be lower than if traps were in all quadrats during all sampling occasions, but because each quadrat has an equal probability of receiving a trap at each occasion, heterogeneity associated with trap placement is reduced. Along with the distribution and density of traps and animals, one also should consider the spatial configuration of groups of traps. In general, placement of multiple traps per home range can be achieved in the interior of a trapping grid, but not on the periphery of the grid. Animals with home ranges overlapping the outer row of grid traps use unsampled areas and therefore tend to have lower capture probabilities than animals with ranges entirely within the grid interior. Heterogeneity associated with grid edges is unavoidable unless the trapping is conducted on a habitat island or other discrete area that can be sampled completely by traps. To alleviate this problem, trap configurations should be used that minimize the ratio of the periphery to area covered by the traps. Thus, a linear transect of traps represents the worst possible configuration with respect to edge problems, and circular arrangement of traps represents the best configuration. For a fixed population size and known sources of variation in capture probability, the magnitudes of the capture probabilities are the primary determinants of estimator precision. Otis et al. (1978) and White et al. (1982) present some computations involving assumed densities and capture probabilities that are useful in determining grid size. Because closed population studies require a relatively small time commitment (e.g., several days; only 2 days with a Lincoln-Petersen study), pilot studies offer an inexpensive way to obtain some idea of the abundance/density and the capture probabilities to be expected. Simulations based on these preliminary estimates then can be used to investigate estimator precision under various designs (grid sizes, numbers of sampling occasions, etc.). Five trapping occasions can be viewed as a minim u m number to estimate population size, and 7-10 often is better (Otis et al., 1978). For grid trapping, trap stations in 10 • 10 grids probably represent a minimum, with larger grids preferable. The necessary capture probability for precise estimation depends on the actual size of the target population. Otis et al. (1978) suggested that a population of size 50 might require an average capture probability as high as 0.40 or 0.50 to produce useful estimates and tests, whereas a popu-
313
lation size of 200 or so might require only an average capture probability of about 0.20. 14.2.9. E x a m p l e
Nichols et al. (1984) trapped meadow voles, Microtus pennsylvanicus, in old field habitat at Patuxent Wildlife Research Center for five consecutive days, 29 August-2 September, 1981. The trapping grid was a 10 • 10 square of trapping stations with 7.6-m trap spacing. A single Fitch trap (Rose, 1973) baited with whole corn and containing hay was placed at each station. The trapping schedule consisted of setting traps for one evening, checking them for animals and closing them the following morning, setting them again in the late afternoon, checking them the following morning, etc. Newly captured animals were marked with individually coded monel fingerling tags placed in their ears. If tags of previously marked animals showed signs of pulling out, a new tag was applied on the opposite ear a n d / o r toes were clipped. Animals were sexed and weighed on each occasion and external reproductive characteristics were recorded. Adults were defined as voles >22 g. The capture-recapture data for adult females (Table 14.9) were analyzed with program CAPTURE (Rexstad and Burnham, 1991). The closure test of program CAPTURE yielded a test statistic of z = 0.43, P = 0.33, thus providing no evidence that the closure assumption was inappropriate. The relatively constant numbers of captures over the 5 days provided little reason to suspect temporal variation in capture probabilities. The discriminant function model selection criteria highlighted M h as a reasonable model for these data (Table 14.10). The test of M 0 versus M h provided strong evidence of heterogeneous capture probabilities, with X2 = 10.0, P < 0.01. The goodness-of-fit test for model M h was X2 = 1.61, P = 0.84, suggesting that the model provides an adequate description of the data. The jackknife estimate (Burnham and Overton, 1978,
TABLE 14.9
Summary of Capture Histories for Adult Female Meadow Voles a
Population data
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (fj)
Measure
1 27 27 0
18
2 23 8 15 15
3 26 9 17 8
4 22 4 18 6
5 23 4 19 5
aCaptured at Patuxent Wildlife Research Center, 29 August-2 September, 1981.
314
Chapter 14 Mark-Recapture Methods for Closed Populations
TABLE 14.10 Closed Model Selection Criteria of Program CAPTURE for the Meadow Vole Data of Table 14.9 Model
M0
Mh
Mb
Mbh
Mt
Mth
Mtb
Mtb h
Criteria a
0.65
1.00
0.23
0.37
0.00
0.29
0.24
0.44
a Model selection criteria are based on the linear discriminant function described by Otis et al. (1978) and Rexstad and Burnham (1991).
1979) of abundance for these data was/~/= 65, SE(/Q) = 5.70, with an approximate confidence interval of (58, 81). The estimated average capture probability was = 0.37. The data in Table 14.9 are not sparse, so we prefer the jackknife estimator to the M h estimator of Chao (1988), although in this case the Chao estimate (N = 63) was very close to the jackknife.
14.3. D E N S I T Y E S T I M A T I O N WITH CAPTURE-RECAPTURE Density is defined as the number of animals per unit area (D = N / A , where D denotes density, N is abundance, and A is area). In attempting to estimate D with trapping data, one typically encounters the problem of not recognizing the area actually used by animals that are subject to trapping. Consider, for example, a small mammal trapping grid that is located in a large area of old field habitat. The use of the area enclosed by the outermost traps of the grid as an estimate of A likely results in an overestimate of density, because the grid traps sample animals whose ranges lie partially outside the grid. This is termed "edge effect," and the estimated abundance/Q actually applies to a larger area of unknown size. Edge effect is more pronounced when home range size is large relative to grid size (White et al., 1982). Recognition of the potential problem in estimating the "effective area" sampled by a trapping grid led Dice (1938, 1941) to recommend the expansion of the sampled area by a boundary strip equal to half the average width of an animal's home range. Called the "extra-grid-effect area line method" by Tanaka (1980), this approach provides a conceptual basis for density estimation, though the problem then becomes one of estimating the width of the boundary strip surrounding the grid. The general approach of using a boundary strip is not restricted to grid trapping, but applies to any situation where sampled animals may come from areas larger than the area in which the sampling actually occurs (e.g., see Karanth and Nichols, 1998). An alternative approach involves direct esti-
mation of density (rather than separate estimation of population and the effective area sampled) using distance sampling or other alternatives (Anderson et al., 1983; Link and Barker, 1994). 14.3.1. U n i f o r m S a m p l i n g Effort (Grid Trapping)
At least three general approaches (Otis et al., 1978) are available to estimate the width W of a boundary strip surrounding a sampled area. One approach uses data on capture locations of recaptured animals to estimate home range size (e.g., Mohr, 1947; Hayne, 1949a; Stickel, 1954; Calhoun and Casby, 1958; Jennrich and Turner, 1969; Van Winkle, 1975; Ford and Krumme, 1979; Dixon and Chapman, 1980; Tanaka, 1980; Anderson, 1982). Half of the average width or radius of the home range estimate is then used to estimate the boundary strip width W. A second approach is to estimate W directly using data from selected subsets of the sampled area (e.g., subgrids), as described by MacLulich (1951), Hansson (1969), Seber (1982), and Smith et al. (1975). Based on this idea, Otis et al. (1978) developed their "nested grid" approach for joint estimation of density D and boundary strip width W. A third approach is based on "assessment lines" designed specifically to estimate both the effective area sampled and the corresponding population size (Smith et al., 1971, 1975; Swift and Steinhorst, 1976; O'Farrell et al., 1977). In what follows we focus on the first two of these approaches. 14.3.1.1. M o v e m e n t D i s t a n c e s B a s e d on T r a p p i n g D a t a
Distances between captures of individual animals have been long used to index the extent of home range (see Stickel, 1954; Brant, 1962). Wilson and Anderson (1985c) investigated a potentially useful approach to estimation of boundary strip width, based on the maximum distance d i between capture locations for each individual i that is captured at least twice. The mean --
1
d=
di
m
of these distances is computed across all m individuals (or all individuals in the age-sex class of interest) caught at least twice, with associated variance m
va'~r(d) --- ~ i = l ( d i _ ~)2
m ( m - 1)
"
Following the suggestion of Dice (1938) that W should be computed as half the average home range width, Wilson and Anderson (1985c) added a boundary strip of width 1~ = d/2 to the perimeter of their simulated
14.3. Density Estimation with Capture-Recapture trapping grids to estimate the effective area sampled. For square trapping grids with sides of length L (see Fig. 14.1), the effective area ~i(l/~ is A(I~V)
proaches to estimation of maximum distances. For example, Jett and Nichols (1987) used an estimator recommended by K. Burnham (personal communication),
-- n 2 if- 4 L I N + -rrl/V2
E(-dj)
with variance v~[A(l/~] = (4L + 2"rrl/~2 var(l/~
(14.18)
(Wilson and Anderson, 1985c), with sampling variance /Q2var[A(l~V)]
va"}(D) =
[A(I/~]4
=
[_1 -
e-(J-SJb]d *,
var(/~r)
-}- [A(I/~]~
given by a delta method approximation (Seber, 1982). A concern with this approach is that the maximum distance moved for an individual animal increases (at least initially) with the number of captures (e.g., Brant, 1962). This has led some to suggest alternative ap-
f
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X
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9 X
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-
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t ~W 2
(14.19)
where dj is the mean maximum distance moved by animals caught exactly j times, d* is the expected maxim u m movement for animals observed a large number of times, and b is a model parameter. Weighted nonlinear least squares can be used to estimate d*, which in turn can be used in place of d in the computations of effective area. Although density estimation frequently is based on data from regular trapping grids, the boundary strip approach applies generally to discrete areas of sampled habitat that are located in the midst of a larger expanse of similar habitat. In particular, the approach is appropriate for irregularly shaped sample areas (Karanth and Nichols, 1998), with estimation differing from that outlined above only in the computation of A(W) and its variance. In using observed movement distances to estimate
based on a delta method approximation (Seber, 1982). Utilizing one of the closed population estimators in Section 14.2 for population size, an estimator of density then is
D = 1cq/A(17V)
315
X
"
X X
X
X
..... X
X
)(
.. t ~
/
LxW
F I G U R E 14.1 Square trapping grid with boundary width W indicating "effective t r a p area." T h e c o m e r s of the effective area are q u a r t e r - c i r c l e s of r a d i u s W.
316
Chapter 14 Mark-Recapture Methods for Closed Populations
boundary strip and density, it is important to try to meet the assumptions underlying the estimation of N in Eq. (14.18), as well as the additional assumptions required for the estimation of A(I~V). One assumption is that the trapping grid does not induce immigration into the study area. The estimator in Eq. (14.18) is biased by movement into the sampled area that is induced by the sampling devices (e.g., baited traps). Efforts to minimize such immigration might include use of capture devices without bait. Another assumption is that one-half the mean of the maximum distances moved is a reasonable estimate of W for the purpose of estimating effective sampling area. We know of no strong theoretical justification for use of this ad hoc estimator and can only note that it seems to have performed reasonably well both in simulations (Wilson and Anderson, 1985c) and in comparisons with estimates obtained using the nested grid approach described below (Jett and Nichols, 1987). Regarding study design, it seems clear that use of observed movement distances are likely to be most useful in situations in which animal ranges are small relative to sampling grid area (also see White et al., 1982). In addition, movement distances are more effectively estimated when most animals are captured multiple times (see Brant, 1962; Tanaka, 1980; Wilson and Anderson, 1985c). For the purpose of estimating movement distance, it thus would be desirable to use at least 10 trapping occasions, but this must be balanced against the need for population closure, which requires shortduration trapping studies (e.g., five occasions). In cases when only a relatively small number of trapping occasions can be used, it may be wise to use a ^quantity such as d* from Eq. (14.19) for estimation of W, rather than d. Radiotelemetry also can provide information about movements for computing W and /~ with the above approach. 14.3.1.2. Nested Grid Approach
Direct estimation of W based on subgrids has been discussed by MacLulich (1951), Hansson (1969), Smith et al. (1975), and Seber (1982). The nested grid method proposed by Otis et al. (1978) and White et al. (1982) and implemented in program CAPTURE is the most widely used of these methods. It utilizes the fact that a large trapping grid can be subdivided to delineate smaller subgrids nested within the original grid. A 10 • 10 grid of trapping stations, for example, can be viewed as containing subgrids of dimension 8 • 8, 6 x 6, and 4 • 4 (Fig. 14.2). Denote the different subgrids by i (i = 1, ..., k), with subgrid i = 1 representing the smallest subgrid and subgrid i = k the largest. Capture-recapture models
in A
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F I G U R E 14.2 Nested trapping grids: (a) 4 x 4 grid; (b) 6 x 6 grid; (c) 8 x 8 grid; (d) 10 x 10 grid. After Otis et al. (1978).
such as those described in Sections 14.1 and 14.2 can be used to estimate abundance for each subgrid, where /~i is the abundance estimate obtained using data only from subgrid i. If A i denotes the area of subgrid i (the area covered by the traps), then a naive density estimate for each subgrid is b i -
Ni/Ai
(14.20)
with associated standard error S-E(/~i)- S~E(1Cqi)/ Ai for i = 1.... , k. Animals with ranges lying partially outside each subgrid are included in the estimated subgrid abundance, leading to positive bias in the density estimates. The idea underlying estimation with the nested subgrid approach is that biases in the naive density estimates [Eq. (14.20)] should be ordered from largest for the innermost subgrid (i = 1) to smallest for the entire grid (i = k). To see how, assume constant population density over the entire grid. For this sampling situation it is reasonable to consider a boundary strip of width W to be added to each subgrid to compute the effective area sampled. If W is known and Pi denotes the perimeter of subgrid i, then we can write the effective area sampled as Ai(W) = A i + P i W / c + -n-W2/c,
14.3. Density Estimation with Capture-Recapture where c is a conversion factor to express Pi W or W2 in the same units as A i (e.g., Fig. 14.2). Given the assumption of constant animal density D over the entire grid, and hence over the subgrids, the expected number of animals at risk of capture on each subgrid is E(Ni) = [ A i ( W ) ] D = [A i + P i W / c + ,rrW2/c]D,
so that the naive densities associated with each subgrid are
F)~ = N i l & = D[1 + a i W + bi W2] for i = 1, ..., k, where a i = P i / A i c and b i = ~r/Aic. Substituting an estimate of N from Section 14.1 or 14.2, we then can write E)i = Fqi/Ai
(14.21)
317
Tests for closure have been described in Section 14.2. One approach to testing for induced immigration and for density gradients involves contingency tests for uniform density by rows, columns, and rings of the trapping grid (Otis et al., 1978). These tests are based on the total captures at each grid point and are computed by program CAPTURE when grid location data are included in the input data. Induced immigration often is accompanied by increased numbers of captures in the outermost ring(s) of traps. Tests for uniform density by rows and columns provide evidence of density gradients. The nested grid approach carries substantial data requirements. Although Wilson and Anderson (1985a) concluded that the approach was theoretically sound, their simulation results indicated that it can be unreliable unless sample sizes are large. The field comparison of Jett and Nichols (1987) involved good sample sizes, and the approach appeared to perform well.
= D[1 + a i W + bi W2] + 8i, i = 1, ..., k, where 8i is a random error term with expectation E(~) = 0 and covariance matrix E(88') = ~. Because the subgrids are nested, any pair of density estimates /~i and Dj exhibits nonzero correlation p(/~i,/~j), which Otis et al. (1978) assume to be equal to the proportion of overlapping area between the two subgrids (including the boundary strips). Generalized nonlinear least squares can be used with the estimates /~/ and the covariance matrix with elements r = SE(/~i)S"E(/~j) P(/~i,/~j) to estimate directly the density and strip width in Eq. (14.21). As with the previous approach to density estimation, the nested grid approach requires population closure, which limits the study duration to, e.g., perhaps 5-10 days with small mammals. In addition, one should guard against inducing immigration during the study period. Removal trapping is known to create ecological vacuums and induce movement into trapped areas, and is thus not recommended for density estimation with the nested grid approach. The use of nested grids also assumes that population density is uniform in the sampled area (Otis et al., 1978), i.e., there is no density gradient over the trapping grid. When true densities are similar over the grid, differences among naive density estimates reflect only the differences in bias associated with a biased measure of the sampled area. One thus should select areas of homogeneous habitat for trapping. This general approach need not be restricted to a single grid, and Dooley and Bowers (1998) used multiple grids of different sizes within the same landscape. Uniform density and homogeneous habitat are especially important when multiple grids are used with this approach.
14.3.2. Gradient Designs (Trapping Webs) Distance sampling methods such as line transect and point sampling (see Chapter 13) were developed to estimate density in the presence of spatial variation in detection probability. However, Anderson et al. (1983) developed a distance sampling approach using capture-recapture data. Their idea was to distribute traps (or other sampling devices) so as to induce a spatial gradient in sampling effort and detection probability, which then can be exploited to estimate density.
14.3.2.1. Trapping Web and Distance Sampling The trapping web of Anderson et al. (1983; also see Buckland et al., 1993) consists of rings of increasing radius from the web center. Traps are placed at equal distances along the m lines of equal length, radiating from a randomly chosen central point (Fig. 14.3). Each line contains T traps, usually (though not necessarily) located at a fixed distance interval 0, starting at distance oL1 = 0/2 from the web center. The distance from the web center for any trap i is given by OLi - - 0 ( i - - 0 . 5 ) , i = 1.... , T, with points b i along each line midway between consecutive traps. Thus, point b0 is the web center, and point b T represents the boundary of the web beyond the last trap. All captures in ring i of the web occur at distance o~i from the web center and are treated as grouped data from the distance interval (bi_l, bi). The total area of the web out to interval i is given by c i = 'rr(bi) 2, and the area associated with the ring i of traps is A i - - Ci -Ci_ 1. This design yields a gradient in capture probability corresponding to the gradient in trap density,
318
Chapter 14 Mark-Recapture Methods for Closed Populations
F I G U R E 14.3 Schematic diagram of a trapping web with 16 lines, each of total length A T with T = 20 traps per line (after Anderson et al., 1983). Traps are equally spaced along each line. Points equidistant between traps are denoted by bi, with b0 representing the center of the web and b T located just beyond the last trap. Captures in the eleventh ring of traps are assigned to the annulus All, which has area -rr(b121 - b120). After Anderson et al. (1983).
with probability the highest in the first ring at the web center and lowest in the outer ring T. The typical field procedure for use of a trapping web involves setting out the traps in the web design, prebaiting and leaving the traps open for several days (this step is unnecessary for unbaited capture devices such as pitfall traps) and then setting and running the traps for several consecutive days. The trapping web typically utilizes only initial captures, so captured animals can be removed or marked with either batch or individual marks. The data resulting from a single web are the number of initial captures uij occurring in traps from ring i of the web on day j of trapping, i = 1, ..., T and j = 1, ..., K. These data are pooled over the days of trapping to yield the total number K Ui = ~ ldq j=l
of first captures in each ring of traps throughout the study, as well as the total number
of individuals caught in the study. The values u i are used in conjunction with standard point transect estimation methods (Buckland et al., 1993) to determine f' (0), the slope of the estimated density of capture distances evaluated at zero (see Section 13.3.2). Estimation of f' (0) from the capture data u i is carried out by program DISTANCE (Buckland et al., 1993). If the population is distributed randomly, then Wilson and Anderson (1985b) recommend using [cv(u)] 2 = 1 / u , whereas under situations with spatial aggregation, [cv(u)] 2 = 2 / u or 3 / u may be more appropriate. The necessary assumptions for analysis of trapping web data using distance sampling methods are (after Buckland et al., 1993): 1. All animals at the center of the web are captured during the study. 2. Distances moved by animals during the study are small relative to the size of the web, and migration through the web does not occur. 3. Distances from the web center to each trap are measured accurately.
T
U--~U i=1
i
In addition to these assumptions, the proportion of captures in a given ring is assumed to be the same as
14.3. Density Estimation with Capture-Recapture the proportion of captured animals whose locations were closest to that ring (the "closest trap assumption") (Link and Barker, 1994). Assumption (1) is analogous to the assumption in point counts that all animals located on the sampling point are detected. When it is not true, density estimates will be negatively biased. The number of new captures near the web center provides information about this assumption. If one captures no new animals in the innermost ring(s) of traps for 2 or 3 days in a row, it is reasonable to assume that most or all animals near the center have been caught. However, if the study lasts too long, then the possibility arises that animals initially located away from the web center move to the center and are trapped there. This possibility relates to assumption (2) that movements are relatively small. Thus, directional movement toward the web center (or any particular location) can produce biased estimates. Designing the trapping web relative to animal movements is important in determining whether the trapping data mimic the assumptions of point transect sampling (Buckland et al., 1993). The trapping web is likely to perform well when home ranges are small relative to web trap spacing; however, there are few guidelines for the desired relationship between trap spacing and animal home range size.
14.3.2.2. Trapping Web and Geometric Analysis Link and Barker (1994) considered a different approach to density estimation with a trapping web by focusing on the geometry of the web as a determinant of the degree of competition among traps They noted that the "closest trap assumption" implies that the number of captures at a particular trap in the web should be determined by the size of the region closest to that trap (i.e., an animal within this region would be closer to the trap in question than to any other trap on the web). They refer to this region as the "maximum locus" of the trap. Link and Barker (1994) also define a trap's "locus of radius y" as the collection of points within distance y from the trap that are closer to the given trap than to any other. This locus of radius y is the intersection of a circle of radius y and the maximum locus of the trap. The shape and area of this intersection are determined by web geometry and, for a given web, by the ring in which the trap is found. Link and Barker (1994) enumerated 17 different forms for the locus of radius y and computed the area associated with each form. The area associated with the locus of radius y is denoted by Ai(y), where i denotes the trap ring. Link and Barker (1994) then focused on the location of each individual trap, rather than on the web center as in the distance sampling approach of Anderson et
319
al. (1983). Let trap location be denoted as t 0, and the distance from a randomly selected animal to t o be denoted by X, with Y = ~rX2. Finally, let
g(y) = Pr {captured at
to l Y = y,
no competition between traps}. They modeled g(y) as a step function, taking the value 1, (k - 1)/k, (k - 2)/k, ..., 2/k, 1/k, 0 for distance intervals (measured from the trap) of [P0 = 0, Pl), [Pl, P2), "', [Pk-1, Dk), [[3k, Pk + 1 "- OO). The number of steps used to approximate g(y) is thus specified by k. As with the distance sampling approach, the data used for estimation are the numbers of animals caught for the first time in each ring, ul, u2, ..., UT. Link and Barker (1994) modeled these data as a multinomial random variable conditional on the total captures u with associated cell probabilities defined by k
Zj---1 Ai(f3j) o
~k
Ai(Pj)
Estimation of the parameters pj can be accomplished using maximum likelihood. The expected values for number of animals caught at a given trap [E(uis), where i denotes a trapping ring and s denotes a trapping radius], ring of traps [E(ui)], and the entire web [E(u)l are given by
D k E(uis) = -ff Z Ai(Pj)' 1=1
E(ui )
=
m D k k ~ Ai(PJ)' j=l
and
E(u) - m D T
k
k Z Z Ai(Dj)" i=1 j = l
where m is the number of spokes of the web and D is the (unknown) density of animals. Based on the above expectations, D is estimated as (Link and Barker, 1994)
D=
ku Z T Z k Ai(~)j)" m i=1 j--1
Link and Barker (1994) recommend using the delta method to compute v~(/)lu), the estimated variance o f / ) , using the estimated information matrix for the 6j (conditional on u). This method has seen little use but seems to hold promise. The geometric approach also lends itself to considerations about spatial configurations of traps
Chapter 14 Mark-Recapture Methods for Closed Populations
320
other than the web and permits consideration of optimal configurations.
14.4. REMOVAL METHODS As with capture-recapture methods, removal methods for closed populations involve multiple samples in which animals in the population of interest are captured. As the term implies, however, captured animals are not returned to the population but are removed, thus distinguishing removal sampling from capturerecapture. We include these models in a chapter on closed-population models because the removals are under the control of the investigator and are assumed to be known. In this sense the population can be viewed as open with respect to investigator removals, but closed with respect to natural processes. The relevant population model is Xi+l
in capture probability, thus defining models in which sampling intensity is not equal for all sampling periods. The overparameterization of model Mtb w a s handled in Section 14.2.3 by taking advantage of a presumed mathematical relationship between initial and recapture probabilities. In this section the approach is to utilize auxiliary information, namely, timespecific measures of sampling effort that are assumed to be directly related to the capture probabilities. An assumed direct relationship between effort and capture probability has led to the wide use of "catch-per-uniteffort" statistics as indices to abundance (e.g., Schnute et al., 1989; Richards and Schnute, 1992; Schnute and Hilborn, 1993). For reasons presented in Chapter 12, we do not discuss these indices here and instead focus on statistically reliable procedures for estimation of population parameters. The models in this class are typically referred to as "catch-effort" models.
"- X i ~ Flit
with time-specific removals rl i reducing the population monotonically over the course of the study. Removal methods are most commonly used to estimate abundance for exploited populations; for example, fisheries applications are common (e.g., Hilborn and Walters, 1992). Removal models can be conveniently placed into two categories, the first of which imposes equal sampling intensity at every sampling period. Models for this situation have been described in Section 14.2.2 as behavioral response models, with the idea that removal is an extreme "behavioral response" for which the probability of recapture vanishes. Sufficient statistics for abundance estimation under two of the behavioral response models (Mb, Mbh) described in Section 14.2.2 are the number of animals caught for the first time in each sampling period. Thus, estimation under any behavioral model proceeds as with a removal model, with initial captures essentially "removed" from the population (recaptures are not used to estimate population size). Constant-effort removal models were introduced by Zippen (1956, 1958), but the estimators described in Section 14.2.3 and computed by program CAPTURE are now the preferred means of analyzing such data. Because the removal (behavioral response) models for equal sampling intensity have been described in Section 14.2, we will not discuss them further here. The other class of removal models permits variation in sampling intensity over time and requires additional structure in order to estimate parameters. Note that the other behavioral response models discussed in Section 14.2, Mtb and Mtbh, include time as a source of variation
14.4.1. Sampling Scheme and Data Structure We again assume that animals are captured on K different sampling occasions, and captured animals are removed from the population. The focus of estimation is on initial population size. We denote this as N 1, where the subscript 1 serves as a reminder that the population is changing throughout the sampling as a result of removals. Define the following statistics: ni
i-1
Xi -- ~ j = l nj
fi
i-1
Fi -- ~ j = l
)~
The number of animals removed from the population at sample period i. The cumulative catch prior to sampling period i (i = 2, ..., K + 1, Xl = 0). The units of effort expended on sample i. The cumulative effort prior to sampling period i (i = 2, ..., K + 1, F 1 -- 0).
The basic model parameter is k, the catchability coefficient or capture rate for a particular animal for one unit of effort. As in previous sections of this chapter, let Pi denote the capture probability for period i and define qi = 1 - Pi. Under the assumption of a Poisson sampling process (see Appendix E), the relationship between capture probability and effort can be written as qi = e - k f i
and Pi = 1 -- e -kfi.
(14.22)
14.4. Removal Methods In the development below, we assume that both catch (n i) and effort (fi) are known. However, when this approach is used with harvest data, neither catch nor effort is likely to be known with certainty. The consequences of measurement error for catch-effort estimation have been investigated by Gould et al. (1997), who suggested a simulation-extrapolation method of inference (Cook and Stefanski, 1994) as a means of adjusting for resulting bias. The following assumptions often are specified for this approach: (1) sampling is a Poisson process, with all animals having the same probability of capture per unit of sampling effort; (2) units of sampling effort are assumed to be independent and additive in their effect on catchability; (3) all removals from the population and the level of effort expended in each sample are known; and (4) the population is closed both to gains and to losses other than known removals.
As background, we begin by presenting three different least-squares approaches that follow the historical development of catch-effort estimation. We then describe the general development of Seber (1982) and Gould and Pollock (1997b) for an approach using maxim u m likelihood estimation. For a general treatment of catch-effort estimation via least squares, we recommend the work of Bishir and Lancia (1996). The idea in all approaches is to characterize captures at each point in time in terms of sampling effort and the size of the population exposed to capture.
14.4.2.1. Approach of Leslie and Davis (1939) Under the "Leslie" method, removals from the population are viewed as conditionally binomial, with probability of capture given by Eq. (14.22). The joint distribution of removals thus is modeled as K
pni qNl-Xi+l
(14.23)
1)!
Under this model, the conditional expectation of the catch at time i can be written as E(ni[xi)
=
(N 1 -
xi)Pi.
(14.24)
Define a catch-per-unit-effort statistic as Yi = ni/fi (this statistic frequently is used as an index to abundance). If Pi is small then Pi "~ kfi, and substituting this expression into Eq. (14.24) and dividing each side by fi we obtain the regression model E(YiIxi) ~, k N 1 -- kxi,
where k N 1 is the intercept and - k is the slope. The parameters of Eq. (14.25) then are estimated using leastsquares methods based on Eq. (14.25). 14.4.2.2. A p p r o a c h of D e L u r y (1947)
DeLury (1947) considered the expected catch, E(ni) = Nlqlq2 "'" qi-lPi
(14.26) = Nle-kFipi ' i-1 for sample i, where F i = ~j=l ~. Thus, in order to be caught in sample i, an animal must be missed (not caught) in the previous i - 1 samples. The Pi are again assumed to be small, permitting the approximation Pi kfi. In addition, DeLury (1947) used the approximation E[ln(ni/fi) ] ~ ln[E(ni/fi) ].
Taking logs of both sides of Eq. (14.26), dividing by fi, and substituting the approximation yields the regression model
14.4.2. Models and Estimators
g({ni}) = I-[ (Xl Xi)! i=1 ni! (N1 - x~ +
321
(14.25)
E(yi]Fi) ~, ln(kN 1) - kFi,
(14.27)
where Yi = ln(Yi) = ln(ni/fi). Under this approach, the catch-per-unit-effort (actually its natural log, Yi) is related to cumulative effort rather than cumulative catch. The slope of the regression line, Eq. (14.27), is again -k.
14.4.2.3. Approach of Ricker (1958a) Ricker (1958a) viewed the entire study as consisting of FK+I samples, each of which represented a single unit of effort. Thus, the expected population size at the time of each sample can be written as E(N 1 - xi) -~ N1(1
-
k) Fi.
(14.28)
Given equality (14.28) and the approximation in Eq. (14.25), Ricker (1958a) derived the model E(y i) ~ ln(kN 1) + Fi[ln(1 - k ) ] ,
(14.29)
where Yi is again defined as Yi = ln(Yi) = ln(ni/fi). Expression (14.29) also can be obtained directly from Eq. (14.27) by utilizing the approximation ln(1 - k) ~ -k.
14.4.2.4. Comments on the Three Least-Squares Approaches All of the above three approaches are based on regression models for which least-squares estimation typically is recommended. As discussed by Gould and Pollock (1997b), the approaches all rely on the approximation Pi ~" kfi, which is reasonable only when Pi is small. However, it is also true that the reliability of
322
Chapter 14 Mark-Recapture Methods for Closed Populations
these catch-effort methods depends on a substantial proportion (usually >30%) (Gould and Pollock, 1997b) of the population being removed during sampling. A reliance on large catches is not consistent with approximations that assume small Pi. An additional objection to the DeLury (1947) and Ricker (1958a) approaches involves the approximation in which the expected value of a logarithm is equated with the logarithm of the expectation. Finally, the regression assumption of constant variance structure (of Yi or Yi) is unlikely to be reasonable, because the catch-per-unit-effort (Yi) should decrease as the population is reduced. For these and other reasons presented by Gould and Pollock (1997b), we favor their recommendation to focus on maximum likelihood methods for modeling and estimation in catch-effort problems.
14.4.2.5. M a x i m u m Likelihood Approach Seber (1982) and Gould and Pollock (1997b) wrote the joint distribution of the catch statistics r/i as the multinomial distribution
NI!
P({ni}]k, {fi}) =
P~'(qlP2)n2""(qlq2" "qK - lP K)nK
(l-I/K= 1 ni!)(Nl--XK+l)' x (1 - Pl -
(14.30)
qlP2 . . . . . qlq2""qK-lPK )N'-xK+'
Rather than using this distribution directly in estimation, Gould and Pollock (1997b) recommended rewriting Eq. (14.30) as the product of two distributions:
P({rli}]k, {fi})
=
Pl(XK+l[k,
{fi})
(14.31)
X P2({ni}]XK+l, k, {fi}). Expression (14.31) decomposes the distribution (14.30) of the catch statistics into two components. Component P1 models the total catch for the entire study, XK+1, as a binomial random variable:
Pl(XK+l]k, 0ci}) =
NI! (1 XK+I!(NI--XK+I)!
-
Q ) XK+I Q N1 -XK+l I
where Q is the probability of not catching a member of N 1 during the entire study. Q is written as the product of the probabilities of not catching an animal at each of the K sampling periods:
Q = 1 - Pl - qlP2 . . . . .
qlq2 . . . .
qK-IPK"
The second component of expression (14.31) then conditions on the total number of animals caught throughout the study and models their distribution over the K sampling periods:
p2({ni}lXK+l,k,{fi})__
XK+I! ( P l ) n l l-Ii K, hi! 1 - Q
(14.32)
X(qlP__b)na...(qlqai"q-~-lPK)nK The actual modeling of capture probability as a function of effort can use any reasonable function. For example, Gould and Pollock (1997b) selected the linear logistic form
ef3O+f31fi Pi = 1 + e ~~ for their examples, which has the advantage of being sufficiently flexible to incorporate other covariates in addition to effort in the modeling of capture probability (see example in Section 14.4.5) (see Pollock et al., 1984; Gould and Pollock, 1997b). Estimation proceeds by conditional maximum likelihood (Sanathan, 1972), using P2 to estimate catchability. The resulting estimate of catchability then is used with P1 to estimate N 1 using the familiar form of the canonical estimator [Eq. (12.1)]:
1~1 "- XK+ 1//9,
(14.33)
where/3 = (1-(~), and (~ is estimated using the catchability from P2 [Eq. (14.32)]. One advantage of this two-step approach is that it avoids the difficulty in numerical maximization with a discrete-valued parameter (N1), the magnitude of which is very different from that of the catchability coefficient. Gould and Pollock (1997b) provide further discussion motivating this approach. Variances can be estimated using Taylor series approximations, although Gould and Pollock (1997b) recommend use of the parametric bootstrap (see Appendix F). Pollock et al. (1984) applied maximum likelihood methods to catch-effort problems as well, but they used an unconditional approach. They included N 1 in the likelihood and used a two-step iterative process to obtain estimates numerically (Hines et al., 1984). Because the conditional approach of Gould and Pollock (1997b) is easier to implement numerically, we recommend it for most uses.
14.4.3. Violation of Model Assumptions The assumption that all animals have the same probability of capture per unit of sampling effort throughout the entire study can be violated in numerous ways. In fisheries, for example, different size or age or sex classes of fish may have different susceptibilities to particular fishing methods, causing heterogeneity in catchability coefficients among individuals. In the face
14.4. Removal Methods of such heterogeneity, the more catchable animals are likely to be caught early on, leaving the less catchable animals to comprise larger and larger portions of the remaining population (N1 - xi). Such a pattern should lead to decreases in average catchability over time. Trends in catchability because of environmental conditions also can result in violation of the equal-catchability assumption. In this case, the time trend could be either positive or negative, depending on the trend in the environmental parameter and its effect on catchability. Intuition suggests that a negative trend in catchability over time should lead to a more negative slope in the relationship between catch-per-unit-effort and cumulative effort. This should produce an estimate of the catchability coefficient that is positively biased and an abundance estimate that therefore is negatively biased [e.g., see Eq. (14.33)]. Conversely, a positive trend in catchability should lead to a less negative slope in the relationship between catch-per-unit-effort and cumulative effort. This should produce negative bias in the catchability estimate and positive bias in the estimate of abundance. Simulation results of Gould and Pollock (1997b) confirm these expectations and indicate that the biases can be substantial. Failure of the closure assumption should affect catch-effort estimates in a manner similar to that of temporal trends in the catchability coefficient. For example, consider a population exposed to losses between sampling occasions but no gains (or a completely open population with losses exceeding gains). The number of animals exposed to sampling efforts at each occasion will be smaller than N 1 - xi, because of the losses in addition to the known removals. This should yield a more negative slope of the relationship between catch-per-unit-effort and cumulative effort than if there were no losses, producing negative bias in the estimate of N 1. If the population is exposed only to gains, or if gains exceed losses, then we speculate that the slope of the relationship between catch-per-unit-effort and cumulative effort should be less negative, producing positive bias in the abundance estimate N 1. Populations experiencing fluctuations in the relative magnitudes of gains and losses between the different sampling occasions should lead to biased estimates, although the directions and magnitudes of bias will depend on the pattern of population change. A lack of influence of sampling effort on the resuiting capture probability is speculated to be a common reason underlying the lack of fit of catch-effort models to actual catch data. This assumption violation causes model-based variance estimates to be too small, necessitating use of variance inflation factors (see below).
323
The assumption that removals and units of effort are known with certainty is likely to be violated when catch-effort models are applied to data for harvested populations (e.g., fisheries). Gould et al. (1997) explored the consequences of measurement error for both catch and effort via computer simulation. They investigated the performance of their maximum likelihood approach as well as that of the approaches of DeLury and Leslie. The maximum likelihood approach performed the best, but estimates nevertheless were positively biased by measurement error, with biases becoming substantial with large measurement error variances. Gould et al. (1997) thus recommended a simulation-extrapolation inference method (Cook and Stefanski, 1994) for reducing bias of estimates in the presence of measurement error. Because assumption violations can lead to substantial bias, efforts to assess model fit to the data are important. Pearson chi-square goodness-of-fit tests based on a comparison of observed catches against their expectations under the model frequently are used to assess model fit. When there is evidence of lack of fit, and when it is believed that lack of independence may be responsible, it is reasonable to use a quasilikelihood approach (Pollock et al., 1984; Burnham et al., 1987; Lebreton et al., 1992; Burnham and Anderson, 1998). Estimators for model parameters frequently remain unbiased in the face of overdispersion caused by lack of independence, but model-based variances tend to be too small and should be inflated (McCullagh and Nelder, 1989). If a Pearson chi-square goodness-of-fit test is used to assess fit, and if it provides evidence that the most general model in the model set does not fit the data adequately, then the fit statistic can be used to compute a variance inflation factor (e.g., Burnham et al., 1987) by =
x21df,
where X2 and df correspond to the goodness-of-fit test of the global model (Cox and Snell, 1989) or the most general model in the model set. Model-based variance and covariance estimates then are multiplied by ~ to obtain estimates that properly account for overdispersion. The variance inflation factor also can be used to adjust likelihood ratio test statistics and Akaike's Information Criterion for the purpose of selecting from among competing models (see Section 17.1.8).
14.4.4. Study Design As was the case for closed-population capturerecapture models, study design in catch-effort studies should include efforts to minimize assumption violations and maximize estimator precision. Regarding the
324
Chapter 14 Mark-Recapture Methods for Closed Populations
assumption of equal catchability for all animals in the population, we noted in Section 14.4.3 that different size, age, or sex classes of animals may exhibit different susceptibilities to capture. Stratification is an obvious way of dealing with this problem, so study design should include recording of auxiliary data that can be used to classify animals to strata. Variables that might be associated with capture probability and are measurable must be selected before the initial capture sample is taken. In the data-analytic stage, models permitting different catchability coefficients for different classes then can be compared against models that do not include such variation. If the models incorporating variation among animal classes are selected, then separate estimates of abundance can be obtained for each class and summed to obtain an overall estimate. In studies of limited size, it may be possible to ensure through investigator effort that numbers of removals and units of sampling effort are known. However, in large studies involving harvest situations, it may not be possible to enumerate directly removals or units of effort. In such cases it is important to use an estimation method that provides unbiased estimates of removals and effort. Many survey methods have been developed for estimation under these conditions. These methods are beyond the scope of this book, but we recommend Pollock et al. (1994) for an introduction to the angler survey methods that are commonly used in fisheries investigations. Methods for removals and effort that permit estimation of associated sampling variances are desirable, because variance estimates can be adjusted to deal with measurement error (Gould et al., 1997). The primary aspects of study design that are relevant to the closure assumption involve time and space. Relatively short studies provide the greatest likelihood that the studied populations are closed to gains and losses other than known removals. It is wise to restrict catch-effort studies to times of the year when population processes such as migration, reproductive recruitment, and mortality are likely to be minimal. Similarly, the closure assumption is more likely to be met in spatially restricted study areas (e.g., small to moderately sized ponds or lakes) than in areas lacking spatial restrictions (rivers and oceans). When the population is found to be open despite study design, special catcheffort models for open populations can be used (Section 19.5.3) (also see Seber, 1982; DuPont, 1983; Bishir and Lancia, 1996; Gould and Pollock, 1997a). Precision and bias of abundance estimates resulting from catch-effort studies are heavily dependent on the fraction of the population that is removed, with larger proportional removals yielding more precise and less biased estimates. The number of sample occasions is one element of study design that determines the re-
moval fraction. Gould and Pollock (1997b) presented simulations with catchability coefficient k = 0.01, yielding the following proportions of the population removed: 19% for K = 3 sampling occasions, 30% for K - 5, and 51% for K = 10. Median negative bias of the abundance estimate for these three scenarios was 51%, 20%, and 2%, respectively. As in capture-recapture studies for closed populations, the selection of the number of occasions represents a tradeoff between efforts to approximate the closure assumption (emphasis on closure will lead to fewer occasions) and efforts to obtain precise estimates by removing more animals (leading to more sampling occasions). The magnitude of the catchability coefficient is very important, with higher catchability leading to more precise abundance estimates. Catchability should be a direct function of sampling effort and is thus an important aspect of study design. Finally, for fixed catchability and number of sampling occasions, measures of relative precision are smaller for large population sizes and larger for small populations (Gould and Pollock, 1997b). Choice of study area boundaries may partially determine the size of the studied population.
14.4.5. Example We present the analyses of Pollock et al. (1984) and Gould and Pollock (1997b) for the classic catch-effort data set of Paloheimo (1963). The data correspond to 2-week periods and include estimated catch (number of legal-sized lobsters removed), effort (in number of trap hauls), and a potential covariate [average water (bottom) temperature in ~ for a Canadian lobster fishery at Port Maitland, Nova Scotia, 1950-1951 (Table 14.11). In the most general model, capture probability is modeled as a linear-logistic function of effort fi and the environmental temperature t i for sampling occasion i: e ~o+ f31fi+ B2ti Pi =
1
+ e f~~
We also consider reduced parameter models in which capture probability is modeled as a constant (131 = ~2 = 0) and a function only of effort (132 = 0). The models were fit using the unconditional approach of Pollock et al. (1984), and Pearson goodnessof-fit tests provided strong evidence that none of the models fit the data well (Table 14.12). This was expected because of the extremely large sample sizes, the likely nonindependence of lobster captures, and the likely influences of factors other than effort and temperature on capture (see Pollock et al., 1984). As indicated by the magnitudes of the residuals (deviations between observed and predicted values), the models
325
14.5. Change-in-Ratio Methods
TABLE 14.11 Catch, Effort, and Temperature Data for a Commercially Harvested Lobster Population a Period
Catch (n i)
Effort (fi)b
Temperature
1
60,400
33.664
2
49,500
27.743
7.7
3
28,200
17.254
6.3
4
20,700
14.764
3.5
5
11,900
11.190
3.1
6
15,600
16.263
2.9 3.1
( t i)
7.9
7
13,200
14.757
8
25,400
32.922
3.25c
9
29,900
45.519
3.4
10
32,500
43.523
3.6
11
24,700
37.478
4.0
12
27,600
43.367
5.9
13
22,200
37.960
6.1
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b). bEffort is in thousands of trap hauls. CThis value was missing, so we used the average of the two adjoining periods.
TABLE 14.12 Comparison of Residuals for Three Models for a Commercially Harvested Lobster Population a Residuals b
Period
Observed catch
1
60,400
2 3
Effort plus temperature Constant (p.) Effort only (Pt)
(Pf+t)
+116.8
+27.6
- 10.8
49,500
+72.8
+51.5
+13.2
28,200
- 30.7
+20.9
+5.7
4
20,700
- 63.5
- 4.3
+22.1
5
11,900
- 106.0
- 43.6
- 22.9
6
15,600
- 77.7
- 31.1
- 3.5
7
13,200
- 85.2
- 34.3
- 14.9
8
25,400
- 2.7
- 31.5
+4.8
a c t u a l l y s e e m e d to p e r f o r m r e a s o n a b l y well. T h e variance inflation factor w a s c o m p u t e d u s i n g the X2GOF a n d a s s o c i a t e d df f r o m the m o s t g e n e r a l m o d e l (Pf+t) as ~ = 300.44, a n d this v a l u e w a s u s e d to a d j u s t v a r i a n c e e s t i m a t e s a n d to c o m p u t e AQAICc v a l u e s ( B u r n h a m a n d A n d e r s o n , 1998) (also see Section 17.1.8 a n d Table 14.13). T h e m o s t g e n e r a l m o d e l h a d the l o w e s t QAICc v a l u e a n d w a s j u d g e d m o s t a p p r o p r i a t e for the d a t a (also see Pollock et al., 1984; G o u l d a n d Pollock, 1997b). T h e e s t i m a t e d linear-logistic coefficient ~1 a s s o c i a t e d w i t h effort w a s p o s i t i v e as p r e d i c t e d , as w a s the coefficient ~2 a s s o c i a t e d w i t h t e m p e r a t u r e . H i g h e r t e m p e r a t u r e s w e r e p r e d i c t e d to p r o d u c e g r e a t e r lobster activity a n d t h u s g r e a t e r p r o b a b i l i t y of b e i n g c a u g h t . T h e coefficient of v a r i a t i o n for a b u n d a n c e u n d e r the g e n e r a l m o d e l w a s small [CV(/~) = 0.087].
14.5. C H A N G E - I N - R A T I O METHODS .
.
.
.
.
.
.
Change-in-ratio m e t h o d s originally were d e v e l o p e d for u s e w i t h h a r v e s t e d species, b a s e d o n the o b s e r v a tion t h a t differential h a r v e s t a m o n g g r o u p s of a n i m a l s in a p o p u l a t i o n can p r o d u c e c h a n g e s in p r o p o r t i o n a t e r e p r e s e n t a t i o n of the g r o u p s . For e x a m p l e , a d e e r harv e s t d i r e c t e d at m a l e s s h o u l d lead to a r e d u c t i o n in the p r o p o r t i o n of m a l e s in the p o s t h a r v e s t p o p u l a t i o n . Kelker (1940, 1944) r e c o g n i z e d t h a t i n f o r m a t i o n o n the ratios of different t y p e s of a n i m a l s (e.g., sex ratio) before a n d after h a r v e s t , c o m b i n e d w i t h i n f o r m a t i o n o n the n u m b e r of a n i m a l s of e a c h type, c o u l d be u s e d to e s t i m a t e a b u n d a n c e . C h a p m a n (1954, 1955) d e v e l o p e d the first stochastic m o d e l s for a b u n d a n c e e s t i m a t i o n ,
TABLE 14.13 Comparison of Parameter Estimates (Standard Errors) for Three Models for a Commercially Harvested Lobster Population a
9
29,900
+34.7
- 55.2
- 17.1
10
32,500
+61.2
+1.8
+22.8
549,974
- 3.94
0.030
0.11
11
24,700
+18.4
+15.4
+12.9
(47,780)
(0.182)
(0.004)
(0.02)
12
27,600
+47.0
+26.7
- 10.8
Effort model (pf)
13
22,200
+17.7
+42.4
- 3.9
472,270
- 3.29
0.037
(21,840)
(0.108)
(0.0036)
X2GOF
df AQAICc
56,282 11 28,600
14,706 10 6200
2704 9 0
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b). bResiduals computed as (0 i - Ei)/V~i , with 0 i the observed catch and E i the expected catch under the model.
Effort-plus-temperature model (Pf+t)
Constant probability model (p.) 716,860
- 2.89
(84,200)
(0.172)
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b).
326
Chapter 14 Mark-Recapture Methods for Closed Populations
and the general approach has since been reviewed by Paulik and Robson (1969) and Seber (1982). The approach has been extended to incorporate more than two types of animals and more than one removal period (e.g., Otis, 1980; Pollock et al., 1985b; Udevitz and Pollock, 1991, 1995). The usual implementation of change-in-ratio methods involves a combination of observation data and removals from managed hunting areas. Returning to the example of deer sex ratio, observation-based methods (e.g., spot lighting) frequently are used to estimate the sex ratio before and after harvest, whereas managers at hunter check stations record the number of deer of each sex removed by hunting. The methods used to estimate the ratio of types of animals in the population before and after harvest need not involve direct observation but may involve trapped samples or any other means of assessment. The utility of the approach (as with that of the methods of Section 14.4) is based on its exploitation of data that are collected routinely in local management programs. The methods to be discussed here assume that, with the exception of known removals, the population is closed to gains and losses. It is important to note that the method is useful only when removals are selective with respect to the different types of animals in the population, because selectivity is the basis for the change in ratio that is exploited in estimation. For situations in which removals are not selective, the investigator should use constant-effort removal models (Mb and Mbh) of Section 14.2 or the catch-effort models of Section 14.4.
14.5.1. Sampling Scheme and Data Structure Although the original use of change-in-ratio methods involved two types of animals and a single removal period bracketed by two observation periods, we present here the notation for the more general case. We distinguish between sampling periods, in which the ratios of different types of animals in the population are assessed, and removal periods, during which the removals occur. Let K be the number of sampling periods in the study. Then there will be K - 1 removal periods, one following each sampling period except the final one. Define the following notation, adapted from that of Seber (1982) and especially Udevitz and Pollock (1991 ):
N/,
The number of individuals in the population of type i in sampling period j.
k
Xj
k
= ~i=1 Nij
rij
k /'j = ~i=1
rij
nij g
nj = ~i=1 nij
The number of types of animals in the population. The total number of animals in the population at sampling period j. The number of individuals of type i removed from the population in sampling period j between sample periods j and j+l. The total number of animals (all types) removed from the population between sampling periods j and j +1. The number of individuals of type i encountered in sampling period j. The total number of individuals (all types) encountered in sample j.
The Nj are the quantities of interest to be estimated. As with the removal approaches of Section 14.4, the relevant model here is
for j = 1, 2, ..., K - 1. However, change-in-ratio models are distinguished from the removal models of Section 14.4 in that change-in-ratio models recognize different animal types in the population and include additional (nonremoval) observations over the course of the study. Several assumptions underlie most change-in-ratio estimation methods (e.g., see Conner et al., 1986; Udevitz and Pollock, 1991). The population is assumed to be closed except for the removals, and the numbers of removals for animals in each type are assumed known. Sampling is with replacement, or else the sampling fractions are negligible. Encounters of animals during sampling periods are independent, with a probability Pij of encountering an individual of type i in sampling period j. Some approaches are based on the assumption that during any sampling period j, the probability of being encountered is the same for individuals in all types, i.e., Pij = Pi'j. Other approaches are based on the less restrictive assumption that the ratio Pij/Pi'j of encounter probabilities for individuals of type i and i' is constant over all sampling periods. The necessary assumptions for encounter probabilities are specified with the models and estimators described in Section 14.5.2. The change-in-ratio approach provides useful estimates only in the case in which animals of different types are not removed in proportion to their original abundance in the population. If removals are not selective with respect to animal type, then the ratio of types in the population is not expected to change, and the sample estimates of these ratios will provide no information with which to estimate abundance.
327
14.5. Change-in-Ratio Methods 14.5.2. M o d e l s and Estimators
/~1 = /~11 -}- /~21
We begin this section with a description of the standard (or at least original) application of two sampling periods separated by one removal period, with two types of animals. We then present the more general cases considered by Udevitz and Pollock (1991).
14.5.2.1. Two Sampling Periods, One Removal Period, Two Types of Animals We begin with an intuitive derivation of an estimator for the sampling situation originally considered with the change-in-ratio method. Assume that nll and n21 animals of two types are counted during an initial sampling period, with/'11 and/'21 known removals. In sampling period 2, n12 and 11122animals of the two types are counted again. If animals of both types are counted in proportion to their abundance in the population (i.e., if underlying detection probabilities are equal for the two types), then the following approximate expectations hold:
(14.38) /'1111122 -- /'21//12
//1 11111//22 m 1112111112
for sampling period 1, and /~2 ~" /~12 q- /~22 -- (/~11 -- /'11) if- (/~21 -- /'21) =
NI
-
(14.39)
ri
for sampling period 2. The abundance estimator for sampling period 1 also can be derived by writing the proportion of type I animals in the population at sampling period 2, N12/N2, as a function of the removals and the proportions of type I animals in the population at sampling period 1, Nil~N1" N12
N l l - rll
N2
N1 - r 1 (N11/N1)N1
-
rll
N 1 - r1
E(n111 ~ \11121/
Pll Nll P21N21
(14.34)
Nll N21
Substitution in the above expression of the estimators nil~n1 and n12/n2 for the proportions Nil~N1 and N12/N2 of type 1 animals in the population at the two sampling periods [see Eqs. (14.34) and (14.35)] yields the intuitive estimator
and
E[n12~l l ~" P l 2 ( N l l \/'/22./
/~1 = /'11 -nil/n1
r11)
P22(N21 - r21 )
(14.35)
Nll - rll N21 - r21"
Note that expressions (14.34) and (14.35) require equal encounter probabilities (Ply = P2j) for the two types within each sample, but allow for different encounter probabilities for the two sampling periods. The expressions can be combined to yield the following estimators for the number of animals in each type in the population before removal: *'/~111 /'1111122- /'2111112 -11111 1111111122 -- 1112111112
(14.36)
/~/21
(14.37)
and =
/ ' 1 1 n 2 2 - /'21n12 n21. 1111111122- //21//12
Expressions (14.36) and (14.37) are equivalent to the intuitive estimators of Kelker (1940; see Udevitz and Pollock, 1992). Based on Eqs. (14.36) and (14.37), the estimators/~1 and/~2 of abundance are simply
/'1(n12/n2 )
(14.40)
-- //12///2
(e.g., see Paulik and Robson, 1969; Seber, 1982; Pollock
et al., 1985b). It is not difficult to show that the estimators in Eqs. (14.38) and (14.40) are mathematically equivalent. The estimators in Eqs. (14.38) and (14.40) are dependent on reasonable estimators for the proportions of type 1 animals in the population at each sampling period. If the sample counts nj represent random samples of predetermined size taken with replacement (this corresponds to the usual case in which the counts are based on observations of unmarked animals), then the type-specific counts nij can be modeled as binomial random variables conditional on the sample counts and the true numbers of animals in the population of each type at each sampling period:
f({nij}l{Nij'nj}) = ~l (\nlj,n2j,) (Nlj~Nj lj(N2j~]2j\ \ Nj / "
(14.41)
Under this product-binomial model, the proportion nlj/n2j of type I animals in the observed sample is the maximum likelihood estimator for the true proportion in the population (e.g., see Chapman, 1954; Seber, 1982; Pollock et al., 1985b).
Chapter 14 Mark-Recapture Methods for Closed Populations
328
If the sample sizes nj are not fixed, then the counts nij can be modeled using the Poisson distribution. Under this model, expression (14.41) becomes the conditional distribution for the nij (Seber, 1982; Pollock et al., 1985b), and estimation proceeds in the same manner as above. The above estimators are based on an assumption that individuals of the two types are detected in proportion to their true abundance in the population (i.e., have equal detection probabilities), although these probabilities are permitted to differ between the two sampling occasions (i.e., we might encounter 15% of the population in one sample and only 10% in another). Now consider the case of equal detection probabilities for the two sampling periods, but different detection probabilities for animals in the two types. In this situation, the ratio r/il/ni2 provides an unbiased estimate of the ratio of type i individuals in the population in sampling periods 1 and 2 (see Udevitz and Pollock, 1992). Equating these sample ratios with the corresponding true values for the population yields E[n11] ~ Ln12J
Nll Nll
-
rll
and
E[H211 ~ Ln223
N21
N21
Y/11r11 /'/11
--
The basic change-in-ratio method was extended by Otis (1980) to include the identification and possible removal of three types of animals (e.g., adult males, adult females, young). Later, Pollock et al. (1985b) focused on two types of animals that are sampled at three sampling occasions separated by two removals. This additional sampling permits robust estimation of abundance in the face of unequal sampling probabilities and provides the data required to test assumptions about the sampling probabilities. Here we present the general formulation of Udevitz and Pollock (1991) for K -> 2 sampling periods, K - 1 intervening removal periods, and k >-- 2 types of animals. Define cij as the probability that a given encounter in sample j will be with a particular animal from type i. The probability that a given encounter in sampling period j will be with any individual from type i is cijNij. These probabilities are conditional on an encounter and hence sum to 1" k
(14.42)
/'/12
and /~21 =
14.5.2.2. Generalization to Multiple Samples, Removals, and Animal Types
-- r21
This system of two equations with two unknowns can then be solved to yield the intuitive estimators /~11 --
We obtain the estimated number/~il of animals in type i at sampling period 1 by dividing the number ril of animals removed by this probability. This procedure again yields the estimators in expressions (14.42) and (14.43).
n21F21 /'/21 -- /'/22
s cijNij-i=1
1
(14.44)
for all j. As an illustration of this general expression, return to the initial example of two sampling periods and two types of animals. The assumption of equal encounter probabilities for the two types in the estimators of Eqs. (14.36)-(14.40) can be written as
(14.43)
Cll -- C21'
(14.45)
C12 = C22.
(e.g., Udevitz and Pollock, 1992). Abundances at sampling periods 1 and 2 then can be estimated in the general manner illustrated in Eqs. (14.38) and (14.39). Note that the estimators of expressions (14.42) and (14.43) can also be viewed in the context of the canonical estimator (12.1). Under this perspective, we view ril as a sample from the type i animals in the population at sampling period 1. Under the assumption of equal detection probabilities in the two sampling periods, the estimated probability of an animal appearing in the removal sample is given by Hi1 -- Yli2 Hi1
Combining the constraints of Eqs. (14.44) and (14.45), we can write the probability of encountering any particular individual in a study with two types of animals as
Clj = c2j = 1/(Nlj 4- N2j) forj = 1, 2. Udevitz and Pollock (1991) also recommend rewriting the constraints of expression (14.45) as C21/Cll -- C22/C12, C21/Cll ":
(14.46)
1,
in order to emphasize the two components of the equal encounter probability assumption underlying the esti-
14.5. Change-in-Ratio Methods mators of Eqs. (14.36)-(14.40). The first equality in Eq. (14.46) expresses the assumption that the ratio of encounter probabilities for individuals of the two types remains constant over time. The second component specifies the actual value of this ratio (1 in this case). The assumption of equal encounter probabilities, expressed by both equalities in Eq. (14.46), is stronger than the "constant probability ratio" assumption of the first component of Eq. (14.46). The general approach developed by Udevitz and Pollock (1991) for multiple types and sampling periods also uses the constant probability ratio assumption, expressed generally as Cil/Cll "-- Cij/Clj
= hi, i = 2, ..., k, j = 2, ..., K, where the parameters h i are defined as the ratio of encounter probability for individuals in type i to that of individuals in type 1, with K1 = 1. Then the probability distribution of the {nij} under the general model of Udevitz and Pollock (1991) can be written as
f({nij}l{Nij, hi, nj}) - .il~~
---
X ./I~1~2k= 1 hi Xij
(14.47)
.
This general formulation can be used to derive estimates under the special cases described thus far. If there are only K = 2 sample occasions (the original change-in-ratio design), then the parameters of expression (14.47) are not identifiable without an additional constraint. For example, if we assume that K2 -- a2 (some positive constant), then maximum likelihood estimates of remaining parameters can be obtained. If a 2 -- 1 for the sampling design with K = 2 periods and k - 2 types, then the maximum likelihood estimates based on expression (14.47) are the intuitive estimators of expressions (14.36) and (14.37). Similarly, the special case of K = 3 periods and k = 2 animal types yields the estimators first derived by Pollock et al. (1985b). Other models can be developed and tested by imposing constraints on the h i. Udevitz and Pollock (1991) provide computer code for obtaining estimates using iteratively reweighted nonlinear least squares. The key assumption underlying the general model in Eq. (14.47) is that the ratios h i of encounter probabilities remain constant over the different sampling periods. Udevitz and Pollock (1995) developed an approach that uses additional information in order to relax this assumption (also see Chapman and Murphy,
329
1965). The additional information is the effort expended to obtain each set of sample counts (the nij for each sampling period j). A catch-effort modeling approach (Section 14.4) then can be used to generalize the model of expression (14.47) in a manner that permits various forms of temporal variation in the relative encounter probabilities of the different types.
14.5.3. Violation of Model Assumptions The assumption of population closure except for known removals can be violated by any gains to the population via reproductive recruitment or immigration and by losses from deaths or emigration. If the studied population is not closed, then the sample counts for periods following the initial sample will be improperly modeled. Consider the situation in which there is mortality between the first sampling period and the period of removal for the standard two-sample, two-type, change-in-ratio study. If mortality rates are the same for both types of animals, then the abundance e s t i m a t o r / ~ 1 of Eq. (14.39) now estimates abundance after mortality and just before the removals (see Paulik and Robson, 1969; Seber, 1982). The closure assumption also can be violated by unknown removals associated with illegal or otherwise unreported harvest, or with crippling loss of animals that are not retrieved. For the standard case of two samples and two types of animals, the bias of the estimates for abundance at the time of each sampling period can be evaluated using expressions provided by Paulik and Robson (1969; also see Chapman, 1955). If the proportions of unreported kills are the same for both types of animals, then abundance estimates for times 1 and 2 will be negatively biased. For example, if the reported kills of animals of both types is 15% too low, then the true abundance at time 1 will be approximately 15% larger than the estimated value (see Paulik and Robson, 1969; Conner et al., 1986). Most of the estimators discussed in this section were derived assuming that either sampling is with replacement, or else the sampling fractions are negligible. However, when sampling is carried out without replacement for the two-sample, two-type, change-inratio method, the resulting hypergeometric model (see Appendix E) yields the same maximum likelihood estimates for abundance as does the binomial model described above (Eq. 14.41). However, the asymptotic variances do differ for the two modeling approaches (Seber, 1982; Pollock et al., 1985). Encounters of animals during sampling periods are assumed to be independent. Although we are aware of no work on effects of violation of this assumption (e.g., when animals travel as pairs or family groups
330
Chapter 14 Mark-Recapture Methods for Closed Populations
such that encounters are not independent), we suspect that it will not lead to biased estimates of abundance but will instead produce negatively biased variance estimates. As noted above, the two-sample, two-type, changein-ratio method was developed initially assuming equal encounter probabilities for animals of the two types (Kelker, 1940; Chapman, 1954, 1955; Seber, 1982). When this assumption is not true (i.e., when )~ 4: 1), then n11/n I and n12/n 2 will be too small or too large when viewed as estimators for N 1 1 / N 1 and N12/N2, yielding biased estimates of abundance [see Eq. (14.39)]. On the other hand, when all removals are of a single type, the abundance estimate for the type removed and its estimated variance are unbiased, even in the face of different encounter probabilities for animals of different types (e.g., see Seber, 1982; Conner et al., 1986).
14.5.4. Study D e s i g n The design of studies utilizing change-in-ratio methods should include efforts to minimize the probability of violating model assumptions and to maximize estimator precision. As with other estimation approaches, the assumption of population closure except for known removals is best met by restricting the temporal extent of the study. The longer the study, the more likely that numbers of animals will be influenced by movement, deaths, and recruitment. Similarly, it is desirable to restrict studies to seasons of the year when migration, mortality, and reproductive recruitment are minimal. With respect to geographic closure, studies carried out on areas with clear boundaries over which movement is rare are most likely to be successful. In the two-sample, two-type situation, animals of the two types must be encountered in samples in proportion to their abundance in the population, so sampiing methods should be selected with that assumption in mind. If it cannot be met, then the ratio of encounter probabilities sometimes can be estimated independently with a separate experiment, e.g., based on a marked subsample or the use of a double-observer approach (Section 12.6). These independent estimates then can be used in the estimation of abundance (see Chapman, 1955; Seber, 1982). Perhaps the best way to deal with the assumption of equal encounter probabilities for the different animal types in the population is to implement a study design that does not require it. The designs based on more than two samples permit differences in detection probabilities of the different types and require only that the ratios of detection probabilities for the different types remain constant over all sampling periods (Pollock et
al., 1985b; Udevitz and Pollock, 1991, 1992). Even this assumption can be relaxed when the sampling design includes the recording of the amount of effort expended on the different samples (Udevitz and Pollock, 1995). In many cases, the number of removal periods will be dictated by the management program (e.g., when the removals are via sport or commercial harvest). However, whenever there is design flexibility regarding the numbers of sampling and removal periods, studies can be designed in ways that require minimal assumptions about type-specific and temporal variations in detection probabilities. Expressions and associated figures relating sample sizes (e.g., n I and n 2) to accuracy of resulting abundance estimates are presented for the two-sample, twotype situation by Paulik and Robson (1969) and Conner et al. (1986). The graphs in these papers are especially useful in planning a change-in-ratio study under the traditional design. A quantity of critical importance to estimation is the difference in the proportional composition of the population between the first and second sampling periods, i.e., the magnitude of the change in ratio of the types:
Nll
N12
ml
N2
- AP.
Paulik and Robson (1969) declared &P < 0.05 to be "almost worthless as a means of determining population abundance." They questioned the use of the change-in-ratio method for situations in which AP < 0.10, although Conner et al. (1986) obtained reasonable results with an estimated change in ratio of &/5 ~ 0.07. For a given sample size and change in proportions, the precision of estimates is higher when the total proportion of the population removed is higher and when the initial type proportions are more dissimilar (Paulik and Robson, 1969; Udevitz and Pollock, 1992).
14.5.5. Example We report a hypothetical example used by Udevitz and Pollock (1991) to illustrate their general approach. They assumed a population with three animal types, sampled with replacement at sampling period 1 to obtain n I -- 500 encounters consisting of nll = 128 animals of type 1, n21 119 animals of type 2, and n31 = 253 animals of type 3. The first sampling period was followed by the type-specific removals of rll = 140, r21 = 280, and r31 = 560. A second sample then yielded n12 = 227, n22 = 167, and n32 = 106. Using the constraint that the ratio of individual encounter probabilities for types 1 and 2 are equal, )k2 = a 2 = 1 (see Section 14.5.2), the estimated ratio of =
14.6. Discussion encounter probabilities of type 3 to type I individuals is K3 = 2.58 (SE = 1.55). Udevitz and Pollock (1991) presented the following type-specific abundance estimates for sampling period 1: /~/1~ = 912 ^
(S'E = 632), A
N21 = 848
(SE = 495),
and ^
A
N31 = 700
(SE = 43).
Despite the removal of fairly large numbers of animals, the above abundance estimates are very imprecise, illustrating a feature of nearly all change-in-ratio estimates.
14.6. D I S C U S S I O N In this chapter we have described methods for estimating abundance based on captures of animals. Sections 14.1 and 14.2 concerned capture-recapture methods in which animals are caught, given individual marks, and then recaptured, all over relatively short time periods. A short period for the investigation increases the likelihood that the population remains closed to gains and losses over the period of sampling. The resulting data can be written as individual capture histories, vectors of ls and 0s indicating the sequence of captures for each individual during the study. Closed models do not require parameters for gains and losses, so the modeling of capture history data involves only capture probability parameters. These parameters can be defined in terms of three potential sources of variation (time, heterogeneity, and behavioral response), and models were developed to include one or more of these sources. The model underlying the two-sample LincolnPetersen estimator of Section 14.1 permits only temporal variation in capture probability. This model is useful in many field situations (e.g., Seber, 1982; Menkens and Anderson, 1988) but also provides an intuitive foundation for the use of capture-recapture models to estimate population parameters. This foundation underlies all of the more complicated capture-recapture models for both closed (Section 14.2) and open (Chapters 17-19) populations. The consequences of the violation of model assumptions for Lincoln-Petersen estimation were discussed thoroughly, as a basis for deducing consequences to estimators in more complicated models. The K-sample closed-population models of Section 14.2 form a complete set of models for estimating abundance in the face of the three sources of variation in
331
capture probability listed above. However, the model testing and selection tools developed for likelihoodbased models (Chapter 4) are not uniformly available because the models incorporating individual heterogeneity have too many parameters, and estimation utilizes ad hoc approaches such as the jackknife, bootstrap, and sample coverage. As noted throughout Section 14.2, the finite mixture models of Norris and Pollock (1995, 1996) and especially Pledger (2000) provide a solution to this problem. Once software becomes widely available for them, we expect these heterogeneity models to see substantial use. Section 14.3 describes some approaches to the difficult problem of estimating density from capturerecapture data. One approach involves first estimating abundance (e.g., as in Sections 14.1 and 14.2) and then estimating the area from which captured animals are sampled. This approach usually involves grid sampling, in which a boundary strip of estimated width W is added to the perimeter of the study area in order to compute the area sampled by the grid. Another approach involves the use of a gradient in trap density, via distance sampling (Chapter 13) or the geometric approach of Link and Barker (1994). Because the latter approaches to density estimation have been infrequently used, we have limited experience with their performance. The capture-recapture methods presented in Sections 14.1, 14.2, and 14.3 are likely to be useful for animals that are secretive, nocturnal, or simply difficult to observe. In general these methods should not be considered for animals that are easily observed, because the observation-based methods presented in Chapters 12 and 13 should be preferable. On the other hand, the utility of the removal methods presented in Sections 14.4 and 14.5 is tied less to the observability of the target organisms and more to the existence of harvesting operations (e.g., hunting, fishing, trapping). Catch-effort models and change-in-ratio methods are designed to use catch information (e.g., the timespecific numbers of animals harvested) as a means of estimating population size. Estimators based on both approaches tend to be relatively imprecise unless the harvest represents a substantial proportion of the population. Nevertheless, in the absence of independent monitoring programs, such efforts to estimate population size using information from the harvest may be essential to the success of harvest management programs. Chapters 15 through 20 use variations of the capturerecapture models introduced in this chapter. Unlike population size, which can be estimated with direct counts as well as capture-recapture methods, estimation of demographic rate parameters such as move-
332
Chapter 14 Mark-Recapture Methods for Closed Populations
ment and survival rates typically requires the use of marked individuals. Chapters 17 and 18 thus use capture-recapture modeling for open populations to estimate abundance, survival, and movement. Notwithstanding the need to incorporate additional parameters in models for open populations, the underlying approach with these models is similar to that introduced here for closed models. Chapter 19 de-
scribes a "robust design" in which both open and closed models are used in a single study design, wherein the closed models in this chapter are components of larger, more inclusive models. Finally, in Chapter 20, the closed models of this chapter and the robust design of Chapter 19 are used to estimate parameters at the community level of biological organization.
C H A P T E R
15 Estimation of Demographic Parameters
15.1. DETECTABILITY AND DEMOGRAPHIC RATE PARAMETERS 15.1.1. Population Growth Rates 15.1.2. Survival Rates 15.1.3. Movement Probabilities 15.1.4. Reproductive Rates 15.1.5. Summary 15.2. ANALYSIS OF AGE FREQUENCIES 15.2.1. Life Tables 15.2.2. Survival Estimation from Sample Age-Structure Data 15.2.3. Population "Reconstruction" 15.3. ANALYSIS OF DISCRETE SURVIVAL AND NEST SUCCESS DATA 15.3.1. Binomial Survival Model 15.3.2. Models for Estimating Nest Success 15.3.3. Radiotelemetry Survival and Movement Studies 15.4. ANALYSIS OF FAILURE TIMES 15.4.1. Statistical Models for Failure Time, Survival Time, and Hazard Rate 15.4.2. Parametric Survival Estimation 15.4.3. Nonparametric Survival Estimation: Kaplan-Meier 15.4.4. Incorporating Explanatory Variables: The Proportional Hazards Model 15.4.5. Assumptions of Failure Time Models 15.4.6. Design of Radiotelemetry Studies 15.5. RANDOM EFFECTS AND KNOWN-FATE DATA 15.6. DISCUSSION
gued in Part II and elsewhere that a focus on abundance and density is both useful and natural, in that these quantities are often the state variables of interest in models of population dynamics. However, the investigation of population dynamics frequently is not restricted to an assessment of population size alone. Depending on study objectives it is useful, and often essential, to include information about the biological processes that influence population dynamics. In Chapters 15-19 we turn our attention to the rates of survival, reproduction, and movement that ultimately are responsible for changes in abundance. An emphasis on estimation of demographic parameters, and on quantifying variability in these parameters, is important for several reasons. First, estimates of abundance at a single point in time obviously provide no information about population dynamics, though a series of estimates of abundance may provide insights about the trajectory of the population. However, even a time series of abundance estimates provides only limited information about which demographic processes contribute to the observed dynamics and thus about why the population behaves as it does. Second, demographic rates provide a more detailed picture of the "health" of the population and, when used with population models (Chapter 8), may be useful in forecasting future population growth. Third, most of wildlife management seeks to control populations at desirable levels, which in turn requires an understanding of the factors that influence survival and reproduction rates. Although management objectives frequently are expressed in terms of population size, management actions often focus on the control of demographic pa-
In Chapters 12-14 we described methods for estimating abundance or density of a population. We ar-
333
334
Chapter 15 Estimation of Demographic Parameters
rameters associated with survival, reproduction, and movement in order to bring about desired changes in abundance. An understanding of how these demographic parameters vary in space and time, and in relation to environmental and management factors, is fundamental to the understanding and proper management of animal populations. With respect to estimation methodology, the methods of Chapters 12-14 were based on the assumption that the population is both geographically and demographically closed. By geographic closure is meant that (1) the population is immobile or (2) the geographic area or time scale of the study is such that movements into and out of the population need not be considered. By demographic closure is meant that neither births nor deaths occur over the period of study (or the numbers of births and deaths are negligible). Taken together, these assumptions imply that abundance is constant and can be represented by a single parameter N over the course of the investigation. We note that the creative use of, e.g., stratification in space and time sometimes allows one to use the methods of Chapters 12-14 even if the assumption of closure is somewhat relaxed. In this and the next several chapters, we remove the assumption of closure altogether and explicitly estimate the demographic rate parameters associated with population dynamics, i.e., rates of survival, reproduction, and movement. In this chapter we begin a general exposition of methods for estimating both abundance and demographic parameters for open populations. The methods described here for investigation of demographic parameters follow the same principles that guide the development of estimation methods for closed populations. Thus, the methods (1) are based on sound statistical estimation and sampling procedures, (2) rely on as few assumptions as possible, with sampling and estimation schemes that are robust to assumption violations, and (3) make effective use of limited resources for sampling and estimation. We start with a discussion of general principles, emphasizing the importance of detectability in the estimation of demographic rate parameters. We then discuss several methods that require assumptions of perfect detection. In particular, age frequency analyses are described under the rubric of "life table analysis." We then cover methods of analysis of discrete nest success and survival data, with sampling methodologies that are designed to meet the assumption of perfect detection (for example, by means of radiotelemetry or the monitoring of sessile objects such as nests). Finally, we describe methods for analysis and modeling of failure times, the principal applications of which involve the analysis of data from radiotelemetry studies.
15.1. DETECTABILITY
AND DEMOGRAPHIC RATE PARAMETERS In Section 12.2 we described a canonical estimator for abundance that incorporated two main sources of variation in animal count data, namely, spatial variation and detectability. Both sources are relevant to the estimation of demographic rate parameters, though detailed studies incorporating estimation of rate parameters frequently concern populations at single locations and thus do not involve spatial sampling. Exceptions include programs such as Monitoring Avian Productivity and Survival (MAPS) (DeSante et al., 1995) and the North American waterfowl banding program (Anderson and Henny, 1972; Nichols, 1991a). These large-scale monitoring programs involve estimation of demographic rate parameters at regional and national scales, though point estimates of the rate parameters are obtained at the level of the local sampling unit. If the selection of sampling units is based on an appropriate sampling design (see Chapter 5), these point estimates can be combined to form an estimate that corresponds to the entire area of interest using the approaches of classical sampling theory (e.g., Cochran, 1977; Thompson, 1992) (see also Chapter 5). In the remainder of this section we defer further discussion on spatial variability and focus instead on the much more frequently encountered problem of detectability. Demographic rate parameters include descriptors of overall population change (e.g., population growth rates such as the finite rate of population increase), as well as fundamental demographic parameters, such as rates of survival, reproduction and recruitment, and movement, that are responsible for population change. Estimates of these rate parameters, like those of abundance, nearly always are based on some sort of count statistic and thus require one to account for detectability. Before proceeding to detailed estimation methods for estimating these parameters, we provide a brief motivation for the need to consider detectability in their estimation.
15.1.1. Population Growth Rates Because population growth rate is a function of abundance at two or more points in time, it should be clear from Chapters 12-14 that detectability is an important consideration in its estimation. To see how, define the finite rate of population increase for a population of interest as the ratio of abundances in successive time periods:
Xi-- Ni+l/Ni.
15.1. Detectability and Demographic Rate Parameters As in Section 12.2.1, define C i a s the count statistic (number of animals detected by the survey method, e.g., capture, visual observation, and auditory detection) and ~i a s the associated detection probability (probability that a member of N i is detected and thus appears in Ci). The count can be viewed as a random variable, with expectation given by (15.1)
E(Ci) = N i ~ i.
One approach advocated by many biologists is to view Ci as an index (Section 12.7) and thus to use the ratio (15.2)
~i = C i + l / C i
of counts as an estimate of k i. The expectation of this estimator can be approximated as
periods, then Eq. (15.3) is recommended. A related discussion is presented in Section 14.1.2 on estimating relative abundance under partial detectability (also see Skalski and Robson, 1992).
15.1.2. Survival Rates Consider a study in which R i animals are caught, marked, and released at time i, with a goal of estimating the probability that a member of R i survives until i + 1 (denote this survival probability as q~i). Denote as M i+1 the number of marked animals (members of R i) that are still alive and in the population of interest at time i + 1. This number can be modeled as a binomial random variable [i.e., Mi+ 1 "" Bin(R/, q~i)], SO that the proportion of survivors estimates q~i: ~Pi -- M i + I / R i .
E(~i) ~ E(Ci+I)
E(Ci) Ni+l~3i+l Ni~i
9
From this approximate expectation it can be seen that the ratio of count statistics provides a reasonable estimator for ~ki only if detection probability does not change over time, i.e., only if f~i+l ~ ~i. The bias in estimator (15.2) is a function of the difference between the two detection probabilities, with larger differences leading to more biased estimates. Even if detection probability is viewed as a random variable, the equality E(~i+I) = E(~3 i) still is necessary for the index-based estimator for Xi to be approximately unbiased. Two reasonable approaches for estimating Ki require the collection of data needed to estimate the detection probability ~i associated with count statistic Ci. The first approach is simply to estimate abundance as advocated in Chapter 12, i.e., 1~i -- Ci/~ir
Although some sampling designs permit direct knowledge of Mi+ 1 (see Sections 15.3 and 15.4), this situation is relatively rare. A more typical situation is that a sample of the population at time i + 1 detects mi+ 1 members of Mi+ 1. In this situation mi+ 1 is simply another count statistic that follows the usual relationship described in Eq. (15.1): E(mi+l) = Mi+lPi+ 1
(here we use Pi rather than ~i to characterize detectability, in keeping with the common use of Pi in capturerecapture literature). Because of the inequalities Pi+l ~ 1 and mi+ 1 ~ M i + l , the naive estimator mi+l/ai
~ ~i
based on the count statistic nearly always is biased low (unless mi+ 1 -- Mi+I). However, if the detection probability associated with m i+ 1 can be estimated, then reasonable estimators for Mi+ 1 and survival can be constructed as ]~Ii+l = mi+l/fii+l
and then to estimate ~-i a s Ki -- /~/i+1/1~i"
335
(15.3)
This approach is conservative in the sense that the estimator requires no restrictive assumptions about the detection probabilities f~i. The other approach is first to test for differences between the detection probabilities for the two time periods (H0: f~i+l = [3i) and, if no evidence of a difference is found, then estimate ~ki as in Eq. (15.2). Because of the assumption of equal detection probabilities underlying the latter approach, it tends to be more precise than Eq. (15.3) (see Skalski and Robson, 1992). However, if test results provide evidence of a difference in detectability ~i for the two time
and ~i -- ] ~ i + l / R i .
Detection probability is thus an important consideration in the estimation of survival probability in field studies.
15.1.3. M o v e m e n t Probabilities Consider a study in which Rli animals are marked and released in sampling period i at location 1 in a system of two habitat patches. Assume that with probability S li these animals survive and remain in the
336
Chapter 15 Estimation of Demographic Parameters
study system until sampling period i + 1. Denote the total number of survivors as M]~_1 [as above, this is a binomial random variable, M~_ 1 --- Bin(R~, S~)], with 11 M i + 1 located in patch I at time i + 1 and M ] 2 1 i n patch 2. Thus, 11
1.
two patches. If P~+I and p2+1 can be estimated, then movement probability can be estimated as "12 = M" i1+2 1 / M ~ + I , t~i+l
where
12
M i + 1 = M i+ 1 + M i+ 1 .
Denote as t~ 2 the probability that a surviving member of R~ moved from patch I to patch 2 during the interval i to i + 1 and is thus present in patch 2 at i + 1 [M12i+1 is a conditional binomial random variable, with M~21 " Bin( MIi+I, ~]2)]. If the numbers Mi+111 and 12 M i + 1 of surviving members of R~ at period i + 1 are known, then the movement probability can be estimated as ~ 2 __ Mi+112/MJ~_I 9
(15.4)
Equation (15.4) is reasonable when all surviving animals can be detected at i + 1 (i.e., if detection probabilities are equal to 1), as in some radiotelemetry studies (Nichols, 1996; Nichols and Kaiser, 1999; Bennetts et al., 2001). However, the more frequently encountered situation involves sampling that records mi+ 112 animals 111 animals reto have moved from patch 1 and mi+ maining in patch 1. These animals are detected with probabilities p2+1 and Pi+l 1 for the two patches. Once again, the numbers of detected animals are count statistics (see Section 15.1.1), and their expectations can be written as functions of the numbers of animals in the two patches and the associated detection probabilities: E(mll 11 1 i+1) = Mi+1P1+1
" 11
11
and
15.1.4. Reproductive Rates Reproductive rate frequently is defined as the number of young animals at time i + & that are produced by an adult at time i, with A typically a relatively small time step. For example, reproductive rate for mallard ducks might be defined as the number of young fledged female mallards in August of year i per adult female mallard in the breeding population in May of year i (at the approximate time of breeding). Age ratio at a particular time of the year often is used to approximate or index reproductive rate. Thus, the number of young mallards per adult in August of year i is used to index the reproductive rate of year i (e.g., see Anderson, 1975a; Martin et al., 1979; Johnson et al., 1997). If N (~ i and NI 1) are the true numbers of young (age = 0) and adult (age = 1) animals in the population at time i, then we can define the age ratio as Ai
9~12 2 -- IVIi+lPi+I.
A naive estimator for t~ 2 frequently is constructed
N ~-i! ~
E(n!%, , = NI~
(1))
12
12
2
Mi+lPi+I
11
i
9
(~
and
12 / m 1. mi+l i+1,
where mi+ 11" = mi+111 + mi+1,12 with expectation approximated by E(t~ 2)~
1
Mi+I = mi+l/Pi+l.
as ~2=
12
Of course, we seldom know the numbers of animals in any age class in the population; instead, the population must be sampled to obtain the numbers n~~ and n! 1) of young and adult animals detected at time i. The expectations for these random variables are
and
E(m i+1) 12
" 12
M i + 1 = mi +l / ~2 +l
1
Mi+lp2+I + Mi+1P1+1
9
E(n i
(1)p(1)
i ,
where P i(0) and pl 1) 9 are the age-specific detection probabilities associated with the count statistics. A naive estimator for age ratio is constructed as ai
It is clear from this approximate expectation that the naive estimator is biased if Pi+I 1 ~ p2+1. Thus, the move1 < p2+1 and ment probability is overestimated for Pi+l underestimated f o r P~+I > p2+1. As in Sections 15.1.1 and 15.1.2, the ability to estimate movement probability thus depends on the estimation of detection probabilities associated with the
= Ni
=
(0)-
ni
/ni
(1)
9
However, the approximate expectation of this estimator can be written as E(ai) ~ Xl~176
,~. i
9
and thus is a function of not only the actual age ratio a i but also the ratio of age-specific detection probabili-
15.2. Analysis of Age Frequencies ties. As with the previous sections, the naive estimator m a y not perform well if pl 1) :~ Pi(2) 9 Estimation of the detection probabilities permits unbiased estimation of the true n u m b e r s of animals in each age class. Thus, one can use /~!o) ,
=
n
I0)
/fii
(o)
and (1)/ (1) /~I 1) = ni -fii , to estimate the age ratio as Ai
/~/!0)//~(1)
15.1.5. Summary
Like the estimators of abundance, estimators of demographic rate parameters typically are based on count statistics and thus are functions of both n u m b e r s of animals and the detection probabilities associated with sampling. Naive estimators of demographic rate parameters typically are constructed as ratios of count statistics and therefore are biased unless detection probabilities are either equal to 1 (as with survival rate estimators) or are equal for different groups of animals (as with rates of increase, m o v e m e n t probabilities, and reproductive rates). Like abundance estimation, a key to estimation of demographic rate parameters is to collect the data needed to estimate detection probabilities associated with the count statistics. These data permit the testing of critical assumptions that underlie the naive estimators. If testing provides evidence that the assumptions are indeed true, then the estimators based solely on count statistics m a y perform well (see Skalski and Robson, 1992). If the tests fail to provide such evidence, the investigator should use estimators that directly incorporate detection probabilities. In either case, the key to successful estimation of rate parameters is to obtain the data needed to make inferences about detection probability.
337
a set of parameter values, these models can be used to project the numbers of animals in each age class through time. In this section we address essentially the reverse problem, i.e., to make inferences about demographic parameters, particularly survival rates, given the observed fates of cohorts of individuals, patterns in age structure, or combinations of both. The types of data used for these inferences are organized in a format generically k n o w n as a life table. As seen below, u n d e r certain circumstances, life tables can be used to obtain valid estimates of survival or other parameters. Using the notation of Section 8.4 we consider a population consisting of k age classes, with population growth according to a birth-pulse model. Start with an assumed cohort of birth-class individuals at N0(0). The n u m b e r of individuals in this cohort that survive to subsequent ages can be obtained by repeated application of
Ni+l(t + 1 ) = Si(t)Ni(t),
where Si(t) is the survival rate from time t to time t + 1 of individuals in age cohort i at time t. Over the cohort's full life cycle, age-specific survival is given by
Si(t) = Ni+l(t + 1)/Ni(t).
15.2.1. Life Tables
In Chapter 8 we considered population models that incorporate age structure, whereby the projection of population growth is a function of age-specific survival and reproduction rates (see Section 8.4). In these models the transition of age cohorts through time is a function of fixed survival and reproduction parameters. A s s u m i n g an initial population age structure and
(15.6)
These calculations are illustrated in an artificial example from Seber (1982) and presented in Table 15.1. In the example, N0(0) = 1000 animals are followed from birth at t = 0 until all have died. Thus, survival from birth to age 1 over (0, 1) is S0(0) = NI(1)/No(O) = 250 / 1000 = 0.25. On the other hand, a different cohort of 1200 animals, born in the next year (i.e., at age i = 0 in year t = 1),
TABLE 15.1
Example Cohort Life Table a
Cohort t = 0 15.2. A N A L Y S I S O F AGE FREQUENCIES
(15.5)
Cohort t = 1
Year (t)
Age (i)
[Ni(t)]
Si(t)
Age (i)
[Ni(t)]
Si(t)
0 1 2 3 4 5 6
0 1 2 3 4 5 6
1000 250 40 10 3 1 0
0.25 0.16 0.25 0.30 0.33 0.00 m
m 0 1 2 3 4 5
__ 1200 400 125 50 40 30
m 0.33 0.31 0.40 0.80 0.75
aAlso known as age-specific (horizontal) life table (Seber, 1982).
Chapter 15 Estimation of Demographic Parameters
338
has age-specific survival calculated over the interval (1,2) as
It follows that
= 400/1200 = 0.33. Age-specific survival m a y or m a y not be the same for different time intervals. The example in Table 15.1 illustrates a case in which it is not the same, i.e., survival is both age specific and cohort specific. If the n u m b e r s surviving in each of a series of cohorts are available, one can determine survival over each interval (t, t + 1) for each age class i and thus separate temporal (cohort-specific) variation in survival from age effects. In practice, multiple-cohort data are seldom available, and assumptions must be m a d e about the nature of age or cohort specificity in order to estimate parameters uniquely (Udevitz and Ballachey, 1998). These assumptions have serious implications as to the generality of life table approaches, as illustrated below and more fully in Section 15.2.2. Information on fates from a series of cohorts is sometimes called an age-specific or horizontal life table and is obtained in one of two ways: either by recording the ages of all the individuals at death (d x series) or by recording the numbers still alive at each time (age) x (l x series) (Seber, 1982). Both types of data m a y be difficult to collect and in practice both are obtained via sampling procedures that m a y lead to serious bias, as discussed further in Section 15.2.2. On assumption that the population is (1) at a stable age distribution and (2) stationary (i.e., K = 1), it m a y be possible to use the standing age distribution, also k n o w n as a time-specific or vertical life table, to calculate age-specific survival rates. To see w h y these assumptions are needed, consider the age distribution Ni(t), i = 1, ..., k for a single year t. To obtain the vertical life table estimate of age-specific survival one calculates the ratio of successive age frequencies at the same time t: S*(t) = N i + l ( t ) / N i ( t ) .
(15.7)
The numerator of Eq. (15.7) can be expressed via Eq. (15.5) as Ni+l(t) = S i ( t -
1)ci(t-
1 ) N ( t - 1),
(15.8)
where ci(t) = N i ( t ) / N ( t ) is the proportion of the entire population at t in age class i. Substitution of Eq. (15.8) into Eq. (15.7) then produces S*(t) =
Si(t-
1 ) N ( t - 1) ci(t)N(t)
1)ci(t-
1)
S*(t) = S i ( t -
S0(1) = N 1 ( 2 ) / N o ( 1 )
only w h e n the population is both stable [i.e., ci(t) = ci(t - 1) = c i for all i] and stationary [N(t) = N ( t - 1)], which in turn requires stationary age-specific survival: Si(t) = S i ( t - 1 ) = S i. Example
Assume that a population is both stable and stationary, with 1000 individuals entering the population each year (Table 15.2). Under conditions of stationarity and stable age distribution, the same n u m b e r of individuals is in each age class each year, and the vertical age structure is constant over time. By year 5 the n u m b e r of individuals from an initial cohort of 1000 that are still alive each year is fully described, and the vertical and horizontal life tables have converged. Thus, the standing age distribution is an accurate representation of survival rates from each of the original cohorts, and age-specific survival rates from Eq. (15.7) are accurate (Table 15.2a). N o w relax the assumption of stationarity to allow a stable age distribution but nonstationary growth. With an increasing population (Table 15.2b), the calculations from Eq. (15.7) no longer faithfully represent survival, but instead are distorted by the increasing population size [N(t) = N ( t - 1)M. As a result, Eq.
TABLE 15.2 Relationship between Cohort (Horizontal) and Time-Specific (Vertical) Life Tables Yea~
Age
1
2
3
4
5
1000 250 40 10 3
1000 250 40 10 3 0
1728 360 48 10 3
2074 432 58 12 3 0
Stable, stationary~ 0 1 2 3 4 5
1000
1000 250
1000 250 40
Stable, nonstationaryb 0 1 2 3 4 5
1000
1200 250
1440 300 40
aStable age distribution with stationary population. bStable age distribution with nonstationary population (~ -- 1.2).
15.2. Analysis of Age Frequencies (15.7) underestimates survival by 1/)t. For example, in year 4 S~(4) = 360/1728 = 0.21, whereas actual survival (constant for all years, because this population is at stable age distribution) is S0(4) = SO = 0.25. In cases where the population is not at a stable age distribution, there is no guarantee that the survival rates based on Eq. (15.7) will be reliable even as indices. In general it is not possible to use age frequency data alone to both estimate age-specific survival and to test the assumptions of age stability and stationarity (Seber, 1982). The assumption of stationarity may be relaxed if independent data on population growth rate ()~) or age-specific reproduction rates (F x) are available to allow estimation of age-specific survival rates (Caughley, 1966; Michod and Anderson, 1980); however, the assumption of age stability is still required. If age distributions are recorded for a number of years and do not appear to be temporally varying, it may be possible to infer age stability and to compute survival estimates from the standing age distributions, again provided that estimates of )~ or F x are available. Finally, we note that Eq. (15.7) assumes known age distributions, even though information on age distributions typically is obtained via sampling methods with agespecific detection probabilities (Section 15.1.4), leading to additional problems. Unfortunately, age distribution methods are in common use, with little heed paid to these critical assumptions. In keeping with the general philosophy of this book, we strongly recommend the use of methods such as radiotelemetry (Sections 15.3 and 15.4) and capture-recapture (Chapters 16-18), which do not require assumptions such as age stability or stationarity that are unlikely to be met in practice, particularly for populations that are harvested or are subject to environmental variation.
15.2.2. Survival Estimation from Sample Age-Structure Data Though the development above is strictly deterministic, it can be extended to allow for the stochastic nature of birth-death processes, still under the assumption that either a complete accounting of the fates of all cohorts (horizontal approach) or of the entire age profile (vertical approach) is available. The usual situation involves data that arise from both a stochastic demographic process and a sampling process. Seber (1982; also see Seber, 1986, 1992) reviewed models that deal with one or both processes and provided estima-
339
tors based on either horizontal or vertical data structures. Most of these models, though of historic interest, are not considered in a general framework such as maximum likelihood estimation and depend to varying degrees on assumptions that often cannot be evaluated. Udevitz and Ballachey (1998) provided a unified framework for survival estimation from agestructured data, which allows for maximum likelihood estimation, model selection, and model evaluation, utilizing sample data from standing age distributions and ages at death. The development below is based on their framework, with modifications for notational consistency. In what follows, likelihoods are developed separately for each type of data structure, under the very general assumptions that the age structure may not be stable and population growth rates are unknown. From Eq. (15.8) the number of individuals in age class i - 1 at time t - 1 that survive to t is given by Ni(t) = N ( t -
1)Ci_l(t-
1)Si_l(t-
1).
(15.9)
By subtraction, the number of individuals in age class 0 at time t is k
No(t) = N ( t ) - N ( t -
1) ~ C i _ l ( t -
1)Si_l(t-
= N(t-
[
1) (15.10)
i=1
k
1) M t - 1) - ~ C i _ l ( t - 1)$i_1(t- 1)
1
i=1
[because )~(t - 1) --- N ( t ) / N ( t - 1)]. From Eq. (15.9) it is easy to see that the number of individuals in age class i that die between times t - 1 and t is N(t-
1 ) c i ( t - 1)[1 - S i ( t -
1)].
(15.11)
15.2.2.1. Model Likelihoods
Assume that we have a random sample xi(t), i = O, ..., k, from the age frequencies in the population at time t. Then by Eqs. (15.9) and (15.10)
I
k
]
E[xo(t)] = oLN(t - 1) )t(t - 1) - ~ Ci_l(t -- 1)Si_l(t - 1) i=1
and E[xi(t)] = e ~ N ( t -
1)Ci_l(t-
1)Si_l(t-
1)
for 0 ~ i -< k, where oL is the probability of sampling any individual from the population age distribution, assumed to be independent of age. Conditional on
340
Chapter 15 Estimation of Demographic Parameters
the total sample size n(t), a multinomial likelihood for these data is given by
P[{xi(t)}ln(t)l =
n(t)!
I-Iik xi(t) ! X(t
1/
k(Ci_l(t- 1)Si_l(t- )xi(t) X ( t - 1)
where n(t) = ~ i xi(t). Note that the conditioning on n(t) removes the need to consider o~ in the likelihood. Assuming a r a n d o m sample of natural deaths between t - 1 and t, from Eq. (15.11) we have
E[yi(t)] = f 3 N ( t - 1 ) c i ( t - 1)[1 - S i ( t -
Si -~ Yi+l(t))k xi(t ) ,
(15.12)
1)
x ~.=
Under the assumption of age stability, the time index for the parameters is eliminated from both likelihoods. Thus simplified, the MLE for age-specific survival can then be obtained from the sample standing age data
as
k I)k(t -1) - ~,i=l Ci_l(t -1)Si_l(t-1)l x~ x
15.2.2.3. Known Stable Age Distribution
1)],
where yi(t) is the n u m b e r of animals of age i at time t - I that die between t - I and t and [3 is the probability of sampling any individual from the population of ages at death, assumed to be independent of age. The conditional likelihood for the ages at death is
(15.14)
i = 0, ..., k - 1, where ~ is known. Under the special case of X = 1, Eq. (15.14) is the naive estimator of survival from age distribution data; the more general estimator has been described by Caughley (1977), among others. On condition that X has a k n o w n value that is not unity, Udevitz and Ballachey (1998) provide an estimate of variance for this estimator using the delta method (see Appendix F): va'~r(Si) = [C32/n(t)] [1/c,i(t) + 1/c,i+l(t)],
(15.15)
i = 0, ..., k - 1, where
k n(t) = ~, xi(t) i=0 and
p[{yi(t)}lm(t) ] =
k I-[
m(t)! i=0 yi(t)! i=0
IF
?,i(t) = xi(t)/n(t). (15.13)
[ c i ( t - 1)[1 - S i ( t - 1)] I y;(t) x L~ki=oCi( t _ 1)[1 - S i ( t - 1)] with m(t) = ~ i yi(t), where again the sampling probability [3 disappears because of this conditioning. If independent data are available from both a standing age distribution and ages at death, a joint likelihood is formulated as the product of Eqs. (15.12) and (15.13) (see Udevitz and Ballachey, 1998).
15.2.2.2. Parameter Estimation The parameters for either of the above likelihoods are the population growth rates X(t), age-specific survival rates Si(t), and age class proportions ci(t). In the usual case where both the survival rates and growth rates are assumed to be time independent [i.e., Si(t) = Si; X(t) = ~] there are still 2k+1 parameters to estimate under either data structure. These parameters are not identifiable without additional assumptions. The usual assumptions are (1) that the age distribution is stable and (2) that ~ is known. Udevitz and Ballachey (1998) show that if both data structures are used with the joint likelihood [product of Eqs. (15.12) and (15.13)], then these assumptions can be relaxed one at a time.
For )~ = 1 this expression simplifies to v~r(Si) = (C32/n) (1/ci
-}- 1/ci+1),
(Seber, 1982), because the assumption of stationarity allows one to estimate age-specific survival from a single age frequency distribution. If ~ is independently estimated rather than known, Eq. (15.15) must be modified by adding the term
(Si/~)2 var(~). Similarly, the MLEs for age-specific survival from the ages-at-death data are
Si- 1 -
yi(t))k i
k
~'j=i yj(t) xj
,
(15.16)
i = 0, ..., k - 1, and are the usual estimators (e.g., Caughley, 1977) w h e n X is known. A variance expression for this estimate is provided by Udevitz and Ballachey (1998):
v~r(Si) = ~/2m(t)di(t)2~ j=i+l
[ ~tj(t)[1-~lj(t)]xai+2j~]2
+ 2~tj(t))ki+J1
2~i(t)2 k-1 ~ clj(t)~tl(t))k2i+j+! ~/4m(t)j=i+l l=j+l + 82di(t)[1 - di(t)] m(t) '
(15.17)
15.2. Analysis of Age Frequencies i = 0, ..., k - 1, where
Section 4.3.4). Other comparisons (e.g., between a model assuming known K but not stable age distribution, against a model with stability alone; models in which survival rates are not age specific) cannot always be tested by likelihood ratio because nonnested models are involved. For these situations, AIC or other criteria can be used to discriminate among models.
k
m(t) = ~,. yj(t), ]=1 k
y = s
341
dj(t)aJ,
j=i
and ~.i y
15.2.2.5. Assumptions about Sampling Effort
Cli( t ) ~ 2i y2 "
Again, if the value of a is independently estimated rather than known, then Eq. (15.17) must be adjusted by adding the term 2
[1-Si
k
i)]
var (~).
When both data structures are available and the joint likelihood (simplified by assuming stability) is used, both )t and the k - 1 survival rates can be uniquely estimated via numerical optimization of the joint likelihood. The maximized likelihood obtained by this approach then can be compared via likelihood ratio tests to a product likelihood based on specific values for growth rates ()~ = ~0; e.g., )~0 = 1).
15.2.2.4. Age Stability Unknown If both data structures are available the joint likelihood formed by the product of Eqs. (15.12) and (15.13) is maximized by
S i ( t - 1)= Ci+l(t)h(t- 1)
(15.18)
~i+l(t)Mt - 1) + di(t){1 - Mt - 1)[1 - ~0(t)]}' i = 0, ..., k - 1, where
~,i(t) = xi(t)/n(t ) and
cli(t) = yi(t)/m(t), i = 0.... , k, provided the K(t) values are known. The variance of this estimator is quite complicated and is not presented here [the interested reader is referred to Udevitz and Ballachey (1998), Appendix B]. The model is saturated, that is, there are no degrees of freedom for a goodness-of-fit test. Stability in the age distribution can be tested by constraining the parameters of the product likelihood from Eqs. (15.12) and (15.13) to be equal over time and by comparing this maximized likelihood to the unconstrained product likelihood (see
The above expressions for the likelihoods of the age frequency data, ages-at-death data, and combined data structures make it clear that strong assumptions are invoked regarding the sampling process. The principal assumption is that sampling probabilities (c~, ~) in the likelihoods are constant over time and among age classes, which is required to allow parameter identifiability under any of the data structures. In practice this assumption is likely to be violated, particularly in cases where the standing age structure is obtained from a harvested sample. Heterogeneity in the rate of harvest should be expected a priori. For example, younger age classes of gamebirds typically are more vulnerable to harvest, and fishing gear often is configured so as to exclude fish below or above certain size limits. Under these conditions the sample age frequencies likely do not reflect the population age structure. Sometimes auxiliary data are available to provide independent estimates of age-specific sampling rates (e.g., relative vulnerability to harvest) (Martin et al., 1979; Miller, 2000a), and these data can be used to adjust the sample frequencies accordingly. Of course, it then is necessary to incorporate the additional component of sampling error in the estimated sampling rates used to estimate survival and age distributions [as in the case of incorporating estimates of ~ in Eqs. (15.15) and (15.17)]. All too often, unadjusted standing age frequencies or ages at death are used without critical evaluation of the underlying assumptions, including that of homogeneous sampling from the population. Age data are relatively easy to collect, and a multitude of estimators and models are available that will produce apparently reasonable estimates. It is likely that many, perhaps most, uses of these estimators are based on unverified assumptions and thus are of dubious reliability.
Example This example is from a sample of ages for moose
(Alces alces) harvested in N e w Brunswick during 1980-1984 (Boer, 1988). The authors used auxiliary data from aerial surveys and analysis of a sequence of harvest age ratios to support their claim that the assump-
342
Chapter 15 Estimation of Demographic Parameters
tions of a stable and stationary population are warranted. We have analyzed their data according to Eqs. (15.14) and (15.15), setting )~ = 1. The results are reported in Table 15.3 and are similar to those reported by Boer (1988) (which were reported as l x rather than Sx estimates). Note, however, that the precision of the estimates is poor, with reasonably narrow confidence intervals only for the first few age classes. A reanalysis of this problem under a constrained model involving fewer age-specific estimates, or a parametric form for patterns in age-specific survival, might improve these results. We note that the data in Table 15.3 are not "pure" age frequencies (as suggested by their noninteger values) but are in fact adjusted frequencies based on estimates of age-specific vulnerability to harvest (Boer, 1988). Because they likely are sample-based estimates, an additional component of sampling variability should be included in the variance terms for the survival estimates. Boer (1988) alludes to harvestand survey-based estimates for determining age stability and stationarity, and these components of variability contributed to the sampling variances of the estimates, but were unaccounted for in the variance computations. These remarks are not made as a criticism of the study, but rather to point out the difficulty of using age frequency methods, even under
T A B L E 15.3 Age class
the best of circumstances, to provide reliable estimates of survival rates.
15.2.3. Population "Reconstruction" Population reconstruction is a technique for calculating the size and age composition of a cohort at some initial time from subsequent mortalities of the population. A rationale for the method is "if an animal was killed in a given year at four years of age, then it was a three-year-old the previous year, a two-year-old two years earlier .... and a fawn four years earlier" (McCullough et al., 1990). In theory, if all the mortalities in the population can be observed (e.g., all deer are killed by hunters, and all hunter kills are reported), then an accurate picture of the population can be reconstructed and sometimes is referred to as a "virtual population." The "data" thus reconstructed then are used in population models and statistical procedures, for example, to calculate estimates of age-specific survival and population growth rates. The method rests on several assumptions that we believe are unlikely to be tenable in practice, with potentially grave consequences in terms of the reliability of the "estimates." The claim sometimes is made that reconstructed populations correspond to a "minimum known alive" population, a
Estimation of Age-Specific Survival Rates for M o o s e in N e w Brunswick a
Frequency
ci
Si
S'E(Si)
Li
lIi
0
128.82
0.281
0.676
0.094
0.492
0.860
1
87.10
0.190
0.697
0.116
0.468
0.925
2
60.67
0.133
0.711
0.142
0.434
0.989
3
43.15
0.094
0.728
0.171
0.393
1.063
4
31.41
0.069
0.746
0.204
0.347
1.145
5
23.44
0.051
0.771
0.241
0.298
1.244
6
18.07
0.039
0.776
0.276
0.235
1.318
7
14.03
0.031
0.802
0.321
0.173
1.431
8
11.25
0.025
0.820
0.364
0.106
1.535
9
9.23
0.020
0.841
0.409
0.038
1.643
10
7.76
0.017
0.870
0.458
-0.027
1.767
11
6.75
0.015
0.874
0.493
-0.091
1.840
12
5.90
0.013
0.903
0.540
-0.155
1.962
13
5.33
0.012
0.925
0.578
-0.208
2.058
14
4.93
0.011
457.84
1.000
Total
a Based on age distribution in the harvest (Boer, 1988). Parameter ci is the proportion of the population in age class i, and Si is the survival probability for age class i; ~/i and /~i are u p p e r and lower confidence limits, respectively.
15.3. Analysis of Discrete Survival and Nest Success Data notion that is similar to one invoked (and rebutted) in the case of capture-recapture sampling (Jolly and Dickson, 1983; Nichols and Pollock, 1983b; Pollock et al., 1990; Efford, 1992). In most cases the basis of the reconstruction is the harvest of known-age animals. There are at least two serious difficulties with the use of these data. First, even if all harvest mortality (legal and illegal) can be accounted for, the method will exclude deaths due to other mortality sources. To the extent that these constitute a significant fraction of mortality (which is usually unknown), the harvest-based reconstruction will produce an increasingly skewed picture of the population through time. Second, harvest is almost certainly biased toward certain age and sex components of the population, further distorting the relationship between the data and the actual population structure. The only scenario in which one might expect population reconstruction to provide an accurate picture of population structure involves a random sample of the population that has been destructively sampled (deliberately, or as the result of a catastrophe), thus providing an accurate ages-at-death sample. However, this would not allow reconstruction of even one cohort, unless it could be repeated through time. Even with this sort of sampling we advocate the use of estimation methods [e.g., Eq. (15.14)] that utilize statistical likelihoods and are based on clear (and testable) assumptions. We strongly discourage the use of "virtual data" from reconstruction as if they are actual data, for the purpose of statistical estimation of demographic parameters. For instance, harvest data could be used to reconstruct the number of animals alive in each age class in each of several previous cohorts, and these data in turn used to calculate age-specific survival rates via, e.g., Eqs. (15.14) and (15.16). This approach is flawed on three grounds: (1) the quantities used in the "estimates" of survival were never observed; they were inferred from a model of the population; (2) all biases inherent in the reconstruction will propagate in the estimates; and (3) even if the assumptions of reconstruction can be met, estimates of sampling variation in these estimates will not take into account sampling error in the harvest or other data on which the reconstructed "data" are based. It is a common practice to impose assumptions about mortality (especially natural mortality) and other demographic rates, in order to make reconstruction provide "reasonable" virtual populations (e.g., account for nonharvest losses). It also is unfortunately common for the resulting "virtual data" to be used to "estimate" these same population parameters or to test the assumptions (e.g., of age sta-
343
bility) thus imposed, an exercise in circular reasoning whose futility should be obvious to the reader.
15.3. A N A L Y S I S OF D I S C R E T E SURVIVAL AND NEST SUCCESS DATA In contrast to some of the approaches described in the previous section, designed studies that include maximum likelihood estimation methods and statistical models that account for random variation in the data offer a statistically reliable alternative for inference about the demographic parameters of a population. Here we describe methods that are appropriate when subjects can be visited repeatedly in the course of an investigation, as in the monitoring of nests at known locations or animals that are radio marked. We assume initially that the fates of individual subjects in the study can be determined with certainty during the study, i.e., their probabilities of detection are 1. We later will include features that accommodate the censoring of individuals, and in subsequent chapters the assumption of perfect detectability will be dropped altogether.
15.3.1. Binomial Survival M o d e l The binomial model (see Chapter 4) is appropriate for processes that have two mutually exclusive outcomes, such as occur in simple capture-recapture studies. In this section we use the binomial model for estimating survival from data structures arising in, e.g., radiotelemetry and nesting studies, in which the investigator is able to classify unambiguously the fates of individual subjects (individual animals, nests, etc.). For a sample of n subjects, the binomial probability function can be used to describe the number of these individuals that survive (x) or die (n - x), where survival is ordinarily defined as occurring over a fixed interval of time. If S is the probability that an individual subject survives, then the binomial distribution of the number of survivors is
S =tntSX k the products S k fk+l, S k S k + l f k + 2, ..., and Sk "'" S l - l f l can be estimated, but not the individual parameters S k, ..., S l_ 1 and f k + l , -.., fl. Thus, only the parameters Si, i = 1 . . . . , k - 1, and fi, i = 1, ..., k, can be separately estimated. Estimation of the l - k products mentioned above brings the total to l + k - 1 estimable parameters for each year of banding. We assume that the year-specific distributions of band recoveries are independent, so that the joint distribution for data across all years is simply the product of the three distributions: 3
P ( m l , m__2, m__3ITll, '112, "IT3) -- H
P i ( m i "rri).
i-1
When data are substituted into this expression, we have the likelihood function for a band recovery experiment involving three banding periods and four recovery periods. The model generalizes in an obvious way to include additional banding years a n d / o r additional recovery years. Thus, recoveries for the general case can be put in a rectangular array that includes k rows, l columns (an l + 1st column could be included to account for individuals never recovered), along with a leading colu m n designating the number banded in each year (Table 16.3). The sum of counts for row i (up to the last column) is the total number of recoveries from the ith cohort. The sum of counts for column j is the total number of recoveries from all cohorts in year j. The general likelihood function for this situation is expressed as the product k
C('n'l,
"", 'rl'kIml . . . . , m__k) = I-[ Pi(mi]'rri) i=1
(16.2)
of the multinomial likelihood functions for each banding period. As before, multinomial probabilities for the general model have a rectangular form with k rows and l + 1 columns (the last column is the probability
'~With i = 1 ..... k b a n d i n g y e a r s a n d j = i..... l r e c o v e r y years.
of a band never being recovered). The sum of probabilities for row i (up to the last column) gives the probability of recovery at some time during the study for individuals banded in period i. Expected cell counts for the general model are displayed in Table 16.4. The expected values for single age-class band recoveries illustrate some of the similarities and differences between the modeling of mark-recapture (Chapters 14 and 17) and band recovery data: 9 Both approaches allow for time-specific probabilities -rrq of recapturing or recovering individuals that are released after initial capture and marking. In mark-recapture experiments of closed populations survival rates are not components of these probabilities, because population closure implies a survival rate of 1. In band recovery studies, on the other hand, survival rates are included in the probabilities ~rq, because they are necessarily less than 1 (the population is harvested) and individuals recovered j periods after banding must have survived j - 1 periods prior to harvest and band recovery. 9 Though both situations include individuals that are never seen, the size of this cohort can be important in capture-recapture studies but is irrelevant in band recovery studies. In the case of some capture-recapture models, the objective is to use information in the capture histories of individuals seen, to determine the number of individuals never seen. In band recovery experiments, on the other hand, the goal of the study is to make inferences about survival rates and other sampling parameters, rather than to determine total population size. The number of unbanded individuals in the population is not germane to this goal. However, because not all marked animals are seen again in band recovery studies, the issue of incomplete detectability comes into play just as it does in population estimation, in that only the animals recovered each year are known
Chapter 16 Estimation with Band Recoveries
370 TABLE 16.4
Expected Recoveries for S i n g l e - A g e Band Recovery Data under M o d e l
(St, ft )a
Recoveries in period j
Releases in period i
1
2
3
-"
aI
alf 1
alSlf 2
alSlS2f 3
a2f 2
R2S2f 3 R3f 3
R2 R3
l
Not recovered
...
a l S i S 2.... Sl_lf l
a I - ~j E(mlj)
...
R2S253 ..... S l - l f l
R2 - ~j E(m2j)
...
R3S3S 4.... S l - l f 1
R3 - ~,j E(m3j)
RkSkSk+ 1.... Sl_lf 1
R k - ~,j E(mkj)
Rk
aWith i = 1..... k banding years and j = i..... l recovery years. Parameters are annual survival rate (Si, i = 1..... l - 1) and annual recovery rate (f i, i = 1..... l).
to have been alive u p to the point of recovery (see Section 15.1). As seen below, the s a m p l i n g probability or rate of detection is m o d e l e d by recovery rates, in the s a m e w a y that capture a n d recapture rates are used in c a p t u r e - r e c a p t u r e analysis.
E(mi. ) = E(mii +
16.1.2.1. Estimation The single-age, t i m e - d e p e n d e n t b a n d m o d e l has the identifiable p a r a m e t e r s i = 1,...,k-
Si,
entiation of the log likelihood function, as described in C h a p t e r 4. However, the MLEs for this m o d e l also can be obtained directly from the m o m e n t estimators E(mi.) -- mi. and E(m.j) = m.j of r o w a n d c o l u m n totals for the cell counts, w h e r e
= Ri[fi + Sifi+ 1 -]-...
recovery
1
The p a r a m e t e r s S i for i > k - 1 and fi for i > k are not separately estimable, but the p r o d u c t s S k ... S k + s _ l f k + s ~ ,
q- ( S i S i + 1 i f - . . . - ] -
Sl_lfl) ]
1 ""
Sj_lfj]
if- [ a 2 s 2 " ' " S j _ l f j ]
q - " ' " q- a j f j .
After substitution of mi. a n d m.j for E(mi.) a n d E(m.j), some algebraic m a n i p u l a t i o n leads to a solution of these equations for the MLEs:
',m'[ mi]i.l
s=l,...,l-k
Si = --a-: 1 are estimable a n d therefore are included in the likelihood. Algebraic, closed-form estimators for the p a r a m eters a n d their a s y m p t o t i c variances can be obtained t h r o u g h application of m a x i m u m likelihood m e t h o d s (Chapter 4) or the m e t h o d of m o m e n t s , as s h o w n below. In practice, closed-form solutions are available only for fully p a r a m e t e r i z e d models, a n d estimates can be obtained for r e d u c e d - p a r a m e t e r m o d e l s only by application of n u m e r i c a l algorithms. We p r o v i d e closed-form estimators for the simple e x a m p l e of b a n d recovery for animals of single (adult) age class, w h e r e the data are unstratified by sex or other attributes. Expressions for more complex m o d e l s (e.g., for multiple age classes) are s t r a i g h t f o r w a r d extensions, and references for these are p r o v i d e d for the interested reader. P a r a m e t e r estimates in a fully p a r a m e t e r i z e d b a n d recovery m o d e l can be obtained t h r o u g h partial differ-
rail)
E(m.j) = E ( m l j + m2j + "" + mjj) = [alS
i = 1, ..., k.
+
and
and fi,
m i , i + 1 -}- . . .
,
i - 1 , ..., k - 1
(16.3)
mi+l
and
f
i
m i.m.i -- RiTi ,
i = 1, ..., k
(16.4)
(Seber, 1970b; Robson a n d Youngs, 1971), w h e r e T i is a total (across all b a n d i n g years u p to and including i) of all recoveries in years including and s u b s e q u e n t to i; i.e., Ti = mi. for i = 1, Ti = Ti-1 - m.i-1 + mi. for i = 2 .... , k, a n d Tk+ s = Tk+s_ 1 -- m.k+s_ 1 for s = 1, ..., l -- k if l > k. As m e n t i o n e d above, the p a r a m e t e r s Si for i > k and fi for i > k are not separately estimable. However, estimates of the p r o d u c t s Skfk+ 1, SkSk+lfk+2~, etc. are given by Sk "'" S k + s - l f k + s =
mkmk+s RkTk
(16.5)
16.1. Single-Age Models for s = 1, ..., l - k (Brownie et al., 1985). Though these values usually are not of interest in their own right, they are required for goodness-of-fit testing. Estimation procedures for the fully parameterized band recovery model are discussed in Brownie et al. (1985). Estimation software is noted in Appendix G. To see how band recovery analyses are related to other "incomplete count" estimators discussed in previous chapters, consider a banding study in which animals are banded and released in each of two years in sequence, and recoveries are obtained from postrelease samples over the two years. The data structure for this problem can be described by
al R2
mll
m12 m22
which is represented in terms of the parameters of the time-specific model as al R2
Rlfl
RISlf2 R2f2
Now take M12 as the true (unknown) number of animals from our released sample (R 1) that survive until the beginning of year 2 [by definition E(M12) = alS1]. Consider m12 to be our index or incomplete count statistic representing M12. We need to adjust this statistic by the sampling fraction of M12 that m12 represents, which in this case is f2, the probability of appearing in the second recovery sample, conditional on being alive at the beginning of the sampling period. A natural estimate of f2 is provided by the recoveries of the marked animals released just preceding this period, that is ?2 = m22/R2,
so that
~/I12 = m12/f2 = m12/(m22/R2) ,
which is of the form of the canonical estimator =
in Eq. (12.1), where f~ = f2. Finally, the estimator of $1 derives from the relationship E(M12) -- RIS1, so that
S1 = ~'I12/a1
(16.6)
= m12R2/m22R 1 .
This derivation is recognizable as a special case of the general approach to survival rate estimation outlined in Section 15.1.2. In general the maximum likelihood estimator for survival rate [Eq. (16.3)] is biased. An adjustment that effectively eliminates bias involves the incrementing
371
of Ri+ 1 and mi+l. by one unit in the formula for Si" Thus, an approximately unbiased estimator for S i is given by
mi t m;t ( i'l+l 1 1
Si-- ~
1 -
\mi+l. +
(16.7)
'
i = 1, ..., k - 1 (Brownie et al., 1985). Because the MLE for recovery rate is unbiased (Robson and Youngs, 1971), an analogous adjustment of the computing formula for fi is not required. Variances and covariances for the parameter estimates can be obtained by application of the delta method (Appendix F) to the moment estimators above, with the variances of m.i and mi. obtained from the multinomial distribution (Brownie et al., 1985). Alternatively, variance estimation may be based directly on the likelihood, via the Fisher Information Matrix (Appendix F), the approach used in modern estimation software such as MARK (White and Burnham, 1999) (see also Appendix G). Regardless of the estimation approach, many of the estimators are uncorrelated (i.e., have covariances of 0). However, Si (Si) is negatively correlated with Si+l (5i+1) and fi+l" The magnitude of the covariance is given by the product of the respective parameters, adjusted by a factor that involves the size of the banded cohort and the total number of recoveries. The existence of nonzero sample correlations among these estimators must be kept in mind, especially when making comparisons among estimates, to avoid spurious conclusions resulting from sampling correlation rather than a true association among the underlying parameters. Following general notation of Lebreton et al. (1992) we denote the model described above as model (S t, ft), where the subscipt t denotes variation in the survival and recovery parameters over time, absent any age, sex, or other stratification in parameters. Model (St, ft) is identical to Model I of Brownie et al. (1985). Numerous procedures exist for obtaining maximum likelihood estimates of the parameters of the model and other reduced-parameter models derived from it. For most applications the numerical estimation procedure implemented in MARK (Appendix G) provides a general approach for obtaining MLEs and for investigating sources of variation in survival and recovery rates via goodness-of-fit and model selection procedures (Section 16.1.8).
16.1.3. Reduced-Parameter Models Straightforward restrictions on the parameters in a band recovery model lead to three simplified models:
Chapter 16 Estimation with Band Recoveries
372
9 Constant survival rates across years (Sj = S, j = 1, ..., l - 1) and variable recovery rates, denoted as model (S, ft'J. 9 Constant recovery rates (fj = f, j = 1, ..., l) and variable survival rates, denoted as model (St, f). 9 Constant survival and recovery rates (Sj = S, j = 1, ..., l - 1; ~ = f, j = 1, ..., l), denoted as model (S, f). Models (S, ft) and (S, f) are identical to Model 2 and Model 3, respectively, of Brownie et al. (1985). There is no model in the Brownie et al. (1985) methodology corresponding to (S t, f), because this model is seldom applicable to the waterfowl data sets for which the model set in Brownie et al. (1985) was developed. Each of these reduced-parameter models is specified by a distinct parametric structure for the multinomial cell probabilities. As with capture--recapture models, the standard approach involves an incorporation of parameter assumptions into the model likelihood function, from which m a x i m u m likelihood estimates are derived. The MLEs then are used to examine the goodness of fit of the model and also to compare models as part of a model selection procedure (Section 16.1.8). It is not possible to derive explicit formulas for estimates of the parameters for the models with parameter restrictions on survival and recovery rates, and a numerical procedure is required to solve the likelihood equations. Models (S, ft) and (S, f) include only a single survival parameter that is constant over time periods, with a total of l+ 1 and 2 parameters, respectively. In the usual case of equal time intervals for banding (e.g., every year at the same time), the constant survival parameter has a biological interpretation. However, in general, intervals between banding may be unequal, in which case the assumption Si = S is likely to be violated simply because of variation in the interval over which survival is estimated, irrespective of true temporal variation in survival per unit time. A solution to this problem is to model survival as
the estimation of taxonomic extinction rates from stratigraphic range data, in which the time intervals between sampling "occasions" (i.e., geologic strata) obviously varied. Both of these applications are handled easily in program MARK by specifying the length of intervals between sampling occasions and thus eliminating the need for special models to handle this situation (e.g., Conroy et al., 1989b). 16.1.4. T e m p o r a r y B a n d i n g E f f e c t
A useful generalization of the band recovery model allows for a temporary banding effect, in which newly banded birds experience a probability of being shot and having their bands recovered that differs from that of previously banded birds. This situation m a y apply for bands recovered near banding sites, where the harvest of banded birds consists primarily of newly banded birds and band reporting rates are lower than reporting rates at other locations (perhaps because of the absence of novelty in encountering a banded bird or the absence of curiosity about banding location). To capture this variation in recovery rates, additional recovery rate parameters are necessary. For example, a banding study involving three banding years and four recovery years would have the parametric structure indicated in Table 16.5, where f* is the recovery rate for individuals banded in year i. The parameters f* are k n o w n as direct recovery rates, to emphasize that banding and recovery occur in the same year. On the other hand, indirect recovery rates apply to recoveries in later years, after the year in which banding occurs. A model with different direct and indirect recovery rates for k banding periods and l recovery periods contains 2l + k - 2 parameters: S;,
i = 1.... , l -
fi,
i = 2, ..., 1;
f*,
i = 1, ..., k.
1;
S i = S ti,
where t i is the length of the interval between banding periods i and i+l. The parameter S now refers to survival over a standardized interval of time (e.g., 1 year), which is hypothesized to be constant during periods of variable length t i. The convention of allowing intervals between banding occasions to be of variable length permits ready extension of band recovery models to m a n y "nonstandard" situations. One obvious application involves gaps in an otherwise regular (e.g., annual) banding operation. Another involves sampling intervals that are inherently variable in length. For example, Conroy and Nichols (1984) applied band recovery models to
TABLE 16.5 Expected Recoveries for Single-Age Band Recovery Data with Temporary Banding Effect [Model (St, f~)]a Releases in period i a1 R2 R3
Recoveries in period j 1
2
3
4
Not recovered
a l f ; R1Slf 2 alSIS2f 3 R1515253f 4 R 1 - ~j E(mlj) R2f 2
R252f 3
a25253f 4
R3f 3
R3S3f 4
R2 - ~,j E(m2j) R3 - ~j E(m3j)
With k = 3 banding years and 1 = 4 recovery years. Parameters are annual survival rates (S i, i = 1..... 1-1), annual recovery rates more than 1 year after banding (fi, i = 2..... l), and annual first-year recovery rates (f~, i = 1..... k). a
16.1. Single-Age Models However, only a limited set of these can be separately estimated if l > k:
Si,
i = 1, ..., k - 1;
fi,
i = 2, ..., k;
f*,
i = 1, ..., k.
The generalization of model (St, ft) to include direct recovery rates is denoted as model (St, fD and is equivalent to Model 0 of Brownie et al. (1985). Estimates of parameters in (S t, fD can be computed using program MARK or other numerical procedures (Appendix G).
16.1.5. Multiple Groups As noted earlier, heterogeneity in survival, recovery, or both can be expected when the banded sample consists of animals in different sex or other (e.g., geographic) strata. If groups can be identified on capture and banding, then the recovery samples can be stratified prior to analysis. The only restriction is that stratum membership must be assigned at the time of banding and remain appropriate for the duration of the study. Two basic approaches can be taken in this situation. In the first, the data are stratified and a separate model is fit for each stratum. Goodness of fit tests then can be used to determine whether estimation based on separate groups is appropriate, or whether groups should instead be combined. A contingency table test developed by Brownie et al. (1985) specifically tests the null hypothesis of identical survival and recovery rates for multiple groups (i.e., it tests the appropriateness of pooling data). In the second approach, the stratum identity is incorporated into the model structure and used to estimate stratum- and timespecific parameters. For example, a generalization of model (St, ft) to allow sex- and time-specific variation in both survival and recovery rates, denoted as model (Ss,t, fs,t), would include identifiable survival and recovery parameters
Sij,
i = 1 ..... k - l ,
373
leads to time-specific (but not sex-specific) survival estimates, in which the same parameters Si, i = 1, ..., k - 1, are shared by both sexes. The term fs,t in the above expression indicates that a different recovery parameter is required for each sex-year combination (e.g., different parameters for males in 1998 and females in 1998). The adequacy of this model then can be assessed by goodness of fit, and model selection procedures (Section 16.1.8) can be used to judge the appropriateness of the model relative to competing models. In addition, one can easily describe models allowing for "parallelism," in which a parameter (e.g., survival) varies over time but in a parallel manner for the groups (e.g., sexes). Parallel effects are denoted by " + " rather than "," in the model notation, so that (Ss+t, ft) describes a situation where survival rates are sex specific but vary over time in a parallel manner (see Section 17.1.5 for a more complete description of interactive and parallel effects in the context of mark-recapture models). Program MARK (Appendix G) allows for construction of both types of group effects and provides a more efficient means of estimation and model selection than does separate estimation by groups or comparison of frequency tables.
16.1.6. Covariates The construction of band recovery and mark-recapture models to allow for inclusion of covariate relationships among parameters (typically survival, but also recapture and recovery rates) has seen tremendous progress in recent years, beginning with important work by North and Morgan (1979) for band recovery models, and Pollock et al. (1984) (see Section 17.1.4) and Clobert and Lebreton (1985) for closed and open capture-recapture models. We consider models for both time-specific covariates (e.g., covariates vary with time but not among individuals) and individual covariates (the covariates vary among individuals). The discussion here anticipates a more general treatment of covariates in Sections 17.1.4 and 17.1.7.
j-1,2
16.1.6.1. Time-Specific Covariates and
fij,
i = 1, ..., k, j = l ,
2
for males (j = 1) and females (j = 2). An obvious advantage of this approach is the ability to readily form models involving parameterizations not possible under the "separate models" approach. For instance, the model
(St, fs.t)
Under model (St, ft) both survival and recovery rates vary with time in an unspecified manner. If independent information exists about the time periods under investigation, it may be possible to model parameter variation over time by taking this information into account. Consider a banding study over k = 11 years, for which 10 estimates of survival are possible under model (St, ft). Suppose that we know years 2, 7, and 9 had especially severe winter conditions, whereas years 1, 3, 4, 5, 6, 8, and 10 were years of "normal" or warm
374
Chapter 16 Estimation with Band Recoveries
winters. With this information we could model survivorship by Si = SL,
= SH,
i-- 2,7,9 i = 1 , 3 , 4 , 5 , 6 , 8 , 10,
where SL, SH are parameters in a new (reducedparameter) model under the hypothesis that survival rates differ among years classed as "severe" and "normal" but not otherwise. This model represents an attempt to capture the temporal variation in survival probability in terms of winter conditions. The model could be compared to model (St, ft) and model (S, ft), with the latter comparison equivalent to the hypothesis of no difference associated with weather severity. A slight reparameterization of this model motivates a more general approach to modeling time-specific covariates. Let X i be an indicator variable, with value X i = 0 in "severe" years and X i = 1 in "normal" years. The above model then can be reexpressed as
Si-- ~o q- ~lXi,
(16.8)
where ~0 and ~1 parameterize a linear relationship between the indicator variable and survival. When ~1 -- 0 there is no relationship, i.e., severe and normal years produce equal survival rates, resulting in model (S, ft) (we assume here that the recovery parameter continues to vary in an unspecified manner through time). A somewhat more complex relationship between survival and winter temperature also can be used. Because temperature is measured on a continuous scale a natural extension is to consider values of the covariate X i as continuous, with ~0 and ~1 parameterizing a linear relationship between specific values of winter temperature and annual survival. Thus, estimates of ~0 and ~1 can be used to predict values of survival, i.e.,
Si-~ ~o q- ~lXi given the value X i = x i. One means of obtaining estimates of ~0 and ~1 under the above model is to obtain time-specific survival estimates under, e.g., model (St, ft), and then use these estimates as the response variable in a standard regression analysis (e.g., Nichols et al. 1982b; Sauer and Boyce, 1983). A more efficient alternative is to incorporate relationships like Eq. (16.8) directly into the likelihood function as constraints on the parameters of a more general model and to proceed with maximum likelihood inference (see examples in Conroy et al., 1989b; Dorazio, 1993). This approach has at least four advantages: (1) a single-step estimation is possible, (2) only two survival parameters (~0 and ~1) are estimated, versus an additional k - 1 parameters for the two-step process, (3) the sampling variances
and covariances associated with survival rate estimation are properly accounted for in the direct estimation procedure, and (4) the procedure fits naturally within the model evaluation and selection process described below. In practice, covariate relationships usually are expressed by means of link functions, discussed more thoroughly in Section 17.1.4. For example, the logit function establishes the relationship l~
_Si Si) ~. ~o -}- ~lXi
between survival and the covariate. An advantage of the logit and certain other common link functions is that predicted values for survival are constrained to the unit interval, whereas a linear relationship as above (under an "identity link") may allow predicted survival to take on logically inadmissible values (e.g., Si 0 or Si > 1), depending on the value of the covariate(s). This approach extends to multiple covariates, e.g.,
t Si ) -- ~o q- ~P ~jXij,
log 1 - S i
j=l
where j is an index denoting p time-specific covariates, including possible polynomial and interaction terms and Xij is the value of covariate j in year i. Models with time-specific covariates may be constructed using program MARK (Appendix G) and are evaluated in comparison with a general model a n d / o r other reduced-parameter models as discussed in Section 16.1.8. 16.1.6.2. Individual Covariates
An assumption of band recovery analysis is that survival and recovery rates of individual marked animals within an identified and modeled stratum (e.g., defined by time, age, sex, and location) are identical (see Section 16.1.9). In practice this assumption seldom is justified, and serious violations of it may lead to a lack of model fit and biased estimation. Some variation among individuals in survival, recovery, and other parameters may be explained by measurable covariates such as size, weight, or another characteristic. In addition, the influence of covariates on survival or other parameters may be of inherent interest. In either case, modeling with individual covariates can be accomplished in a manner similar to that used for timespecific covariates as described above, with the qualification that the covariates characterize an individual for the duration of the study and do not vary over time. We describe models incorporating individual covariates in more detail in our coverage of conditional capture-recapture studies (Section 17.1.7).
375
16.1. Single-Age Models
16.1.7. Banding Multiple Times per Year Here we consider a banding study in which bands are placed on individuals at two or more occasions prior to a single recovery period. For example, migratory waterfowl may be captured and marked twice during the year, in the late winter prior to spring migration and again in late summer or early fall just prior to the hunting season. The motivation for such a design is that within-year banding allows one to partition annual survival rates into a component corresponding to the hunting season and a component for the remainder of the year when other sources of mortality besides hunting are operative. In this way it is possible to better isolate the effects of hunting and to investigate possible compensation for hunting by other nonhunting mortality factors. A factor that sometimes complicates such analyses is the dispersal of individuals into different areas during migration. If migratory populations are banded at different times of the year, it is likely that different population cohorts, with different survival and recovery parameters, will be banded. For example, birds from different breeding ground locations may winter in the same area and be indistinguishable. Banding may thus involve a specific group of birds during the late summer, but a mixture of these and other birds during winter. A potential consequence is heterogeneity in survival and recovery rates among the sampled individuals. One way to help reduce this problem is to choose banding periods when animals are most likely to be sedentary [e.g., summer or winter; see Blohm et al. (1987) and LeMaster and Trost (1994)]. We note that these problems with heterogeneity should not arise in resident populations, so that the use of multiple banding periods per year is likely to be most useful for residents. Consider the special case in which resident animals
TABLE 16.6
Period (year) i 1 2 3
are banded twice per year, an early (e.g., spring) banding and a later banding during the period just before the hunting season. A parameterization of annual survival (as measured from preseason of year i to preseason of year i+1) in the one-age band recovery model allows incorporation of data from these two periods: Si ~ "Yi~i+l,
where q)i is the probability that a bird that is alive at the midpoint of the first (e.g., spring) banding period in year i survives to the midpoint of the second banding period, and "~i is the probability that a bird that is alive at the midpoint of the second period [e.g., later summer (preseason)] of banding in year i survives to the midpoint of the first banding period in year i+1. In the second period, all animals (including survivors from the current and previous years' first-period bandings) are assumed to share subsequent survival and recovery probabilities, leading to a matrix of expected recoveries as illustrated for 3 years of bandings and recoveries in Tables 16.6 and 16.7. The model associated with expected values as in Table 16.7 is denoted as model (q~t, ~/t, ft). This model and a number of associated reducedparameter models are provided by program MULT (Conroy et al., 1989b) and may be constructed in MARK. Reduced-parameter models include the following examples: 9 Model (q0t, '~, ft)--Second period survival constant over years. 9 Model (~, ~/t,ft) mFirst period survival constant over years. 9 Model (q~, ~/, ft)--First and second (and annual) survival constant over years. 9 Model (q~t = ~/t,ft) mSecond period survival equal to first period survival (but varying over years for both periods).
D a t a S t r u c t u r e for S i n g l e - A g e B a n d Recovery Data: Banding Two Occasions a
Subperiod h
Releases in period i and subperiod h b
Recoveries in period jc 1
2
3
Not recovered
1
Rll
mll I
mll 2
mll 3
Rll - ~,j m11j
2
R12
m121
m122
m123
R12 - s m12j
1
R21
m212
m213
R21 - ~,j m21j
2
R22
m222
m223
R22 - ~j m22j
1
R31
m313
R31 - ~j m31j
2
R32
m323
R32 - ~j m32j
aWith i = 1, 2, 3 banding years and j = i..... 3 recovery years. The first two bonding periods precede the first recovery period. bRih is the number of animals released during subperiod h of period i. cmihj is the number of recoveries in year j of animals released during subperiod h of period i.
376
Chapter 16 Estimation with Band Recoveries TABLE 16.7 Expected Recoveries for Single-Age Band Recovery Data under a Time-Specific Model with Two Banding Occasions/Year"
Period (year) i 1 2
3
Subperiod h
Releases in period i and subperiod h
Recoveries in period j 1
2
3
Not recovered
1
Rll
a11q~1f 1
a11~plS1f 2
a11q~15152f 3
Rll - ~j E(m11 j)
2
R12
R12 fl
R12S1 f2
R12S1 $2f3
R12 - ~j E(ml2j)
1
R21
R21qo2f2
R21~252f 3
R21 - ~j E(m21 j)
2
R22
R22f2
a22s2f 3
R22 - ~j E(m22 j)
1
R31
Rglq~gf3
R31 - ~j E(m31j)
2
R32
R32f3
R32 - ~j E(m32 j)
~With i = 1, 2, 3, banding years and j = i..... 3 recovery years, aih is the number of animals released during period (e.g., year) i and subperiod h, with i = 1..... k and h = 1, 2; Si = "~iq~i+l is the probability that a bird that is alive at the midpoint of period 2 in year i survives to the midpoint of period 2 in year i + 1; fi is the recovery rate in year i.
Brownie et al. (1985) described a slightly different parameterization for banding twice per year with their models H 7 and H 8, although they did not develop the corresponding reduced-parameter models. Many additional reduced-parameter models are possible--for instance, the above models in combination with time constraints on recovery. These and other models are available in MULT; in addition, MARK provides a more flexible framework for construction and selection of models (Appendix G). The above approach can be extended to situations in which more than two marking periods occur during each year or other interval. In some cases, the intervals between periods are of inherent biological interest (e.g., corresponding to important portions of the animal's life history, such as the reproductive season). The above approach can be extended to cover these situations, by incorporating the appropriate constraints in MARK or SURVIV (Appendix G). 16.1.8. E v a l u a t i o n a n d S e l e c t i o n of M o d e l s
16.1.8.1. Goodness of Fit Once parameter estimates are obtained for a given model, it is possible to evaluate how well the model describes variation in the data set. Goodness-of-fit testing for band recovery models utilizes the multinomial structure of band recoveries in a manner similar to capture-recapture models (Chapter 17). Recall from Chapter 4 that with multinomial data, the observed cell frequencies can be compared with expected counts via the Pearson or log likelihood chi-square statistics (Section 4.3.3), to determine if the model "fits" the data adequately. Large computed values of the goodnessof-fit statistic with correspondingly small probabilities of occurrence provide evidence that one or more of the model assumptions (Section 16.1.9) are violated. Use
of the band recovery model under these conditions may lead to biased estimators of model parameters, with unrealistically low standard errors. Of the many assumptions that potentially can be violated, the most likely candidate is the assumption that all individuals in a banding cohort experience equal survival and recovery rates.
16.1.8.2. Model Selection Assuming one or more models adequately fit the observed data, the issue arises as to which model to select (and thus which parameter estimates to accept). Two basic approaches have been developed, the first of which is based on likelihood ratio tests between models that are "nested," i.e., one model can be formed by constraining the parameter space of another (more general) model. For example, models (St, f~), (St, ft), (S, ft), and (S, f) form a hierarchy, from more general to more reduced parameter structures. Thus, model (S, ft) is a generalization of model (S, f), model (St, ft) is a generalization of model (S, ft), and model (St, f~) is a generalization of model (St, ft). Under these conditions it is reasonable to subject the models to pair-wise comparisons, via the likelihood ratio testing procedure described in Section 4.3.4. Based on test results from the model comparisons (along with information from goodness-of-fit testing), one can identify the most appropriate model for the data. In general the likelihood ratio test statistic is computed as T = -2[ln(L 0) - ln(La)],
(16.9)
where L0, La are the likelihoods (evaluated at their MLEs) under the null and alternative models, respectively. Under the null hypothesis that the simpler model (H 0) describes the data as well as the more complex model (Ha), the statistic T is distributed as
16.1. Single-Age Models 2 X~,ka-kO" where k0, ka are the numbers of estimable parameters for each model (see Section 4.3.4). As a general rule, the model should be used which (1) adequately fits the data (i.e., the goodness-of-fit test does not indicate rejection) and (2) is not rejected when tested as the null model against more complex models [e.g., see Burnham and Anderson (1992, 1998)]. However, this procedure will work only in cases (as above) in which the models under consideration form a nested hierarchy, so that simpler models can be formed by imposing constraints on the parameters of the more complex models. A more general procedure that removes model selection from the framework of hypothesis testing is based on information theory (Akaike, 1973; Burnham and Anderson, 1992, 1998). The procedure is to select the model that minimizes the Akaike Information Criterion (AIC),
AIC = -21n(L) + 2k,
(16.10)
where again L is the maximized likelihood under a candidate model and k is the number of independently estimated parameters. This approach is very general and can be used with any nested or nonnested set of models, as long as they are likelihood based and are evaluated using the same data. Differences in AIC of 2 or less can be expected if models are essentially equivalent. Occasionally AIC values for two or three models will be very close to one another (differences k: S(1) i i
i = 1,
fl 1),
i = 2, ..., k;
fl 1)*,
i = 1, ..., k;
.--I
k-
1;
S(0) i ,
i = 1'
""1
k - 1;
i , f (0)
i = 1,
..-1
k.
Programs BROWNIE or MARK (Appendix G) can be used to obtain estimates for model (Sa,t, f**t) for the two-age case [denoted H 2 in Brownie et al. (1985)], as well as goodness-of-fit tests and likelihood-ratio test statistics for comparison with other models. More general cases with m age classes can be constructed in program MARK (Appendix G).
16.2.5. Unrecognizable Subadult Cohorts 16.2.4. Temporary Banding Effect A straightforward extension of the m-age class model (Sa.t, fa.t) allows for temporary effects of banding on recovery rates, in a manner analogous to the singleage class model (St, f D described in Section 16.1. This can be illustrated by the two-age case, in which it is assumed that adults banded in year i have a different recovery rate f* from that of adults banded previous
It sometimes is possible to distinguish juveniles (v = 0) from older individuals, but impossible to distinguish subadults (v = 1) from adults (v = 2). In this situation the banded cohort of juveniles at the time of banding consists exclusively of juveniles, but the cohort labeled as "adults" consists of both subadults and adults in unknown proportions. If subadults and adults differ in their survival and recovery rates, these differences
388
Chapter 16 Estimation with Band Recoveries Table 16.22
Expected Recoveries for Two-Age Band Recovery Data under Model (Sa. v fa.t) a
Releases in i Age Adult
Recoveries in p e r i o d j Bands
(v = 1)
R{ 1)
1
2 R~1)r ~'1 J 2
1;}(1)((1)* *'1 Yl
n(21)
R (21)f (21)*
R(3~)
9.-
3 a ~ l ) s ~ 1)~(1)((1),-'2/3
(1)c(1)f(31) 2 ~'2 (1)((1)* 3 ]3
. . .
k . . .
. . .
. . .
". . .
(v=0)
R~o)
(O)r
~(o)((o) *~1 11
1 `'1
(O)((o)
R(2o) R(3o)
2 J2
~c(1) J2
-'-'1D(1)c'(1)c'(1)'"S11-)1f11)~'1 `'2
.
D(1)c(1)...c(1) f}l) *'2 '-'2 `'I-1
R(1)
~j
E(m(2}))
.
D(1)r162 *~3 " 3
R(31)
-- E j
E(m(3}))
a(k1)
__ Ej E(m(k}))
"'"
`'1-1
f}l)
~(1)c(1)...c(1) fll) ~" `'k `'I-1
R (o)c(0)r
`'2 J 3
.
.
.
.
.
.
D(0)C(0)C(1)...r *'1 ~'1 `'2 `'l-1
R(o)r 2 `'2
]3
.
.
.
.
.
.
~(o)r x~2 ~"2
.
1 "1
R(o)((o) 3 J3
R(k0)
.
.
.
.
R(~
.
0)
R~1) - Gj E(m~} ))
.
(1)((1)* k Yk
Juvenile
Not recovered
l
_
R~0) __
~j
~(o)~(o). c(1) f}l) ~ 3 `'3 " "'-'1-1
R(2~ R(o)
Ej E(m(2~)) s E(m(3~))
*~kD(0)C(0)"'C(1)`'k `"-lf~1)
a(kO) -- Ej E(m(k~))
c(1)f}l)
"'~
f}l)
_
_
E(m~?))
aWith a g e s v = 0 , 1 a n d i = 1 . . . . . k b a n d i n g y e a r s , j = i . . . . . l r e c o v e r y y e a r s . P a r a m e t e r i z a t i o n s i n c l u d e t e m p o r a r y effect of b a n d i n g . P a r a m e t e r s are r a t e s of a n n u a l a g e - a n d t i m e - s p e c i f i c s u r v i v a l (.q(O) --i , i = 1 ..... k; S i(1) , i = 1 ..... l - 1 ); a g e s v = 0 ( b i r t h / h a t c h i n g y e a r ) , v = 1 (after b i r t h / h a t c h i n g y e a r ) , j u v e n i l e r e c o v e r y r a t e s f l ~ i = 1 ..... k, a d u l t d i r e c t r e c o v e r y r a t e s f l 1)*, i = 1 . . . . . k, a n d a d u l t i n d i r e c t r e c o v e r y r a t e s f l 1), i = 2 ..... l.
will be reflected in the survival and recovery parameters of the "adult" cohort for the banding year (after the banding year all surviving subadults in the cohort will have matured into adults). To model this situation, we include the survival and recovery parameters as before for juveniles (S f(~ ,1 subadults,_i (~q(1) and ---i(~ and Ji f(1)~ and adults (SI 2) and (2) f i )" T o these parameters must J i ,i be added the additional parameters SI 1-2) a n d Ylf9 -2)i where "1-2" denotes the mixture of subadults and adults in an "adult" cohort in the year the cohort is banded. Thus, the suite of parameters for this model for l > k includes SI~
i = 1, ..., k;
fl ~
i = 1, ..., k;
511) 1
i = 2, ..., k + 1;
fl 1),
i = 2, ..., k + 1;
S(2) i ,
i = 2,
"",
l-
f i(2),
i = 2,
""1
l;
S(1-2) i ,
i = 1,
"-',
k;
I I f !l-2)
i = 1t
""t
1;
k;
where the first set of survival and recovery parameters applies to juveniles, the second set applies to subadults, the third set applies to adult cohorts after the banding year, and the last set of survival and recovery rates applies to "adult" cohorts for the year they are banded (Brownie et al., 1985). The array of expected recoveries
for a study involving unrecognizable subadults is portrayed in Table 16.23. Note that there are only two arrays of data for this model, even though it is applicable to populations with three age classes. The reason for the limitation to two data arrays is that subadults and adults, because they are indistinguishable at the time of banding, essentially are included in the same "adult" cohort. The year they are banded, these "adult" cohorts have survival and recovery rates that reflect the mixture of age classes; hence the need for the parameters fl 1-2) and S I]-2) in the formulas for expected recoveries. In the years after the banding of an "adult" cohort, all individuals in the cohort are certain to be adults rather than subadults, so that the adult survival and recovery rates then apply. Note also that subadult parameters must be included in the juvenile array to account for the fact that juveniles pass through a subadult stage before becoming adults. In essence, the juvenile cohorts are assumed to consist only of juveniles, but they must mature through a subadult age class before reaching adult status. For this reason there are parameters ill!l) and SI1) that apply specifically to subadults. Because it is only for individuals banded as juveniles that the subadult age can be unambiguously recognized, these parameters only occur in the juvenile data array. From the model parameterization, it can be seen that banding-year variation in survival and recovery is introduced in both the juvenile and the adult arrays. Recall that the two-age class model accounted for juveniles by means of parameters for the year of banding
16.2. Multiple-Age Models TABLE 16.23 Age
Releases
v
R! v)
(v = 1-2)
R~1-2)
Expected Recoveries for Two-Age Band Recovery Data with Unrecognizable Subadult Cohorts a Recoveries in period j 1
/~(1-2)~:(1-2)
*'1
Jl
R(21-2)
2
1~(1-2)r
*'1
"1
3
J
*-2/;? (1-2) f(1-2)j2
/(31-2)
/~(1-2)r162
*'1
"1
...
"2 J3
"'"
R~o) e (0) a(30)
~}(0)~(0) ~Xl J1
~(0)r ~c(1) ~'1 "1 J2 12(0) ,r (0) *"2 ]2
Not recovered
l
/~(1-2)~(1-2)~(2) c(2),c(2)
R~1 - 2 ) -
*'1
"1
~
""~
~j L~,"'lj " lvt"(1-2)~
2 ) , . . ,f2 *x2~(1-2)(c:(1-2)
-.-
*-2/2 (1-2)c~(1-2)..,,,2.o1_1~2(2)f~2)
R(21-2) - ~j ~,"'2jlE'(1-2),/ /'*"
*'3/~(1-2)(1-2)j3 ~e
""
"3~(1-2)r . , , , 3 . .oi_1c(2)f12)
R(1-2) - ~j E(m(1-2))3j
~'k~(1-2)c(1-2)~...OlC(2)_1Y~C(2)l
R(k1-2) -- Ej L~,,,~kjlV f,,,,(1-2)~j
/~(0)c(0)r }2) 1 ~ "2 "'3 '-'l-lf //~(0)C (0)r (1)C(2)...C(2) ~2) 2 ~ a3 ~ ~ /~(0)c(0)c(1)c(2) c(2) ~2) BOB o 4 o 5 ""Ol_lf
R(2O)
/)(o)c(o)c(1) c~(2) c(2)f12) ~"k ~ ~176176
R(kO) _ Gj E(m(~))
R(k1-2)
(v = 0)
389
l~(o)c(0)c(1)f(2) ~xI o 1 o 2 /2(0)C(0),e(1) "x2 ~ J3 /~(o)~c(o) x-3 J3
R (0)
..... ..... .....
R~o)
--~'j
E(m~?)) E(m(2~))
-- Ej a(3O) -- ~j E(m(~))
aWith i = 1 ..... k b a n d i n g y e a r s , j - i ..... I r e c o v e r y y e a r s , a n d u n r e c o g n i z a b l e s u b a d u l t c o h o r t s (v = 1-2). A n i m a l s b a n d e d as " a d u l t s " are a m i x t u r e of s u b a d u l t s (v = 1) a n d a d u l t s (v = 2), b o t h of w h i c h are d i s t i n g u i s h a b l e f r o m j u v e n i l e s (v = 0) b u t n o t f r o m e a c h other. P a r a m e t e r s are a g e - a n d t i m e - s p e c i f i c r e c o v e r y r a t e s (f!v)) a n d s u r v i v a l rates (SIv)) for j u v e n i l e s (v = 0), s u b a d u l t s (v -- 1), a d u l t s (v = 2), a n d m i x t u r e of a d u l t s a n d s u b a d u l t s (v = 1-2).
of the juvenile cohorts. In the present case, both juvenile and "adult" banding cohorts have parameters for the year of banding, and juvenile cohorts also have additional parameters to account for subadults. Thus, the model can be seen as a fairly straightforward extension of the two-age class model. However, because of the addition of yet another set of parameters into the already parameter-rich model structure of the two-age class model, data requirements for this model are quite onerous. A total of 6k + 2l - 3 parameters are identified for the model. As above, not all parameters can be separately estimated. In fact, the only separately estif ! 1 - 2 ) . .i .= . 1 mable parameters are Jf(O) i ' ~, . , k, and the parameters of primary interest, i.e., survival rates, cannot be separately estimated (Brownie et al., 1985). Thus this model is of little interest for estimation. However, lack of fit of the ordinary two-age model compared to this one is diagnostic of the need for inclusion of a marked subadult class in the data structure. The program BROWNIE (Brownie et al., 1985) includes estimation procedures for parameters of this model (denoted as H3), as well as goodness-of-fit tests and likelihoodratio test statistics for comparison with other two-age class models. Alternatively, the above model structure can be formulated in program MARK (Appendix G), together with constraints to form appropriate reduced parameter models (e.g., constancy of parameters over time). The problem for two age classes is a special case of a problem that occurs any time one seeks to estimate
parameters for m + 1 age classes using band recovery data, but the field situation allows identification of only m (or fewer) age classes on release. As above, differences in survival or recovery rates among unidentifiable cohorts may be expressed as lack of fit when a simplified (but identifiable) model structure is used to estimate parameters. Programs such as MARK can be used to construct models for this situation based on all m + I age classes, but the majority of the parameters are unidentifiable unless further constraints are imposed. However, such models may be useful for comparative purposes, particularly when it is of interest to identify reasons for lack of fit that may occur in simpler models. 16.2.6. G r o u p a n d C o v a r i a t e E f f e c t s
The effects of sex, geographic location, or other strata can be incorporated in a manner analogous to the one-age case (Section 16.1.5). For example, an ageand sex-specific generalization of model (Sa.t, fa.t) would have all combinations of age and sex classes at each sampling occasion for both the survival and recovery parameters, and might be denoted as model (Sa.s.t, fa.s.t). Unique estimation of parameters for this model requires a data structure in which animals of each identifiable sex and age class are released at each period, i.e., RI ~'s) > 0 for v = 0..... m ages, s = 1, 2 sexes, and i = 1.... , k banding occasions. Reducedparameter models can be formed by constraining pa-
390
Chapter 16 Estimation with Band Recoveries
rameter indices over the relevant dimensions. For instance, model (Sa.s.t, fa,t) describes a situation in which sex- and age-specific survival parameters are modeled for each time, but recovery parameters are modeled as age specific but not sex specific. As described in Section 16.1.5, parallelism in parameter variation also can be modeled. For example, model (Sa.s+t, fa.s+t) describes a situation in which all 2(m + 1) combinations of sex and age are modeled, but the respective age- and sexspecific parameters vary over time in parallel. Similarly, covariate effects can be modeled as in Section 16.1.6, by imposing functional constraints on the parameters of a general model such a s (Sa.t, fa.t). For example, the constraint
[
]= oov,+ov,xi
In 1 - S}v)
describes a relationship between age-specific survival and a time-specific covariate (e.g., temperature). Though this example models the covariate relationship uniquely for each age class; parallel relationships could be specified--for instance, by specifying a common slope but different intercepts:
is,v,]: oov,+olXi"
In 1 - S}v)
16.2.7. Banding Multiple Times per Year As was the case with single-age models, it is sometimes useful to band at two or more times per year with multiple-age models. Rather than attempting to characterize the large number of potential applications for this banding scheme, here we provide an example involving models with two age classes (Hestbeck et al., 1989). In this sampling situation, both fledged young and adult birds are banded just before the hunting season. The matrices of expected numbers of recoveries for a study with k = 3 banding years and l = 4 recovery years are a~l)
R(21
a~l)f~l)R(1)q(1),c(1)/~ (1)q(1)~(1),c(1)/~ (1)~(1)~(1)c(lhe (1) *'1 "1 J2 *'1 "1 ~ J3 *'1 ~ ~ ~ J4
R(i1f21
/(31)
R(i1 s 1 f(31
R(21s1,s(31,f(41,
1~(1)~r "~3 j 3
R(31)~(1)~-(1) ~ j4
for adult birds, and R~O)
R (0, R(3o)
R~O)f { o ) l ~ ( o ) q ( o ) ~ r 1 6 2 *'1 ~'1 J2 *'1 ~'1 " 2 J3
R(O)f(o)
a study of mallard ducks in southwestern Saskatchewan by Hestbeck et al. (1989) included birds banded as "locals" [ducklings in broods aged in classes II and III of Gollop and Marshall (1954)] during July and early August (median banding date in mid-July). In this case the two banding periods per year applied only to young birds, and the objective was to estimate not only annual survival of adults and fledged young, but also the probability of local young surviving to become fledged young in mid-September (median banding date for fledged young). Let QI~ denote the number of local young banded _(0) denote the probability during July of year i, and Wi that a local young in July of year i survives to become a fledged young in September of year i. The matrix of expected values for recoveries of birds banded as local young can be written as
Q~O) Q(20, Q(3o)
,~(0} (o)r(o) kdl q}l f l
,~(0) (o)~(o)r(1) k~l q~ D1 f l
Q(20)q~(20)f(20)
,~(0) (0),~(0),~(1)r(1) ,-~(0) (0),~(0),~(1)c,(1)t(1) kdl qOl 131 D2 f3 k~l q01 D1 ~2 ~}3 f4
Q(20)q~(20)s(20)f(31) Q(20)~(20)s(20)s(31)f(41) ,~(0) (0).(0) k~3 ~p3 J3
.~(0) (0),~(0).0) k~3 ~3 ~3 J4
The model described by this array of expected recoveries permits estimation of the 2-month probabilities of surviving from the local young stage to the fledged young stage ~" i (0),}, as well as the annual survival rates of adults and fledged young and their associated recovery rates. The model used by Hestbeck et al. (1989) was developed for a specific sampling design and estimation problem, but other designs using multiple banding periods per year with multiple ages can be envisioned. For example, Reynolds et al. (1995) developed a model to estimate spring-summer survival probabilities for yearling (first year) and adult (after first year) mallard females banded in both spring and late summer in Manitoba, Saskatchewan, and Alberta. Birds were aged and released as either yearlings or adults in the spring bandings, whereas all birds were treated as adults in the summer bandings (all birds were > 1 year old). The model was developed for a specific sampling situation and biological question and will not be described here, but the point is that designs involving multiple banding periods per year and multiple age classes can be tailored to sampling situations and questions of interest using the general modeling principles in this chapter.
(o,q(o,~(1)c(1)~.(1) "'1 ~"1 "2 ~ J4
R(o)S(O)f(1)
R(o)S(o)s(1)f(1)
~,(o)r (0) *~3 ] 3
i~(0) r *~3 ~
(1) ./4
for young birds (these are the same expectations as in Table 16.21). The new element here is that young birds (not adults) also are banded at a younger age and thus in an earlier period during the same year. For example,
16.2.8. Model Comparison and Selection As seen above, models involving two or more age classes have a large number of parameters. In situations where the appropriate data structure exists, i.e., release of banded animals in each of m +1 recognizable age classes, age- and time-specific estimates of survival and recovery rates can be obtained under model (Sa.t,
16.3. Reward Studies for Estimating Reporting Rates
fa.t)
o r the generalization (Sa,t, f**t). Once the MLEs for a model are obtained, they can be used to compute matrices of expected values, which in turn provide a means of evaluating goodness of fit. Assuming that a general model such a s (Sa.t, f*a,t) fits the data, one can be reasonably confident that the MLEs are unbiased estimates of the underlying parameters of interest. However, as a general rule one should endeavor to estimate as few parameters as possible and at the same time fit the data. In many cases, especially when data are relatively sparse, estimates under highly parameterized models will have poor precision. As with oneage models, it is desirable to seek a compromise between model fit, which is increased by adding parameters, and estimator precision, which is increased by eliminating unnecessary parameters. The same basic approaches can be used for both multiple-age models and models with a single age class (Section 16.1.8), the obvious difference being that many more reduced-parameter models can be obtained with the more general parametric structure of a multiple-age model.
16.2.9. C o h o r t M o d e l s a n d Parameter Identifiability
It is important to keep in mind that proper identification and estimation of the parameters of multiple-age models is possible only if sufficient numbers of animals of each age class are released at each sampling occasion. Thus, for two-age classes (m = 1), both juveniles (v = 0) and adults (v = 1) must be marked and released each year. If the primary interest is in adult survival, and few juvenile birds can be marked, then it is probably best to use the one-age models described in Section 16.2 and ignore juvenile bandings. However, the opposite problem is more common, in which juveniles are readily captured and marked but few or no adults are marked. A similar situation in the context of m a r k recapture is addressed by means of cohort-specific Jolly-Seber analyses (Section 17.2.3), wherein the availability of multiple recaptures allows for the estimation of time-specific survival rates for each age and cohort. However, with band recoveries such an analysis is not possible. One approach (Section 16.4.3) has been to ignore time specificity and construct a completely age-specific band recovery analysis. But even with this approach, arbitrary constraints generally are required to obtain identifiable parameters. In addition, the necessary assumptions of time constancy of parameters, especially of recovery rates, frequently are not justified. For many, perhaps most, species both survival and recovery (harvest) rates can be expected to be highly variable through time, and in fact this variability may be of
391
primary interest as it relates to changes in harvest or habitat management. We recognize that some circumstances may warrant the use of cohort or other modifications of band recovery analysis. In Section 16.4, we discuss issues related to analysis of band recovery data from studies that differ from those typically encountered in the investigation of North American waterfowl (e.g., nonharvested species, whereby recovery is principally through incidental discoveries of dead birds).
Example Multiage band recovery models can be illustrated with a two-age example originally presented by Brownie et al. (1985). Adult and young male mallards were banded during July-September, 1961-1973, in the San Luis Valley, Colorado. Band recoveries in each of the 1963-1971 hunting seasons are presented in Table 16.24. Fifteen models were fit to these data using program MARK (Appendix G), ranging in complexity from model (Sa.t, fa**t), which allows time- and agespecific variation in survival and recovery and includes a temporary banding effect on recovery, to model (S, f), which estimates one survival and one recovery parameter for all ages and sampling occasions. In addition, several models were fit that allow a parallel response over time--for example, model (Sa+t, fa+t)" Model (Sa, fa+t), describing age- but not time-specific variation in survival and age-specific, parallel time variation in recovery rate, was selected on the basis of comparison of QAIC c statistics (Table 16.25). Parameter estimates for this model are reported in Table 16.26 and indicate higher survival rates and lower recovery rates for adults than for juveniles.
16.3. R E W A R D S T U D I E S FOR ESTIMATING REPORTING RATES 16.3.1. D a ta S tru cture
As noted earlier, recovery rates for harvested birds implicitly incorporate the rate of mortality from harvest, as well as the probability of retrieval of harvested birds and the probability that a retrieved bird is reported. If we define the rate of harvest H i as the proportion of (legally) killed animals that are retrieved, recovery rates can be reexpressed as fi = ~iHi,
where H i = c i K i is the probability that an animal alive at the midpoint of the banding period in year i is harvested (K i) and retrieved (ci) during the hunting season of year i, and ~'i is the probability that a har-
392
Chapter 16 Estimation with Band Recoveries TABLE 16.24
Recoveries of Adult and Young Male Mallards Banded Preseason (July-September) in the San Luis Valley, Colorado a Recovered during hunting season
Age class
Year
Bands
1963
Adult
1963 1964 1965 1966 1967 1968 1969 1970 1971 1963 1964 1965 1966 1967 1968 1969 1970 1971
231 649 885 590 943 1077 1250 938 312 962 702 1132 1201 1199 1155 1131 906 353
10
13 58
83
35 103
Young
1964
1965
1966
1967
1968
1969
1970
1971
6 21 54
1 16 39 44
1 15 23 21 55
3 13 18 22 39 66
1 6 11 9 23 46 101
2 1 10 9 11 29 59 97
18 21 82
16 13 36 153
6 11 26 39 109
8 8 24 22 38 113
5 6 15 21 31 64 124
3 6 18 16 15 29 45 95
0 1 6 3 12 18 30 22 21 1 0 4 8 1 22 22 25 38
aFrom Brownie et al. (1985).
vested a n d retrieved bird is reported (Fig. 16.1). Frequently, an estimate of the rate of harvest H i is of intrinsic interest a n d m a y be obtained if the reporting rates are either k n o w n or can be estimated with auxiliary data. One a p p r o a c h is to divide the m a r k e d sample into t w o groups, a "control" sample that is m a r k e d using s t a n d a r d m a r k s (e.g., n o r m a l a l u m i n u m leg b a n d s for birds) a n d a " r e w a r d " sample of animals m a r k e d with b a n d s indicating that the finder will be paid a m o n e t a r y or other r e w a r d for reporting the band. The r e w a r d b a n d s are a s s u m e d to have high rates of reporting (ideally, ~ close to 1), a n d the ratio of recovery rates for the t w o samples can be used to estimate )~. This can be seen by c o m p a r i n g the expected values for the direct (first-year) recoveries from a simple r e w a r d study, w h e r e R control b a n d s and R' r e w a r d b a n d s are applied, a n d recoveries are from a single harvest period i m m e d i a t e l y following release:
Number marked
Expected number recovered
Control
R
RKH
Reward
R'
R'H
A simple m o m e n t estimator for )~ is obtained by equating the n u m b e r of control (m) a n d r e w a r d (m') recoveries to their expected values and solving for the u n k n o w n parameters. This leads to H=
m'/R'
and
m/R m'/R'"
A complication occurs w h e n deliberate efforts are u n d e r t a k e n to obtain b a n d s from hunters in the field, usually in the course of law enforcement or g a m e checking activities. These efforts result in t w o types of reports of s t a n d a r d bands: normal, v o l u n t a r y reports by hunters, a s s u m e d to occur at the rate K, a n d reports t h r o u g h solicitation, w h i c h are a s s u m e d to be perfectly reported (K = 1). In this situation the probability that a b a n d on a harvested animal is reported t h r o u g h solicitation (~/) is usually not of intrinsic interest, but m u s t be taken into account in the estimation of reporting a n d harvest rates. The recoveries for s t a n d a r d b a n d s
16.3. Reward Studies for Estimating Reporting Rates TABLE 16.25 Model Selection of Adult and Young Male Mallards Banded Preseason (July-September) in the San Luis Valley, Colorado a Goodness of fit b Model
Parameters
X2
df
P
(Sa, fa+t) (Sa, fa*t) (Sa*t, fa*t) (Sa*t, f*a*t) (Sa*t, fa+t)
12
110.1
76
0.006
0.00
20
95.43
68
0.016
3.05
34
68.49
54
0.089
7.31
42
53.11
47
0.250
9.79
26
97.06
63
0.004
16.53
(Sa+t, fa*t) (Sa+t, fa+t)
AQAICc r
27
114.3
62
0.000
33.82
19
133.2
70
0.000
34.53
4
172.7
84
0.000
39.45
(St, fa+t)
18
142.3
71
0.000
40.52
(Sa.t, fa)
18
148.3
71
0.000
45.89
(S, f~)
3
186.7
85
0.000
49.83
(Sa, f)
3
253.2
85
0.000
108.64
(Sa, f~)
(Sa+t, ft)
18
232.8
71
0.000
120.64
(S, f)
2
278.0
86
0.000
128.59
(St, ft)
17
248.4
72
0.000
132.42
a From Brownie et al. (1985). Parameters are annual survival (S i) and recovery (fi) rates; subscripts a and t denote variation in parameter over age and time, respectively; " , " denotes interactive effects of age and time, " + " denotes additive effects of age and time. b Deviance-based chi-square test; see Section 4.3.3. c Difference between model QAICc (Akaike's Information Criterion, corrected for small effective sample size and adjusted by the quasilikelihood factor ~ = 1.13; see Section 4.4.), and QAIC c value for the lowest ranked model.
now are poststratified by those obtained from hunters (mh) and by solicitation (ms); reward bands are assumed reported by hunters (m'), with K = 1 as before. The expected values for these observed recoveries are Expected number recovered Band type
Number marked
Hunter reported
Solicited
Control
R
RM1 - ~/)H
Reward
R'
R'(1 - ~/)H
R~H R' ~/H
and moment estimators of K, ~/, and H are
H = m'/R', ~/
ms/R m'/R"
and
mh/R m' /R' - ms/R
393
(Henny and Burnham, 1976; Conroy and Williams, 1981). Variances can be derived for these estimators (e.g., Henny and Burnham, 1976) or obtained numerically from the Fisher information matrix, using SURVIV (White, 1983), for example. As noted by Conroy and Williams (1981), the estimator of reporting rate for standard bands is sensitive to the assumption that ~ = 1 for both the reward bands and the bands obtained through solicitation. Positive biases in K occur whenever either the first condition (100% reporting of reward bands) or both conditions (100% reporting of reward and solicited bands) do not hold. Bias can be reduced (but probably not eliminated) by increasing the reward value; alternatively, bands of differing rewards can be used to allow estimation of a response curve for reporting rate of reward bands, and estimation of a reward band reporting rate ()tr < 1) (Nichols et al., 1991). The above estimators form the basis for a more complete investigation of the variation in reporting, solicitation, and harvest rates, as indicated in the following section. 16.3.2. M o d e l i n g S u r v i v a l R a t e s w i t h Indirect and Direct Recoveries
In this situation both control (standard) and reward bands are applied to animals during each banding period, and the data consist of recoveries of both types of bands in the subsequent harvest periods. The data are stratified by the two types of bands (control and reward), with expected values for the recoveries described in terms of the released sample sizes R or R' and time-specific harvest, reporting, and survival rates. To illustrate, consider a study not involving recoveries by solicitation, in which control and reward bands are applied in each of three years, followed by four years of recoveries (Table 16.27). The expected numbers of recoveries for this data structure are shown in Table 16.28. The model allows for time-specific estimation of reporting, harvest, and survival rates. Parameter estimates and tests of hypotheses are programmed in MULT (Conroy et al., 1989b) and also can be constructed using SURVIV (White, 1983). Notice that survival rates are specified separately for the control and reward groups; this is the fully parameterized model for which closed-form estimates are possible. Under most reasonable study designs (e.g., animals randomly assigned to control or reward marking) the model can be simplified under the hypothesis H0: S; = Si, i.e., common survival rates for the two groups. If band recoveries are solicited, then the data must be poststratified by type of reporting (hunter report vs. solicited) and analyzed by accounting for solicita-
394
Chapter 16 TABLE 16.26
Age
Estimation with Band Recoveries
Parameter Estimates for Model (S a, fa+t) for Data and Analysis in Tables 16.24 and 16.25 a
~(v)
Year
~'-E(~(v))
C'~L
CAU
flv,
S"E(f~v))
C'L
CU
0.006
0.042
0.066
Adult
1963-1971
0.654
0.013
0.629
0.678
Young
1963-1971
0.541
0.027
0.486
0.592
Adult
1963
0.053
Young
a
1964
0.087
0.007
0.075
0.101
1965
0.052
0.004
0.044
0.061
1966
0.072
0.005
0.063
0.082
1967
0.059
0.004
0.051
0.067
1968
0.066
0.004
0.058
0.074
1969
0.075
0.004
0.067
0.084
1970
0.076
0.004
0.068
0.085
1971
0.055
0.004
0.047
0.064
1963
0.083
0.009
0.067
0.102
1964
0.135
0.010
0.117
0.155
1965
0.082
0.006
0.071
0.095
1966
0.112
0.007
0.099
0.126
1967
0.092
0.006
0.081
0.105
1968
0.103
0.006
0.091
0.116
1969
0.116
0.006
0.104
0.130
1970
0.118
0.007
0.106
0.132
1971
0.087
0.007
0.074
0.101
Estimates are annual survival (S (v)) and recovery (fi(v)~, rates for young (v = 0) and adults (v = 1).
Example
tion rates as indicated above. For example, in a 3-year banding study with recoveries of both types for each of the three subsequent harvest periods, the data structure and expected recoveries are displayed in Tables 16.29 and 16.30. As above, program MULT provides estimates for this data structure and provides tests of hypotheses of time specificity in each of the parameters )ki, "~i, Hi, and Si. Alternatively, models can be constructed using SURVIV (White, 1983).
TABLE 16.27
Band type Control
American black ducks (Anas rubripes) were trapped in eastern Canada during 1978-1980 following the hunting season (i.e., during late January-March) and banded with control and reward bands (Conroy and Blandin, 1984). Because of the long interval between banding and the subsequent opportunity for harvest (during September-January), the estimated harvest
Data Structure for a Reward Band Study I n v o l v i n g Indirect Band Recoveries in the A b s e n c e of Band Solicitation a Recoveries in period j
Releases in period i
1
R1
mll
R2
3
4
Not recovered
m12
m13
m14
R1 - ~,j mlj
m22
m23
m24
R2 - Xj m2j
m33
m34
m~3 m~3
m{4 m~4
R3 - ~,j m3j a~ - ~,j m{j R~ -- ~ j m~j
m~3
m~4
R[3 - ~j m~j
R3 Reward
R~
m~l
R~
R~ aWith k = 3 banding years and l = 4 recovery years.
m~2 m~2
16.3.
TABLE 16.28
Expected Recoveries for a Reward Band Study I n v o l v i n g Indirect Band Recoveries in the A b s e n c e of Band Solicitation a
Releases in Band type Control
Recoveries in period j
period i
1
2
3
al
Rlf l
alSlf 2
R15152f 3
RISIS253f 4
al - ~j E(mlj)
R2f l
R2S2f 3
R2S2S3f 4
R2
Rgf 3
RBSBf 4
R3 - ~j E(m3j)
R~S{H 2
R~S~S~H 3
R~S{S~S~H 4
R~ - Xj E(m~j)
a~H 1
R~S~H3
R~S~S~H4
R~ - Y_,jE(m~j)
R~H 3
R~S~H4
R~ - ~,j E(m~j)
R2 R3 Reward
R~
R~H 1
a~ R~
4
aWith k = 3 banding years and l = 4 recovery years (Table 16.27). Parameters are annual survival (h i) rates; annual recovery rate is fi = hiHi.
TABLE 16.29.
Band type
395
R e w a r d S t u d i e s for E s t i m a t i n g R e p o r t i n g Rates
(S i
Not recovered
-- ~ j
E(m2j)
and S;), harvest (Hi), and reporting
Data Structure for a Reward Band Study I n v o l v i n g Both Indirect and Solicited Band Recoveries a
R1
Solicited
Recoveries in period j
Recoveries in period j
1
2
3
1
2
3
mh11
mh12
mh13
ms11
ms12
ms13
mh22
mh23
ms22
ms23
Releases in period i
Control
Hunter reports
R2 R3 Reward
ms33
mh33
R~
m~l
a~
m~2
m{3
m~2
m~3
R~
m~3
a With k = 3 banding years and l = 3 recovery years, with poststratification of recoveries by type of reporting (hunter reported vs. solicited). Excludes final column of animals released in i and never recovered, R i - ~,j mhij -- ]~j msij and R; - ~ j m;j. Because expectations are the same for hunter-reported and solicited reward bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
TABLE 16.30.
Band type Control
Expected Recoveries for a Reward Band Study I n v o l v i n g Both Indirect and Solicited Band Recoveries a
Releases in period i R1
1 R l fhl
R2
Hunter reports
Solicited
Recoveries in period j
Recoveries in period j
2
3
1
R1Sl fh2
R I S1S2fh3
alfsl
a2fh2
R2S2fh3
R3 Reward
a~ R~ a~
R3fh3
R~H 1
R~S~H2 R~H2
2
3
alSlfs2
RiS1S2fs3
a2fs2
R2S2fs3 R3fs3
R~S~S~f3 R~S~H3 R;H 3
a With k = 3 banding years and l - 3 recovery years, with poststratification of recoveries by type of reporting (hunter reported vs. solicited). Excludes animals released in i and never recovered: R i - ~,j mhij -- ~,j msq and R~ - "dq m;j. Parameters are annual survival (S i and S;), harvest (Hi), solicitation (~/i), and reporting (Ki) rates; annual recovery rate for hunter reports fhi = Ki(1 -- ~li)Hi, and for solicited reports fsi = "~iHi 9 Because expectations are the same for hunter-reported and solicited reward bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
396
Chapter 16 TABLE 16.31
Estimation with Band Recoveries
Reward Study of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Number of recoveries by year Solicited
Hunter reported Band type
Control
Reward
Year banded
Number banded
1979
1980
1981
1979
1980
1978 1979 1980 1978 1979 1980
925 694 758 208 150 150
20
9 9
16
10 7
11
11 11
17 10 23 6 8 13
1981
a Banded during 1978-1980 following the hunting season (i.e., during late January-March), with control and reward bands (Conroy and Blandin, 1984), and recovered during the 1978-1980 hunting seasons.
rates (/-?/i) included mortality during the period J a n u a r y September. H o w e v e r , estimates of reporting (Xi), solicitation (~/i), a n d survival rates w e r e unaffected. Substantial effort at b a n d solicitation w a s exerted (Conroy a n d Blandin, 1984), so recoveries of control b a n d s w e r e poststratified into those obtained by v o l u n t a r y reports from h u n t e r s a n d those obtained t h r o u g h solicitation (Table 16.31). P r o g r a m MULT (Conroy et al., 1989b) w a s u s e d to fit several m o d e l s to these data, w i t h results s u m m a r i z e d in Table 16.32. All the m o d e l s fit the data (P ~ 0.15), a n d several w e r e r a n k e d nearly the s a m e w i t h respect to AIC scores (AAIC < 2). The m i n i m u m AIC w a s p r o d u c e d for m o d e l ()~, "Yt, Ht, S), with time-
TABLE 16.32 Model Selection Criteria for a Reward Study of American Black Ducks Banded in Eastern Canada a Goodness of fit b Model
0~, ~/t, Ht, S) (Kt, "~t,
Ht, S)
(K, ~/t, Ht, St) (~, ~, H, S) (~'t, "~t, Ht, St) 0~, ~/, Ht, S) Okt, "~t"Ht, St, St)
Parameters
8 10 9 4 11 6 13
X2
df
P
10.013 10 0.439 6.228 8 0.622 9.673 9 0.378 19.027 14 0.164 5.892 7 0.552 16.421 12 0.173 5.095 5 0.404
AAICc 0.0 0.3 1.4 1.6 1.7 3.0 4.9
specific variation in solicitation a n d h a r v e s t rates b u t constant survival a n d reporting rate (Table 16.33).
16.3.3. Modeling Spatial Variation in Reporting Rates with Direct Recoveries
M a n y questions, such as that of g e o g r a p h i c variation in reporting rate, can be a d d r e s s e d by considering only direct recoveries, ideally obtained i m m e d i a t e l y following the m a r k i n g period (i.e., not s e p a r a t e d by a l e n g t h y p e r i o d over w h i c h mortality can occur). This design simplifies m o d e l i n g because survival rates no longer n e e d be considered, a n d it also p r o v i d e s a d d e d flexibility w i t h respect to the testing of hypotheses. As above, the s a m p l e s of control a n d r e w a r d b a n d s (Ri, R I) are again g r o u p e d by the time periods of b a n d ing (i = 1, ..., k). Direct recoveries are stratified by the year of b a n d i n g a n d also by g e o g r a p h i c locations of recovery (t = 1 .... , b), w h e r e "location" signifies a n y arbitrary g e o g r a p h i c stratification (e.g., states, flyways, s t r e a m segments). M o d e l structure can be illustrated w i t h a b a n d i n g s t u d y consisting of three years of control a n d r e w a r d b a n d i n g , a n d three possible areas of
TABLE 16.33 Parameter Estimates for Model 0~, Yt, Ht, S) for a Reward Study of American Black Ducks Banded in Eastern Canada a Year
Banded during 1978-1980 following the hunting season (i.e., during late January-March); with control and reward bands (Table 16.31). Parameters are annual survival (S i and S;), harvest (Hi) , solicitation (~/i), and reporting 0~i) rates; subscript denotes variation in respective parameter over time. bPearson chi-square test; see Section 4.3.3. cDifference between model AIC (Akaike's Information Criterion; see Section 4.4.) and AIC value for the lowest ranked model.
~
~(~)
~,,
S'E(~,,)
I-/i
~(I-/i )
S
S'E(S)
a
1978 0.389 0.071 0.250 0.069 0.066 0.012 1979 0.244 0.063 0.056 0.010 1980 0.113 0.037 0.082 0.015
0.697 0.077
a Banded during 1978-1980 following the hunting season with control and reward bands (Tables 16.31 and 16.32).
16.3. Reward Studies for Estimating Reporting Rates TABLE 16.34
Band type
397
Data Structure for a Reward Band Study with Multiple Recovery Locations a Hunter reports
Solicited
Area of recovery (t)
Area of recovery (t)
Releases in period i
1
2
3
1
R1
mlhlb
m21
rrl31
R2
m~2
m22
m32
a3 R~
mlh3 ml 'c
m23 m 2'
m33 m 3'
ms1 1 ms2 1 ms3
R~
m 1'
m 2'
m 3'
R;
m~'
m 2'
m 3'
Control
Reward
1
2 2
ms1 2 ms2 2 ms3
3
ms1 3 ms2 3 ms3
a Poststratification of direct recoveries by type of reporting (hunter reported vs. solicited) and area of recovery, with k = 3 b a n d i n g years t, and b = 3 recovery areas. Excludes animals released in i and not recovered: R i - ~t mthi -- ~t msit and R; -- ~t mi" b mthi and msit denote the n u m b e r of direct recoveries in location t resulting from birds b a n d e d in year i with standard (control) bands, that are reported by hunters (h) and are solicited (s), respectively. c m~' denotes the n u m b e r of direct recoveries in location t resulting from birds b a n d e d in year i with r e w a r d bands. Because expectations are the same for hunter-reported and solicited r e w a r d bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
recovery (Table 16.34). The expected numbers of control recoveries from such a study are displayed in Table 16.35, based on area- and time-specific harvest, reporting, and solicitation rates. Program MULT provides estimates for several models under different assumptions about variability in parameters over time and space (Conroy et al., 1989b). Reward band models incorporating both spatial and temporal variation and including both direct and indirect recoveries can be implemented using SURVIV (White, 1983; see Nichols et al., 1995b). Example
This example is from the previously mentioned study of American black ducks, conducted in eastern TABLE 16.35
Canada during 1978-1980 (Conroy and Blandin, 1984; Conroy, 1985). As before, bandings were included for all three years of the study, and direct recoveries of control and reward bands were stratified by categories of distance from banding stations: 0-20, 21-100, and > 100 km. At issue was whether proximity to banding stations had an effect on the reporting rate of band, as suggested by Henny and Burnham (1976). Data for this study are presented in Table 16.36. Program MULT (Conroy et al., 1989b) was used to fit four (of 19 available) models to these data, focusing on variation in X over space and time (Table 16.37). Model (ha,t, '~a.t, Ha, t ) in Table 16.37 is saturated, i.e., the number of parameters is equal to the number of independent observations (27). Of the other three models, only (ha,
Expected Recoveries for a Reward Band Study with Multiple Recovery Locations a'b Hunter reports
Solicited
Area of recovery (t)
Area of recovery (t)
Band type
Releases in period i
1
2
3
Control
aI
alflhl
Rlf21
alf31
a l f lsl
R l f s2
a l f s3
R2
a2f12
R2f22
R2f32
R2f12
R2f s2
R2f s3
a 3 f ls3
R 3 f s2
a 3 f s3
Reward
R3
a3f~3
a3f~3
a3f33
R~ R~
R~H~ R~I41
R~I42 R~H2
R~I-I3 R~H3
R;
R;I4~
R;I4~
R;I43
2
a With k = 3 b a n d i n g years and b = 3 recovery areas, with poststratification of recovery by type of reporting (hunter vs. solicited) and area of recovery. Excludes animals released in i and not recovered: R i - Xt E(mthi) - ~Zt E(mtsi) and R; - ~,t E(m~'). b parameters are reporting rate (h~), b a n d solicitation rate (~I), and harvest rates (/-~i); subscripts and superscripts denote variation in respective p a r a m e t e r over time (i) and recovery area (t); annual area-specific recovery rate fthi = h~(1 - - "Yi)Hi t t for hunter reports and t for solicited reports. ftsi = "~iI-~ii
398
Chapter 16 Estimation with Band Recoveries
TABLE 16.36
Numbers of Bandings and Recoveries of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Number of recoveries by distance from banding stations Hunter reported
Band type Control
Reward
Year banded
Number released
0-20 k m
1978
2719
111
26
1979
2809
83
28
1980
3113
142
1978
374
1979
599
1980
627
21-100 km
Solicited >100 km
0-20 k m
21-100 km
>100 k m
61
65
18
5
64
26
15
15
29
100
34
19
10
41
16
34
46
20
52
47
20
74
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands, stratified by distance from banding stations.
"~a,t, Ha.t) fit the data (P > 0.25); the latter model also ranked lowest in AIC score. The model allows for areaand time-specific variation in solicitation and harvest rates, with reporting rates constant over time but varying with respect to distance from banding sites. Parameter estimates are shown in Table 16.38. Further tests of linear contrasts among the distance intervals (Section 16.1.10) indicated that reporting rates were higher (P < 0.05) within 20 km from banding stations, contrary to suggestions by Henny and Burnham (1976) that proximity to stations results in depressed reporting rates for mallards (Conroy et al., 1989b; Conroy and Blandin, 1984).
TABLE 16.37 Model Selection Criteria for a Spatially Stratified Reward Study of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Goodness of fit Model b (ka, %*t, Ha*t) (Ka*t, (K,
"~a*t, Ha*t)
~[a*t, Ha*t)
(Kt, "~a*t, Ha*t)
Parameters
X2
df
pc
&AIC a
21
4.706
6
0.582
0.0
27
me
m
m
7.3
19
24.5712
8
0.0018
16.3
21
22.5033
6
0.001
18.0
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands; recoveries stratified by distance from banding stations (Table 16.36). b Parameters are reporting rates 00, band solicitation rates (y), and harvest rates (H); subscripts denote variation in respective parameter over areas (a), time (t), or both area and time (a't). c Pearson chi-square test; see Section 4.3.3. a Difference between model AIC (Akaike's Information Criterion; see Section 4.4.) and AIC value for the lowest ranked model. e Saturated model.
16.4. A N A L Y S I S OF B A N D RECOVERIES FOR N O N H A R V E S T E D SPECIES In this section we extend the methodology developed in Section 16.1 through Section 16.3, in which recoveries are reported via bands on animals harvested or found dead by sportsmen, to a situation in which the general public reports bands on animals found dead during the entire year. We also deal with two situations that have rendered analysis of band (or ring) recovery data more difficult, particularly in many European bird ringing studies: (1) banding of young of the year only and (2) unknown numbers of banded birds released. 16.4.1. D a t a S t r u c t u r e
The models described above and by Brownie et al. (19~5) emphasize the situation wherein recoveries of marked animals are obtained via reports from hunters or anglers. In this case, the recovery rates fi can be interpreted in terms of harvest pressure, particularly if adjustments can be made for reporting rate or crippling loss (see Fig. 16.1). However, there is no special requirement that the process of band recovery must involve harvest, versus the general finding of bands by the public. This is especially the case for the reporting of bands (rings) from nonharvested birds, and much of the European literature on the subject is oriented toward this type of reporting. 16.4.2. P r o b a b i l i s t i c
Models
In these applications, the basic data structure for band recovery data (Section 16.1.1) still holds, but the
16.4. Analysis of Band Recoveries for Nonharvested Species TABLE 16.38
Parameter Estimates for M o d e l (ha, ~a.t, Ha*t) for A m e r i c a n Black D u c k s B a n d e d in Eastern Canada a
Recovery Year (i)
1978
1979
1980
399
Band solicitation rates
Band reporting rates
stratum (t)
Distance (km)
1 2 3 1 2 3 1 2 3
0-20 21-100 >100 0-20 21-100 >100 0-20 21-100 >100
~t
b
0.549 0.328 0.274 n m m ~ -~
(Anas rubripes)
Harvest rates
~'E(~t)
~/~
~'~(~//t)
/?/~
S'~(/?//t)
0.063 0.064 0.028
0.241 0.178 0.026 0.143 0.151 0.061 0.119 0.181 0.027
0.034 0.049 0.011 0.030 0.045 0.016 0.023 0.048 0.009
0.101 0.038 0.087 0.067 0.035 0.088 0.089 0.033 0.119
0.010 0.007 0.010 0.007 0.006 0.009 0.009 0.006 0.011
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands; recoveries stratified by distance from banding stations (Tables 16.36 and 16.37). bEstimates vary by geographic stratum (distance interval), but not by year: ~/I = ~/t for i = 1, 2, 3.
parameterization of Seber (1970b) replaces that of Brownie et al. (1985), i.e.,
fi = (1
-
Si)ri,
where r i is equivalent to ~'i in Seber (1970b) and is often termed the "reporting rate." This parameter, which is not to be confused with the reporting rate discussed earlier (Fig. 16.1), refers to the probability that a marked, dead animal is found and its band reported by the finder. The parameterization thus differs from that of Brownie et al. (1985) in that the probability 1 Si of a mortality event leading to the recovery is treated separately from the process of recovery (finding and reporting), whereas u n d e r the Brownie et al. (1985) formulation both processes are s u b s u m e d in fi. This distinction is important because in a typical ringreporting study, m a r k e d animals are found dead t h r o u g h o u t the year, i.e., recoveries are not confined to a well-defined harvest period. The contrast between the two m o d e s of recovery is clarified by contrasting Fig. 16.1(a and b), in which animals are b a n d e d and released shortly before the harvest period in each year and are recovered only during the harvest period, and Figure 16.1c, in which animals are b a n d e d on an anniversary date each year i and recovered t h r o u g h o u t the interval [i, i+1]. In the Brownie et al. (1985) parameterization, recovery is viewed as a destructive sample of a population alive at the time of sampling and is thus conditional on survival to the time of recovery. If survival for the interval following b a n d i n g to the first recovery period is nearly 1, as is reasonable in m a n y
preseason b a n d i n g situations, fi is then interpretable as an index to the harvest mortality process (after appropriate adjustment for reporting rate and crippling loss). On the other hand, recovery of nonharvested species is viewed as a sample of a population that is dead at the time of sampling and is thus d e p e n d e n t on mortality (1 - Si) d u r i n g the interval [i, i+1], followed by sampling with probability equal to the "reporting" rate r i. The above a s s u m p t i o n s and interpretation of parameters lead to expected values of recoveries u n d e r the simple, one-age model that are similar to those for the one-age model considered in Section 16.1. For example, with k = 3 b a n d i n g and l = 4 recovery periods, the expected n u m b e r s of recoveries are given by R1
R2 R3
R1(1
-
Sl)rl R1S1(1 - -
$2)~"2
a2(1 - S2)r 2
R15152(1
-
$3)~-3
R1515253(1
-
$4)/. 4
R2S2(1 - S3)r 3
R2S253(1
R3( 1 - S3)r 3
R3S3(1 - S4)r 4
-
S4)r 4
One statistical and computational a d v a n t a g e of the r i parameterization is that because r i is a conditional probability [conditional on death, which occurs with probability (1 - Si)], it can logically a s s u m e any value on the interval [0,1]. In contrast, fi is a probability that includes both death and reporting. Because an animal cannot experience both recovery and survival d u r i n g the same interval, the p a r a m e t e r s fi and Si are implicitly related as fi 1 year). In some species, it is possible to distinguish first-year, second-year, and adult animals. If age specificity in a parameter structure is restricted to age classes that can be distinguished at the time of initial capture, then the models of the type developed by Pollock (1981b) and Stokes (1984) can be used (Section 17.2.2). In other situations, the investigator can only distinguish young from older (> 1 year) animals on capture, but nevertheless wishes to consider age specificity for ages 2, 3, 4, etc. In such a situation, only the animals marked as young will be of known age in any subsequent year. Estimation of age-specific parameters therefore relies on animals marked as young. Cohort models (Buckland, 1982; Loery et al., 1987; Pollock et al., 1990) were developed for use with such data (Section 17.2.3). An important consideration for age-specific models involves the need for correspondence between the time separating sample periods and the time required to
440
Chapter 17 Estimating Survival, Movement, and Other State Transitions
make the transition to the next age class. For example, if one is interested in variation in survival or capture probabilities among annual age classes, then sampling should be conducted (at least) annually. Under the most common sampling scheme, the time separating sample periods i and i + 1 should be the time required to move from age class v to class v + 1. The models to be discussed in this section are based on this common design feature. Inferences about age specificity are possible under other sampling designs, but parameterizations of associated models must correspond to the temporal sampling frame and are best considered on a case-by-case basis. A final consideration relevant to modeling agespecific data involves the ages exposed to sampling efforts. In many seabirds and passerines, both young, first-year ( 0). The number of releases thus typically declines monotonically with age, so that estimates of parameters associated with older animals tend to be less precise. Cohort models also have been used for unaged adults (Loery et al., 1987). In such cases, the superscript for both data and parameters corresponds to the number of time periods since initial capture rather than precisely to age. Such analyses are viewed as being relevant to relative, rather than exact, age.
17.2.3.2. Model Structure Models for cohort data were considered by Buckland (1980, 1982), Loery et al. (1987), and Pollock et al. (1990). Parameters are defined as in the Pollock (1981b) model (Section 17.2.2), with probabilities of capture (p!V)) and survival (q~!v)). Modeling is also similar to that for the Pollock (1981b) models, except that age is defined not only for classes recognizable at capture but for animals of all ages, given that they were initially caught at age v = 0. For example, the probability associated with capture history 011010 for individuals first captured as young in year 2 is Pr(011010 I release at period 2 as young) =
(3) q~(20)p(31)q~(31)(1--P4(2),}q~4(2)I,,(3) /"5 X5 "
Note that unlike the Pollock (1981b) model, every increase in sample period (subscript i) is accompanied by an increase in age (superscript v).
445
As suggested in the data summary of Table 17.11, the general cohort model can be viewed as a series of separate CJS models, one model for each cohort of age0 releases. The modeling of m!~)-array data under the cohort model is illustrated in Table 17.12. Each row of Table 17.12 follows a conditional multinomial distribution (conditional on releases, RlV)), and the probability distribution for the entire array is given by the product of these multinomials. The multinomial associated with each cohort of age-0 releases can be viewed as a separate CJS analysis. This is easily seen in Table 17.12, because the modeling for the three different age-0 release cohorts contain no shared parameters. Each parameter is indexed by both time and age, and specific time-age combinations are unique to particular age-0 release cohorts.
17.2.3.3. Model A s s u m p t i o n s The CJS assumptions about homogeneity of survival and capture probabilities are required for animals of a specific age at a specific time. The homogeneity assumption is much more likely to be met in the standard situation in which all releases are of young (age 0) animals. Occasionally cohort models are used with adults of u n k n o w n age, resulting in a situation for which data and parameter superscripts actually correspond to "time since initial marking" rather than to age (Loery et al., 1987). Obviously, homogeneity is less likely in this case. As with the age-specific model of Pollock (1981b), we also assume that the timing of sampling and age class transition are synchronized, such that an individual of age v in sample period i will be at age v + 1 in sample period i+1. As noted above, this temporal synchronization of sampling and aging may be considered a design restriction rather than an assumption.
17.2.3.4. Estimation Parameter estimation under cohort models is accomplished using maximum likelihood based on multinomial models of capture histories or the m(i~f)array (e.g., Tables 17.11 and 17.12). Estimation can be viewed as a series of CJS analyses, with each analysis based on a specific cohort of age-0 releases, R!~ i = 1, .... K - 1. In each analysis, and thus for each cohort, the initial capture probability p !0) cannot be estimated, and the final survival and capture probabilities can be estimated only as the products q~(K v~ v+l~ . The closed -lP (K form estimators of the CJS model can be used with cohort data (Buckland, 1980, 1982; Loery et al., 1987; Pollock et al., 1990). Any software that computes CJS estimates can be used one cohort at a time to produce estimates under
446
Chapter 17 Estimating Survival, Movement, and Other State Transitions TABLE 17.12
Expected N u m b e r s of R e c a p t u r e s L~mij "~" (v) I R (v) i ) for the Data of Table Recapture period j
Release cohort (i)
Releases in period i
2
1
R{~
R(O) ~(0).(1) 1 '+'1 I-'2
2
a(21~ R(32) R(2~ R(31~ R(3~
3 a
17.11a
3
4
(1) ,,(1)p(32) 2 ~-'2
R(O) ( 0 ) , t _(1)~ (1)~-, _ (2)~ (2)_ (3) 1 q01 ~1 -- P2 Jq02 kl -- P3 JqV3 P4 a ( 1 ) ,,(1)(1 _ (2), (2)_ (3) 2 ~ 2 ' ~ - P3 Jq~ P4
R ( 0 (0) ) (1)
R(0) (0),1 _(1), (1)_(2) 2 q02 ~,1 -- P3 )q~3 P4
,~(0) (0),., /~1 q~l t l
-
_ (1), (1)_ (2) P2 /q~ P3
R(2) (2)_ (3) 3 q~ P4 2 q~2 P3
R(1) (1)_ (2) 3 q~ P4
R(0) (0)_(1) 3 q~3 P4
Under the structure of an age-specific cohort model with all initial releases at age v = 0.
cohort models. Software such as SURGE (Lebreton et al., 1992) and MARK (White and Burnham, 1999) can compute estimates under cohort models.
17.2.3.5. Alternative Modeling The cohort model is quite general in that it permits different survival and capture parameters for each age-time combination occurring in a study. It often is of interest to consider reduced-parameter models in which parameters are constrained to be equal over time or age or both factors. The imposition of such constraints was considered by Buckland (1980, 1982) and Loery et al. (1987), and modern software (White, 1983; Lebreton et al., 1992; White a n d Burnham, 1999) permits direct estimation under reduced-parameter assumptions (see Pugesek et al., 1995; Nichols et al., 1997). One particularly interesting application involves the investigation of senescent declines in survival rates (Pugesek et al., 1995; Nichols et al., 1997). For example, Nichols et al. (1997) modeled age-specific survival probability as a linear-logistic function of age for certain age classes over which senescent decreases in survival were expected: q~Iv) =
exp(c~i + ~v) 1 + exp(oti + ~v)'
(17.26)
where v denotes age. The linear-logistic model was used for some age classes, whereas separate timea n d / o r age-specific survival parameters were established for other ages (those hypothesized to be unaffected by senescence). Note that the above survival model [Eq. (17.26)] permits time specificity, with the oLi parameters scaling the survival probability according to calendar year. This model can be viewed as an additive model permitting a form of parallelism of age-specific survival over time. Nichols et al. (1997) also fit models that assumed the same linear-logistic relationship regardless of year [i.e., Eq. (17.26) was modified so that O~i - - O~for all i].
The above description of the cohort model assumes that the release cohorts are of age 0, the sampling situation most frequently encountered. However, the requirement of known age does not necessarily restrict the cohorts to young animals of age 0. For example, data analyzed by Nichols et al. (1997) for the European pochard (Athya ferina) included young birds that could be aged as either age 0 or age 1. These two age classes were discernible in the field, so release cohorts of both age classes were included in the analyses. Releases of birds of age 1 thus included both previously marked and unmarked birds. This extension of the basic cohort model is easily handled and reinforces the general idea that the statistical modeling should be tailored to the details of field sampling methods. The various kinds of alternative modeling described in previous sections can be applied to cohort models as well. Time-specific and individual covariates can be used to model parameters, and forms of capturehistory dependence also can be introduced.
17.2.3.6. Model Selection, Estimator Robustness, and Model A s s u m p t i o n s As noted for multiple-age models, the discussion of model selection in Section 17.1.8 for single-age models is applicable to cohort models, as is the discussion of estimator robustness in Section 17.1.9. However, some important differences should be noted. In particular, age-specific survival estimators of cohort models tend to be much less robust to heterogeneous capture probabilities compared to the standard single-age CJS estimators. In a simulation study, Buckland (1982) found evidence that heterogeneous capture probabilities can produce substantial negative bias in survival estimates for the first survival probability (q~l~ and last few survival probabilities. Buckland (1982) noted that these biases can be misinterpreted as evidence of lower survival for young and old animals. On the other hand, Loery et al. (1987) used simulation to investigate ex-
17.2. Multiple-Age Models treme heterogeneity in capture probabilities and found evidence of substantial bias in the survival estimator for young animals (age 0) but little evidence of bias in the survival estimators for older age classes. That heterogeneous capture probabilities can produce substantial negative bias in the initial survival estimate of cohort models may seem surprising, in view of the relative robustness of standard CJS model estimates. The initial survival estimate under the cohort model can be written as q~!o) =/~,i(21)/ R~O),
(17.27)
where/~4(21) is the estimated number of marked animals (all are age 1) in the population at sampling period 2. As specified in Eq. (17.9), the numbers of marked animals in the CJS and related models are estimated by essentially equating two ratios: the proportion of marked animals caught at time i (R i) that are recaptured at some later time (r i) and the proportion of marked animals not caught (yet in the population) at time i ( M i - m i) that are recaptured at some later time (zi). With heterogeneous capture probabilities, the average capture probability is higher for the m 2 marked animals recaptured at time 2 than for the (M 2 - m 2) marked animals not recaptured at time 2. Under the CJS model, the number of animals released at time 2 consists of both marked and unmarked animals, a 2 = m 2 4- u 2. Because of the m 2 animals with relatively high capture probabilities, the R 2 animals are expected to have a somewhat higher average capture probability compared to the (M 2 - m 2) group. Thus, relatively more of the R 2 animals are likely to be recaptured than the (M 2 - m 2) animals, yielding a small negative bias in/~42, and hence in q~l. Under the cohort model, the releases at time 2 are the marked animals caught in two consecutive sampling periods. These marked animals are "undiluted" by the new unmarked animals that would be present in the CJS treatment. Thus, the larger negative bias of the initial survival estimate under the cohort model with heterogeneous capture probabilities is to be expected. On the other hand, subsequent survival estimates under the cohort model are of the form
~p(v) l~d(v+l)/l~Iv) i -- ~vli+1
(17.28)
Although the ~'~i]~(v) typically is negatively biased, the negative bias appears in both the numerator and denominator of Eq. (17.28), rather than only in the numerator, as with Eq. (17.27). Thus, subsequent survival estimates are affected by heterogeneous capture probabilities considerably less than is the initial survival estimate.
447
17.2.4. Age-Specific Breeding Models Not all ages may be exposed to sampling efforts under some capture-recapture sampling designs. Young of many colonial breeding bird species depart the breeding ground of origin following fledging and do not return to the breeding colony of origin until they are ready to breed. Thus, prebreeders of age > 0 can be viewed as temporary emigrants with 0 probability of being captured or observed prior to their first breeding attempt. There are two basic approaches to dealing with temporary emigration of this sort. One approach involves the use of the robust design, which will be covered in Chapter 19. The other is to use standard open-model capture-history data, but to develop a model structure that accommodates the absence of prebreeders. Here we focus on the latter approach. Rothery (1983) and Nichols et al. (1990) considered estimation in the situation in which all birds begin breeding at the same age. Clobert et al. (1990, 1994) considered the more general situation in which not all animals begin breeding at the same age. The latter approach is described here, recognizing that the models of Rothery (1983) and Nichols et al. (1990) represent a special case of the Clobert et al. (1994) approach. Although the general model has been used primarily for birds, it may be useful for a variety of other groups, including sea turtles, anadromous fish, some amphibians, and perhaps some marine mammals.
17.2.4.1. Data and Sampling Design Sampling can be viewed as a hybrid between the sampling approaches for the Pollock (1981b) and cohort models. Thus, animals are marked at age 0 on the breeding grounds, so that their ages are known throughout the study. However, adults are treated as in the models of Pollock (1981b), in the sense that age is considered no longer relevant once an animal begins breeding. Thus, releases each year can consist of both young animals (age 0) and adult breeders of u n k n o w n age. As with the previous age-specific models, the time separating successive sample periods must equal the time required to make the transition from one age class to the next. The discussion below will use "year" as the unit of time, as this corresponds to the situation most frequently encountered. Capture history data can be summarized using the notation of Pollock (1981b) (also see Section 17.2.2.). Thus, the number of animals exhibiting each capture history again carries a superscript denoting the age at initial capture and release. Young animals are again denoted as age v = 0; however, animals first caught as breeding adults will be indicated as v = k+, where age k is the first age at which animals can become
Chapter 17 Estimating Survival, Movement, and Other State Transitions
448
breeders (we assume that k is known). For example, assume that the first age of breeding is age 3. Then x(0) 100101 denotes the number of animals released as young (age 0) during the first year of the study that are subsequently caught in years 4 and 6. In this instance the capture histories of all animals released as young necessarily have two 0s following the initial release, corresponding to the fact that animals cannot breed until age 3 at the earliest. Animals with the above capture history attempted to breed and were captured in year 4, were not caught in year 5, but were caught again in year 6. The statistic -~010110~'(3+)denotes the number of animals first caught as adult breeders (hence at least 3 years old) in year 2, not caught in year 3, caught again in years 4 and 5, but not caught in year 6. These statistics are compiled as in Table 17.1, with negative numbers again indicating the number of animals not released following capture. (v) rr ay form The data can be summarized in .,,Lij-a (Table 17.13) in a manner similar to that for the Pollock (1981b) model. Note that all m~ ) = 0 for ages j - i, such that j - i < k (i.e., for all ages less than the age of first possible breeding). As was the case for the Pollock (1981b) model, animals released at age 0 can only appear in a single ml~) statistic. They are recaptured only as breeders, and breeders are released following capture as age k+. Of course, animals may appear in a number of releases (RI k+)) and recaptures (m!~+)) as adults.
(v)-Array Representat'on 1 f or the Data TABLE 17.13 The mij Resulting from a Four-Period Capture-Recapture Study on A n i m a l s Released in Two Age Classes a
Age at release Young
Releases in period i R~0)
2
3
4
5
"(0)b "'12
"(0) "'13 "(0)b "'23
1,1,1(0) "'14 "(0) "'24 .,(0)b "'34
"(0) "'15 .,(0) 1"25 "(0) "'35 .,(0)b "'45
(2+) m12
"(2+) -,13 ,..(2+) "'23
"(2+) '"14 .,(2+) "'24 (2+) m34
"(2+) ,,'15 .,(2+) "'25 .,(2+) ','35
R(3o) R(4o) A d u l t (breeder)
R(22+) R(32+) R(42+)
The following material is based loosely on the approach of Clobert et al. (1994). However, we have modified their approach to permit direct estimation and modeling of breeding probability parameters. Clobert et al. (1994) recognized that with a standard capturerecapture model parameterized by survival and capture probabilities, the information about nonbreeding and temporary emigration is incorporated into the capture probability estimates. They estimated age-specific breeding probabilities as functions of these capture probability estimates. We have applied a direct estimation approach to the model of Clobert et al. (1994), which we use here because we believe it is more easily understood and permits more flexible modeling. Define the following threshold ages, which are assumed to be known:
m
.,(2+) "'45
a A g e classes of release, y o u n g (v = 0) a n d a d u l t s (v = 2+). RI ~) d e n o t e s the n u m b e r of a n i m a l s of age v r e l e a s e d at time i, a n d (v) (v) m i. d e n o t e s the m e m b e r s of R i next c a u g h t at time j. All r e c a p t u r e s , t'v) m i j , are of a d u l t animals. P r e b r e e d e r s of age >0 are not e x p o s e d to s a m p l i n g , a n d the first possible age of b r e e d i n g is 2 y e a r s (hence m(0) i,i+1 = 0). b.. (0) = 0, i = 1, "", 4. mi,i + 1
The first age at which a young animal can breed, and thus the first age at which an animal marked as young (R!~ can be exposed to capture efforts and possibly recaptured. The age by which all animals are assumed to be breeding; i.e., the first age at which breeding probability is known to be 1 (or at asymptotic adult rate--see below).
Define the following model parameters:
p!k+~ ~!k+)
Recapture period j
R(20)
R~2+)
17.2.4.2. Model Structure
o•(V) i
The probability that a marked breeder (denoted as age k+) in the study population at sampling period i is captured or observed during period i. The probability that a marked animal of age - k (regardless of breeding status) survives until period i + 1 and remains in the population. The probability that a young animal (age 0) released at sampling period i survives until sampling period i + k (hence, until age k). The probability of breeding for an animal of age v at sampling period i that has not previously bred.
The above parameters differ from those discussed for previous models and therefore require some additional explanation. Capture probability is defined as conditional on being a breeder (hence, exposed to sampling efforts) so a corresponding parameter is needed only for breeders. Prebreeders of age > 0 are assumed to have capture probabilities equal to 0. The adult or breeder survival parameter q~!k+) is equivalent to the q~l) in Pollock's (1981b) model (see Section 17.2.2) in that it applies to all animals above a threshold age. The young survival parameter q~!0)
17.2. Multiple-Age Models differs from previous survival parameters in that it refers to a multiperiod time interval prior to breeding age. No inference can be drawn about time-specific survival probability of prebreeders before age k because the animals cannot be sampled during this interval [though inferences about average annual survival probability of young prebreeders can be obtained v i a (~}0))(1/k)]. Finally, we note that ~ is needed (and estimated) only for ages v = k, k + 1, ..., m - 1. Breeding probability before age k is known to be 0, and breeding probability after age m - 1 is assumed to be 1 (or at least is assumed to be at some asymptotic adult level). In addition, it is assumed that following the initial breeding attempt, an animal breeds with probability 1. To illustrate, consider a situation in which the first possible age of breeding is k = 2 and the age at which all animals breed is m = 4. Consider capture history 10011 for both young (age 0) and adult breeder (age 2 +) releases. The probability associated with this capture history for young animals can be modeled as Pr(10011 I release at period 1 as young) =
~1
IO~3 1
-- p(2+)
'+'3
],%+:,v42+:,d42+:,pF+:, (2
~4
/4
The survival term q4~ corresponds to the survival of the animals from release in year 1 until sampling in year 3. The large term in braces consists of the sum of two different products of probabilities, each product corresponding to a different sequence of events. In the first component of the sum, the animal breeds in the first available year (year 3) and age (age 2) but is not captured during that breeding season. The animal then survives and is captured during each of the next two breeding seasons. The breeding probability parameter is only needed in year 3, because once the animal breeds for the first time, breeding probability is 1 for subsequent years. In the second component of the sum, the animal does not begin breeding in year 3; hence no capture parameter is needed for this year (because prebreeders are not exposed to sampling efforts). The animal survives and then does breed in year 4 and is caught at that time. The animal then survives until year 5 and is caught again. If we dissect the sequence of ls and 0s that comprise the capture history, we see that the "0" in period 2 is required by the restriction that k = 2. The "0" in period 3 corresponds to an uncertain event, because there are two possibilities: the animal bred in period 3 but was not caught, or the
449
animal did not breed. The sum in the above probability statement reflects this uncertainty, with each side of the sum representing a scenario associated with capture history 10011. Given the "1" in period 4, there was no uncertainty associated with the modeling for the final "1." The probability associated with this same capture history for adults is modeled as Pr(10011 ] release at period 1 as adult) --- q~
(2+)1 , . ( 2 + ) r l p ( 3 2 + ) 1 , . . ( 2 + ) . ( 2 + ) ..(2+)p(52+) 1 -- P 2 iv2 L1 -J~3 /4 ~d4 9
This modeling is more straightforward, because there is only one possible sequence of events leading to (10011) and hence no uncertainty requiring a sum of two possibilities. All survival probabilities from period 1 through period 4 are required. Capture probabilities are used for the periods when the animal was captured, and the complements of capture probabilities are used for time periods of no capture. Thus, modeling for adults is identical to that for the standard CJS model. The probabilities associated with different capture histories again are specified by multinomial distributions that are conditional on the releases of previously unmarked animals of both ages [young (0) and breeding adults (2+)]. These product multinomials are of the same basic form as the multiple-age models of Pollock (1981b) and thus are similar to those shown in Eq. (17.24), with the exception that the cell probabilities for capture histories of animals released as young are different (more complicated) in the age-specific breeding model. The probability distribution for this model also can ,(v)-arra y summary statisbe described in terms of the ,lij tics of Table 17.13. Writing out the expected values or cell probabilities for the entire table can be tedious, so we illustrate with two examples. As with the modeling of capture history data, the probabilities for animals released as young are more complicated than those for animals released as adults. Assume the same age thresholds as above (k = 2, m = 4) and consider the animals released in period 1 as young and next seen in period 5 as breeders: Pr{'(~ ~,(0)} " ~ 1 5 *Xl
:
_
450
Chapter 17 Estimating Survival, Movement, and Other State Transitions
This probability includes the sum of three terms inside the braces. The first term corresponds to an animal that began breeding in the first possible year (3) and was simply not captured until year 5. The second term corresponds to the event of first breeding in year 4, and the third term reflects the event of first breeding in year 5. No breeding probability parameter is needed for period 5 even in this last component of the sum, because all animals are assumed to breed at age m = 4. The corresponding probability for adults released in period 1 and not recaptured until period 5 is given by
pr{2+ "~15
• q~22+)[1 - p~32+)]q~32+)
7. Every marked prebreeding animal of age v, where k ~ v < m, in sampling period i has the same probability ~i-(v)of initiating breeding and becoming a breeder in i. 8. Every marked animal that attempts to breed for the first time in period i breeds with probability 1, or with asymptotic adult breeding probability, at all sampling periods after i. 9. Marks are not lost or overlooked and are recorded correctly. 10. Sampling periods are instantaneous (in reality they are very short periods) and recaptured animals are released immediately. 11. Except for the temporary absences of prebreeders, all emigration from the sampled area is permanent. 12. The fate of each animal with respect to capture and survival probability is independent of the fate of other animals.
• [1-p~42+)]q~42+)p~52+). The above probability is again equivalent to the probability under the standard CJS model. The animal survives and is not recaptured for three consecutive sampling periods, survives, and finally is caught at period 5. 17.2.4.3. M o d e l
Assumptions
The age-specific breeding model described above uses standard open-model capture-recapture data and permits estimation of a kind of temporary emigration associated with prebreeding animals. The ability to estimate these temporary emigration probabilities (actually, their complements, the age-specific breeding probabilities) comes at the cost of some fairly restrictive assumptions about the modeled process. The following assumptions are required by the age-specific breeding probability model: 1. The age k of first possible breeding is known. 2. All animals become breeders by age m. 3. Every young animal released at age 0 in sampling period i has the same probability r ~ of survival until sampling p e r i o d / + k. 4. Every marked animal aged ~k in sampling period i, regardless of breeding status, has the same probability q01k+) of survival until sampling period i+1. 5. Every marked breeding animal present in the population at sampling period i has the same probability p~k+) of being recaptured or resighted. 6. Marked prebreeding animals of age > 0 are not exposed to sampling efforts and have a probability of 0 of being captured in any sampling period.
If the age of first breeding in assumption (1) is not known a priori, the investigator may simply set k equal to the first age at which animals are observed to return and breed. Assumption (2) is met when all animals of age m and greater breed with probability 1. As noted above, however, use of this model is appropriate even if all animals are not assumed to breed with probability 1, but instead breed with some asymptotic adult probability. In this case, the age-specific breeding probability estimates are no longer absolute probabilities but instead reflect age-specific breeding proportions expressed relative to those for adults. Although estimation under a particular model is conditional on a priori knowledge of m, it is possible to fit models incorporating different values of m, and to then use LRTs or AIC to select the most reasonable model and therefore the most reasonable value of m. Assumptions (3) and (4) deal with homogeneity of survival probability within an age class. Of particular importance is the assumption that survival probability of animals of age ~k is the same regardless of whether or not the animal has become a breeder. It does not appear that relaxation of this assumption is possible with single-state, open-model data. Assumption (5) of homogeneous capture probabilities is required in most open-population capturerecapture models. However, assumption (6) of capture probability of 0 for prebreeders is specific for this model. If prebreeders are available for sampling on the breeding grounds, then multistate modeling (Section 17.3) can be used, even if prebreeders (or even nonbreeding adults) have different capture probabilities than breeding adults (see Nichols et al., 1994; Cam et al., 1998).
17.2. Multiple-Age Models Assumption (7) deals with homogeneity of agespecific breeding probabilities for animals that have not bred previously. The discussion of heterogeneity of rate parameters for the CJS model is relevant to this parameter as well. Assumption (8) represents another strong hypothesis about the underlying process of accession to reproduction, an alternative to which might involve animals that, having previously bred only once, breed again with a lower probability than older, experienced breeders. 17.2.4.4. Estimation
Clobert et al. (1994) used maximum likelihood estimation to estimate survival and capture probability parameters for this underlying model. Estimates of breeding probabilities then were obtained as functions of capture probabilities of young animals (the complements of their ~Iv) values include the probability of not breeding and therefore of not being exposed to sampling efforts) and adult breeders (the complements of their ]~!v) values include only noncapture when all adults breed, but also include nonbreeding in the more general case of some adult nonbreeding). We have implemented this model using program SURVIV (White, 1983), because it permits flexible modeling of the agespecific breeding probabilities. The model also can be implemented as a multistate model in MARK (White and Burnham, 1999). As with the CJS and other multiple-age models, capture probabilities for the initial sampling period cannot be estimated, and the final capture and survival probabilities can only be estimated as products. Additional information on estimable parameters is provided by Clobert et al. (1994).
17.2.4.5. Alternative Modeling The discussion of modeling under the previously described age-specific models (Sections 17.2.2 and 17.2.3) is relevant to the age-specific breeding models. Time constraints can be placed on capture, survival, or breeding probability parameters. Because breeding probabilities are often difficult to estimate, it can be both useful and reasonable to assume these probabilities are constant over time. Under many reasonable scenarios, breeding probabilities are hypothesized to increase monotonically with age, so it is useful to model them as linear-logistic functions of age; e.g., as o~(V)_ i -
exp(~/i + ~v) 1 + exp(~/i + f~v)'
(17.29)
where "Yi is a parameter associated with year effects and [3 is the linear-logistic slope parameter (expectation
451
is that ~ > 0). Recall that o~ i - (v) is estimable for ages v = k , k + 1.... , m - l and is defined to be 0 for v < k andlforv>(m1). As noted above, it frequently is useful to construct several different models assuming different values of m. AIC or LRTs then can be used to help decide which model, and thus which value of m, is most appropriate. The above model structure is fairly general, and we note that constraints on this model can produce the models considered by Rothery (1983) and Nichols et al. (1990). In particular, they considered the case in which k = m. Animals released as young (age v = 0) in year i do not return to the breeding grounds until year i + k, but breeding probability at age k is 1 (or at least the same as that of adults). So oLlk-l) = 0 and ~Ik) = I by assumption, and a model in which all birds begin breeding at the same age is obtained simply by removing the breeding probability parameters from the general age-specific breeding model.
Example This example is based on a long-term study of roseate terns, Sterna dougallii, on Falkner Island, Connecticut, in Long Island Sound (e.g. Spendelow, 1982; Spendelow and Nichols, 1989). Falkner Island is a breeding colony site for the terns, and banding of both adults and chicks has occurred there every spring and summer since 1978 [for description of trapping methods and other logistical issues, see Spendelow (1982) and Spendelow and Nichols (1989)]. Because of some problems with band losses (Spendelow et al., 1994), color bands were replaced in 1988 with field-readable metal leg bands designed for reobservation. Data from 1988 to 1998 are used in this example. Very few birds return to the breeding colony as breeders until age 3 years, and some are not seen again until ages 6 and 7. Few nonbreeders are seen at the breeding colony, and only known breeders were used in this example analysis. Thus, the estimation problem is equivalent to one in which nonbreeders are completely absent from the colony. The data are summa(v) ~rr ay format in Table 17.14. Having rized in .m/j-~, previously been unavailable for marking, birds banded as chicks (designated as Y for young) were of course unmarked when captured. Note that the first nonzero entries in the array for young are for ,I(0) " q , i + 3 r reflecting the fact that very few birds breed before age 3. On the other hand, releases of adult birds could be divided into unmarked (not captured previously on Falkner Island) and marked birds. This categorization is useful for models that include certain types of capture-history dependence (Section 17.1.6). The estimation problem involved estimating the survival probabilities for young and breeding-age birds,
452
Chapter 17 Estimating Survival, Movement, and Other State Transitions T A B L E 17.14
T h e .,(v) .,/j - -array for Roseate Terns a Year of next e n c o u n t e r
Age
R e l e a s e s R (v) i
R e l e a s e year
M a r k status b
1988
U
206
1989
U
136
1990
U
142
1991
U
158
1992
U
103
1993
U
189
1994
U
186
1995
U
122
1996
U
82
1997
U
97
1989
90
91
92
93
94
95
96
97
98
0
0
17
9
3
0
0
0
0
0
0
0
9
6
3
0
0
0
0
0
0
9
7
3
2
0
0
0
0
3
0
2
0
0
0
0
17
4
4
1
0
0
26
14
7
0
0
15
8
0
0
10
0
0 0
1988
U
160
20
3
0
2
0
0
0
0
1989
U
136
57
78
9
1
0
1
1
0
0
0 0
1989
M
57
37
4
1
0
0
0
0
0
0
1990
U
108
73
7
0
2
0
0
0
0
1990
M
135
100
3
0
2
0
1
0
1
1991
U
72
37
4
3
1
0
0
0
1991
M
206
115
7
0
1
1
0
1
1992
U
31
16
1
0
0
0
0
1992
M
182
158
6
2
0
0
0
1993
U
72
28
1
0
0
0
1993
M
205
177
5
1
0
0
1994
U
29
11
4
1
0
1994
M
233
182
3
1
0
1995
U
21
7
2
2
1995
M
224
175
15
0
1996
U
39
9
1
1996
M
226
173
7
1997
U
23
5
1997
M
234
176
a Captured and released as both first-year young (Y) and adult breeders (A) and then recaptured in subsequent breeding seasons, 1988-1998, Falkner Island, Connecticut. b u denotes previously unmarked, and M denotes previously marked.
as well as age-specific breeding probabilities. We use the modeling approach of Clobert et al. (1994), which was implemented using a specific version of program SURVIV (White, 1983) developed by J. E. Hines for this purpose. The first possible age of breeding was taken as k = 3 years, and the age by which all birds were assumed to be breeding was taken to be m = 6. Two basic models were parameterized as described above, model (~o), ~3+), p~3+), Ot(3,4)) and model (~f0), q~f3+), P t(3+) ' Ot(3,4,5)) 9 Both models contain timespecific survival probabilities for young and adult
birds, as well as time-specific capture probabilities for adults. The superscripts on the c~ parameter indicate which age-specific breeding probabilities are not equal to either 0 or 1, and hence require estimation. For example, Ot(3'4'5) indicates that separate breeding probability parameters are estimated for ages 3, 4, and 5, with the assumptions that k = 3 and m - 6. The model with Ot(3'4) does not include estimation of ~(5), but instead assumes oL(5) -- 1 and thus m = 5. Both forms of trap dependence (transient parameterization and trap response in capture probabilities)
17.2. Multiple-Age Models were needed in the model to deal with permanent and temporary emigration from Falkner Island to other breeding colonies in the Long Island Sound system (see Spendelow et al., 1995). Some emigration is permanent, whereas some can be viewed as Markovian temporary emigration [see Chapter 19 and Kendall et al. (1997)] in that birds emigrate, stay at the new colony site for some time, and then return to Falkner. A transient parameterization of the models was implemented by rewriting survival for unmarked adults allowing for some proportion of transients [Eq. (17.13)]. Model notation for the transient parameterization includes T~3+), indicating time-specific proportion of transients among adults, e.g. model (q~o), q~3+), ,1.13+), pI3+), O~(3,4)).
In addition, a trap-response model (Section 17.1.6) was developed in which animals caught the previous sampling period had different capture probabilities than animals not caught the previous period. The model notation p~3+) and p(3+), for inclusion of trap response indicates that adult capture probability for animals caught the previous period is time specific, whereas the capture probability for animals not caught the previous period is constant over time. The latter constraint is required for parameter identifiability in this model, as for the simpler CJS-type models [see Section 17.1.6 as well as Sandland and Kirkwood (1981) and Pradel (1993)]. Models with age-specific breeding probabilities, transient response in adult survival probability, and trap response in capture probabilities fit the data well and thus could be used as the basis for estimation (Table 17.15). Both the lack of fit of the original models and the need for trap-dependent models could be at-
TABLE 17.15
453
tributed to the movement of birds among the breeding colonies of the study system. Although the best way to deal with movement is via multistate modeling with multiple sampling sites (Spendelow et al., 1995), such models have only recently been extended to deal with age-specific breeding probabilities. The two models with the smallest AICc values are designated as (q~0), ~3+), TI3+), p~3+), p(3+),, o~(V)),the distinction between them being that one contains parameters for age-specific breeding probability for ages 3-5 (denoted as 0~(3'4'5)), whereas the other contains parameters for age-specific breeding probability only for ages 3 and 4 (OL(3'4)), assuming that ~(5) = 1 (Table 17.15). The age-specific breeding probabilities were modeled as constant over time, as AICc values indicated that such models were preferable to models with time-specific c~v). The Pearson X2 goodness-of-fit statistics for both models indicated reasonable fit (Table 17.15). Parameter estimates for the models were consistent with biological knowledge and a priori predictions. Most of the annual survival probabilities for young were in the interval from 0.50 to 0.70, whereas most of the adult estimates were between 0.75 and 0.95 (Table 17.16). A severe hurricane occurred following the breeding season of 1991, so the 1991 survival probabilities were predicted to be low, especially for young birds [see Spendelow et al. (2002)]. This prediction clearly was supported by the estimates, as evidenced by the very low survival estimates for that year. It should be recalled that the survival estimates presented for young are actually estimates corresponding to the 3-year period following release as chicks, expressed as (0) ~1/3 annual rates ,~,q~i,i+3) . Thus, the survival probability
A A I C c V a l u e s a n d P e a r s o n X2 G o o d n e s s - o f - F i t Test S t a t i s t i c s a Goodness of fit b
Model
(q010), q~?+), Tt(3+), Pt~(3+),p(3+)', ~(3,4)) (q010), q0? +), T~3+), Pt'~(3+),p(3+)', 0((3,4,5)) (q0(0), q0~3+), Tt(3+), p~3+), p(3+)', O~(3,4)) (q0~0), q~3+), Tt(3+), /dt ..(3+) , Or (q010), q~(3+), Tt(3+), Pt~(3+)'pC3+)', O~(3.4)) (q0~0), q0~3+)' Pt*'(3+)'p(3+)', O~(3,4)) (q0~0), q0~3+), pl 3+), OL(3'4'5)) (q0~0), q0?+), Vt 4..(3+)' 0((3,4))
Numbers of parameters
j(2
df
0.00
24.8
25
40
2.15
24.8
24
0.41
32
33.14
68.9
33
K. In his modeling, Barker (1995, 1997) also assumed that resighting occurs throughout the range of the animals, such that all animals are exposed to resighting efforts (including those not at risk of capture during some capture occasions). Barker (1995, 1997) notes that the term "resighting" is intended to be general in that it could actually involve capture or dead recovery. The key feature is that the process associated with resighting covers the entire range of the population of interest. As usual, losses are permitted on capture and are handled by simply conditioning on releases. Losses on resighting (e.g., recoveries of dead animals) are included in the model and incorporated into the likelihood. Barker (1997) defined the following statistics: Ri, c
Ri, r mi,j,c, c
mi,j,c, r mi,j,r, c
mi,j,r, r
ri, c ri, r
mi oi
The number of animals released following capture at occasion i. The number of animals released following resighting in (i, i + 1). The number of Ri, c that are next encountered by capture at occasion j. The number of Ri, c that are next encountered by resighting in (j, j + 1). The number of marked animals last encountered by resighting in (i, i + 1) that are encountered next by recapture at occasion j. The number of marked animals last encountered by resighting in (i, i + 1) that are encountered next by resighting in (j, j+l). The number of Ri, c that subsequently are encountered by either method. The number of ai, r that subsequently are encountered by either method. The total number of marked animals captured at time i. The total number of marked animals resighted in (i, i + 1).
Chapter 17 Estimating Survival, Movement, and Other State Transitions
482
The number of animals removed from the population (not released) following resighting in (i, i + 1). The number of animals marked before i, not captured at i, but subsequently caught or resighted [includes animals observed in ( i , i + 1)]. The number of marked animals in the population immediately before i that are resighted or recaptured at or after i (Ti = zi q- m/). The number of animals in the population immediately after i that are subsequently encountered after sampling time i [includes animals resighted in (i, i + 1) ( V i = z i + ri,c)].
Ii
zi
Ti
Vi
in that the probability of capture at i does not depend on whether the animal was at risk of capture at time i - 1. Barker (1997) defined the following parameters: Si Pi
Oi
Fi
Data also can be summarized in encounter history form, but here we follow Barker (1997) and develop the model using summary m i j - a r r a y data. An example m/j-array representation of data is presented in Table 17.31 for a study with four periods of capture and resightings following each capture occasion and occurring up to hypothetical capture occasion 5. Note that all animals enter the study as members of a release cohort of captured animals, Ri,c, but that they may later become members of a release cohort of resighted animals, Ri, r. Multiple resightings of animals in an interval are ignored. The relevant information is whether an animal was seen at least once during an interval.
Vi
Thus, the random emigration assumption is given by the constraint F i = 1 - F;. As is the case in the CJS model (Burnham, 1993; Kendall et al., 1997), F i is confounded with Pi+l so that only the product P~+I -FiPi+ 1 can be estimated. Note that the parameter f differs in its meaning here from previous uses in this book, in particular in Chapter 14 where it is used to denote capture frequency, and in Chapter 16 where it denotes recovery probability. We have retained the use of f for different attributes in order to facilitate cross-
17.5.2.2. Model Structure
Barker (1997) presented a model that corresponds to the random emigration model of Burnham (1993), TABLE 17.31
The probability that an animal alive at time i is alive at i + 1. The probability that an animal is captured at time i, given that it is at risk of capture at time i. The probability that an animal is resighted in the interval (i, i + 1), given that it is alive at time i. The probability that an animal alive at time i is not resighted in the interval (i, i + 1), given that it is alive at i + 1. The probability that an animal alive and at risk of capture at i and alive at i + 1 is at risk of capture at i + 1. The probability that an animal alive and not at risk of capture at i and alive at i + 1 is not at risk of capture at i + 1. The probability that an animal is released, given that it is resighted in (i, i + 1).
The m/j-Array Representation for Data Resulting from a Study with Four Capture Periods and Ancillary Observations Occurring after Period 4 a Period of next encounter j
Release
Number
cohort i
released
Recapture 2
Resighting
3
4
1
2
3
4
ml,3,c, c
ml,4,c, c
ml,l,c, r
ml,2,c, r
ml,3,c, r
ml,4,c, r
m2,3,c, c
m2,4,c, c
m2,2,c, r
m2,3,c, r
m2,4,c, r
m3,3,c, r
m3,4,c, r
Released following capture 1
R1, c
2
R2, c
3
a3, c
4
a4, c
ml,2,c, c
m3,4,c, c
m4,4,c, r
Released following resighting 1
al, r
2
a2, r
3
a3, r
ml,2,~c
aUp to hypothetical capture period 5.
ml,3,~c
ml,4,~c
m2,3,~c
m2,4,~c m3,4,~c
ml,2,~r
ml,3,~r
ml,4,~r
m2,3,~r
m2,4,~r m3,4,~r
17.5. Mark-Recapture with Auxiliary Data referencing b e t w e e n material in this book and the biological literature, where f is similarly used. In constructing the probability model it is i m p o r t a n t to recognize that the survival probability over (i, i + 1) for a m e m b e r of ai, r is not Si, but should be larger than Si because these animals have been seen after i. Here we follow Barker's (1995) d e v e l o p m e n t of an expression for
483
s u m m a r y of Table 17.31 are presented in Tables 17.32 and 17.33. Table 17.32 includes expectations for animals that are released following capture. Some of these releases are next encountered as captures and others are encountered as resightings. As an example, consider the expected value for the entry ml,3,c,c. The expectation begins with the n u m b e r al, c of releases following capture in period 1. Animals associated with this statistic then survive until period 2 (probability associated with this event is $1), are neither seen b e t w e e n 1 and 2 (01) nor captured at 2 (q~), survive until 3 ($2), are not seen b e t w e e n 2 and 3 (02), but are caught at 3 (p~). On the other hand, some animals released following capture are next encountered as resightings, as with m3,4,c, r. The expectation for this statistic begins with the n u m ber R3,c of releases following capture at period 3. In order to a p p e a r as a m e m b e r of mg,4,c,r, a n animal m u s t survive from occasion 3 to 4 ($3), not be seen d u r i n g that interval (03), not be caught at 4 (q~), but then be resighted following capture occasion 4 (f4). Expectations for animals released following resighting are presented in Table 17.33. Thus, the entry m2,4,r, c represents animals released following resighting b e t w e e n periods 2 and 3 and next encountered by capture at period 4. The expectation begins with the n u m ber of releases following resighting b e t w e e n 2 and 3, R2,r. Animals associated with this statistic m u s t then survive until period 3, given resighting b e t w e e n 2 and 3 and release, and the probability for this event is
Pr[individual survives from i to i + 1 ] it was seen in (i, i + 1) and released]. Begin by noting that an animal seen in (i, i + 1) m u s t be released in order to survive until i + 1. Thus, Pr[(survives from i to i + 1 and seen in (i, i + 1)] = Pr[survives from i to i + I and seen in (i, i + 1) ] released] Pr(released). Using the above expression, we obtain Pr[survives from i to i + 1 ]seen in (i, i + 1) and released] Pr[survives from i to i + 1 and seen in (i, i + 1) and released] Pr[seen in (i, i + 1) and released] Pr[survives from i to i + 1 and seen in (17.48) (i, i + 1) I released] Pr(released) Pr[released ] seen in (i, i + 1)] Pr[seen in (i, i + 1)] Pr[survives from i to i + 1 and seen in [i, i + 1)] Pr[released ]seen in (i, i + 1)] Pr[seen in (i, i + 1)]
(1
"
Animals in m2,4,r, c a r e not caught at occasion 3 (q~), survive until occasion 4 ($3), are not resighted b e t w e e n 3 and 4 (03), but are then captured at 4 (p~). The analo-
Using the above notation, and also defining q* = 1 - p*, the expected values for the mq-array data
TABLE 17.32 Release cohort i
02)S 2
f2v2
(1 -- Oi)S i
ivi
-
Expected N u m b e r s of Recaptures and R e s i g h t i n g s for A n i m a l s Released F o l l o w i n g Capture (Ri, c) a Number
Period of next encounter by recapture j
released
1
Rl,c
2
R2, c
3
a3, c
2
3
Rl,cS101p 2
4
Rl,cS101q2S202p3
a1,cS101925202q35303P4
R2,cS202P3
a2,cS202q2SgOgP4 R3,cS303P4
Period of next encounter by resighting j 1 1
a I ,c
2 3 4
R2, c R3, c
Rl,cfl
2
3
4
Rl,cSlOlq2f2
Rl,c51019252q3 f3
R2,cf2
a2,cS20293 f3 RB,cf3
Rl,c510192520293530394f4 a2,cS20293530394f4 Rg,cSgOBq4f4
R4, c
a Under the random emigration model of Barker (1997) (see upper half of Table 17.31).
R4,cf4
484
Chapter 17 Estimating Survival, Movement, and Other State Transitions
TABLE 17.33
Expected Numbers of Recaptures and Resightings for Animals Released Following Resighting (Ri, r) a Period of next encounter by recapture j
Release cohort i
Number released
al,r 2
R2,r
3
R3,r
2
Rl,r
I(1
3
- 01)S11 flVl
P~
4
R1r[ ( 1 - 01)S1] ' flY1 j q~S202p~ Rare(1 - 02)$2~ ,
'[
72q
JP3
al,rl (1 - 01)S1]
71Vl Jq~S202q~S303p~
[(1 ~ 0_2)$2]
R2"r
f2v2 j q~S303p~
R3,r[( 1 - 0__3)$3]
f3v3 JP~
Period of next encounter by resighting j 2
1
R1,r
2
R2,r
3
R3,r
R 1 rl (1- 01)$1]
,
71Vl jq'~f2
3
4
al,r[(1- O1)Sllq'~S202q'~f3 flY1
R2,r{(l[ - 02)$2 ] , 72~
Jq3f3
Rl,rl (1 --flVl01)51]q,~S202q~S303q~f4 - 02)5 2 R2,rl (1 72~2 ] q~S303q~f4 R3r[( 1 - 03)$3-] ,t , [ 73V~ ]94./4
aUnder the random emigration model of Barker (1997) (see lower half of Table 17.31).
gous statistic for animals released following resighting between periods 2 and 3 and next encountered by resighting following period 4 (rather than capture at 4), m2,4,r,r, has a similar expectation, differing from that of m2,4,r,c in that the final capture probability of E(m2,4,r,c) is replaced by q~,f4 (these animals are not caught at occasion 4, but are instead resighted following 4). As under previous models of this chapter, each row of the mij -array can be modeled as a multinomial distribution, and the likelihood for the entire study is given by the product of these multinomials. 17.5.2.3. M o d e l A s s u m p t i o n s
In addition to the usual CJS assumptions listed in Section 17.1.2, Barker (1997) lists the following assumptions underlying his general approach: 1. All animals have the same resighting probabilities 0i and fi at time i. 2. All animals at risk of capture at i and alive at i + 1 have the same probability F i of being at risk of capture in i + 1, and all animals not at risk of capture at i and alive at i + 1 have the same probability F; of not being at risk of capture at i + 1 (this assumption can be modified depending on model specifics; e.g., recall that F i = 1 - F; under the random emigration model). 3. Resightings occur throughout the animals' range, but capture only occurs at a specific location within
the range, so study animals may or may not be at risk of capture at any time. 4. Survival probability does not depend on location within the range, so that all animals alive at i survive with probability Si, regardless of whether they are at risk of capture. Assumptions 1, 2, and 4 listed above are additions to the homogeneity assumptions of the CJS model. In addition to capture and survival probabilities, resighting and emigration probabilities must also be the same for all animals in the study population. As noted in the discussion of CJS assumptions, these probabilities may vary as a function of state variables associated with individual animals. Stratification and the incorporation of age and multiple states provide possible means of dealing with homogeneity assumptions. Assumption 3 is as much a statement about the sampling design as it is an assumption. 17.5.2.4. E s t i m a t i o n
As with previous models, estimation is accomplished by using the mij-array data in conjunction with the product-multinomial likelihood function to obtain maximum likelihood estimates. Barker's random emigration model is an option of program MARK (White and Burnham, 1999), and estimates can be easily obtained in this manner. For the random emigration
17.5. Mark-Recapture with Auxiliary Data model with full time specificity, Barker (1997) presents the following closed-form estimators:
fi
--
oiri'c
i = 1, ..., K;
Ri,cVi, mi
i = 2, ..., K;
I
Riczi , ri,c Vi -
+ m i
oi
Zi+l
Yi,c ai,cWi
r
Ri+l,cZi+l
i = 1 , . . . , K - 1; + mi+l
ri+l,c
,
i = 1 .... , K - l ;
and ~;i -
Ri'r,
i = 1, ..., K.
(17.49)
oi
Additional confounded parameters representing functions of the above parameters can be estimated as well (Barker, 1995). Asymptotic variances and covariances are provided by Barker (1995, 1997). 17.5.2.5. A l t e r n a t i v e
Modeling
Barker (1995) considered several alternative models, all of which make use of auxiliary observation data. One such model is analogous to the permanent emigration model of Burnham (1993), in that animals can depart the location where they are at risk of capture (with probability 1 - Fi), but this departure must be permanent. Barker's model is a generalization of Burnham's (1993) model because animals can be released following resighting. The probability structure looks similar to that of Burnham (1993), and the parameters Fi, i = 1 . . . . , K - 2, can be estimated. Closed-form estimators do not appear to exist (Barker, 1995), and estimates must be computed numerically using MARK (White and Burnham, 1999) or perhaps SURVIV (White, 1983). Another alternative discussed by Barker (1995) involves stationary Markov movement. Under this model, animals may move between the locations where they are at risk of capture and locations where they are not at risk, according to a first-order Markov process (movement of an animal between i and i + 1 depends only on its location at i). If these movement probabilities are assumed to remain constant over time (a stationary Markov movement model), then parameters of interest appear to be estimable numerically (Barker, 1995). Barker (1995) also described the use of auxiliary data with multiple-age models of the form described
485
in Section 17.2.2. He presented a detailed structure for the random emigration model with age specificity and derived closed-form maximum likelihood estimators with asymptotic variances and covariances. Numerical estimates are possible under the age-specific analogs of the permanent emigration and stationary Markov movement models. Barker (1995) also considered models for auxiliary observations with capture-history dependence for both single- and multiple-age models. Finally, Barker (1995) outlined an approach for the use of auxiliary observations in multistate models, although additional work is required in this area. Modeling with multiple groups and time-specific covariates should represent straightforward extensions of the models presented by Barker (1995, 1997) for auxiliary data. Certainly, reduced-parameter models will be useful as well, and the additional parameters of the auxiliary-observation models present many opportunities for potentially useful and interesting constraints. 17.5.2.6. M o d e l
Selection, Estimator
Robustness, and
Model Assumptions The approach to model selection described in Section 17.1.8 is applicable to Barker's models using auxiliary observations. Specific goodness-of-fit tests were developed by Barker (1995, 1997) for the random emigration model as well as for many of the alternative models considered above. Discussions of estimator robustness and model assumptions presented in Section 17.1.9 should be relevant to these models, because the modeling and estimation are similar. The use of two distinct sampling methods (e.g., capture and resighting) should reduce problems associated with heterogeneous capture probabilities. Higher capture probabilities also tend to result in less bias in parameter estimates in the face of heterogeneous capture probabilities (Carothers, 1973; Gilbert, 1973), and the additional information provided by auxiliary observations should similarly lead to reduced bias. In general, estimator robustness has not been addressed with these models, though the topic is worthy of future investigation. Barker (1995) specifically discusses tag loss and notes the potential for using double tagging to model tag loss and estimate parameters in the face of this problem.
17.5.3. Capture-Recapture with Radiotelemetry Radiotelemetry has proved to be useful in studies of animal populations, and biostatisticians have devel-
486
Chapter 17 Estimating Survival, Movement, and Other State Transitions
oped useful approaches for estimating survival and movement probabilities from radio-marked (radioed) animals (see Sections 15.4 and 15.5). Telemetry frequently is used in conjunction with other forms of marking and sampling animals. For example, radios are expensive relative to other kinds of tags, so it has become fairly common to conduct a standard capturerecapture study, but also to release a small group of animals marked with radios. In some of these studies, observations of radioed animals are simply used to interpret the estimates of demographic parameters obtained from the capture-recapture data. For example, low survival rates from capture-recapture studies can result from either high mortality or high permanent emigration, and telemetry with even a few animals can be used to judge the likely importance of permanent emigration. In studies using both radiotelemetry and standard capture-recapture with reasonable samples of marked animals having each type of mark, it is common to compute different estimates (e.g., survival probability) using each set of data separately. When resulting estimates are similar, it is generally concluded that both estimation approaches are performing reasonably. However, when they differ, a posteriori stories are developed to explain why and to infer which estimate is "right." Because of the greater sampling intensity that is possible with radioed animals (detection probability approaches 1 in many studies), estimates resulting from telemetry data frequently are assumed to be more accurate (or at least reasonable). However, some researchers have cited possible radio effects (e.g., reduction in survival probability associated with the attached radio) and possible associations between censoring (Sections 15.4 and 15.5) and animal fate as reasons to distrust telemetry-based estimates. For example, Bennetts et al. (1999) found evidence of strong year-to-year variation in survival probabilities of juvenile snail kites (Rostrhamus sociabilis), as estimated from radiotelemetry. Survival estimates were low (and consistent with capture-recapture estimates) for one year in which search effort for dead radioed birds was especially intensive, but high (and different from capture-recapture estimates) for two years of reduced search effort for dead birds. Bennetts et al. (1999) concluded that many of the "censored" birds in the two years of low search effort were actually dead, leading to telemetry-based survival estimates that were biased high. Our purpose in this section is to suggest an alternative to the simple comparison of estimates resulting from two different groups of marked birds (radioed and otherwise marked). Thus, it is possible to combine data from both groups into a single likelihood and
utilize both data sources for estimation simultaneously. In most cases the modeling of such a situation is likely to be tailored specifically to the sampling design of interest [see Powell et al. (2000a) for an application with wood thrushes (Hylocichla mustelina)]. This section is more abbreviated than previous sections in this chapter, primarily because there has been little development of models that combine capture-recapture and radiotelemetry. 17.5.3.1. D a t a Structure
In the previous models in this section, a single type of mark is used on all animals, but the mark can be detected in two different ways, with different detection probabilities and different sampling periods. The situation is somewhat different with radiotelemetry, which involves the release of animals with two different types of marks that then are detected with different kinds of sampling and with different detection probabilities. Even if sampling for radioed animals and animals marked with other sorts of tags (the latter will be referred to as "tagged") is conducted at the same time periods, radioed animals will be detected with much higher probabilities compared to tagged animals. The detection probability typically is assumed to be 1 for radioed animals. There may be cases in which this assumption is not justified, and Pollock et al. (1995) developed a capture-recapture modeling approach for this situation. Data for tagged animals can be summarized in either capture-history form or mq-array form. These data summaries are identical to those presented in the sections above. Data for radioed animals require different summary statistics, because simple detection or nondetection (or in multistate models, state-specific detection or nondetection) is not the only possible fate for radioed animals. Even in a single-state, single-location study, it usually is possible to detect radioed animals that die on the study location. Depending on the area searched for radioed animals, it may be possible to locate animals that have emigrated from the area over which capture efforts occur. As an example of a data structure, we consider the sampling situation of Powell et al. (2000a). They worked at a study location that could be subdivided into two sections, a core section (denoted as area 1) in which all capture (mist-netting) efforts occurred, and a peripheral area (denoted as area 2) that could be searched for radioed animals but in which tagged and radioed animals had zero probability of being captured. Denoting location with superscripts, Powell et al. (2000a) defined the following summary statistics:
17.5. Mark-Recapture with Auxiliary Data
A r
rl ai,i+l birS+l ar
The n u m b e r of tagged animals released in the core area following capture occasion i. The n u m b e r of tagged animals released in the core area following capture occasion i that are next captured in the core area at occasion j. The n u m b e r of radioed animals released in area r (r = 1, 2) following capture or radiolocation at capture occasion i. The n u m b e r of members of A r captured at time i + 1 on the core study area. The n u m b e r of individuals of A r radiolocated (but not caught) at time i + 1 on area s. The n u m b e r of individuals of A r that die between i and i + 1.
The above quantities are defined for r = 1, 2 and s = 1, 2. The first two statistics listed above are used in standard capture-recapture modeling for a multistate (in this case, two locations) system. However, note that the only m~js statistic listed is m~j1. Animals are caught only in the core area (location 1), so releases and recaptures of tagged animals can only occur in area 1. Radioed animals can be captured only in the core area, but can be radiolocated and released in either area, and can be found dead following release in either area. It should be clear that m a n y different designs for such a study are possible.
17.5.3.2. Model Structure Parameterization of models for combined capture-recapture and telemetry data depends heavily on the sampling design. Here, we focus on the model of Powell et al. (2000a) for illustrative purposes. Powell et al. (2000a) defined the following parameters" ~ s is the probability that an animal in location r at time i is in location s at time i + 1, given that the animal is alive at time i + 1; S r(b) is the probability that an animal with m a r k type b (b = 1 indicates a standard tag, b = 2 indicates a radio) that is alive on area r at time i is still alive at time i + 1; p ](b) is the probability that an animal with m a r k type b in the core study area (location 1) at time i is captured at i. Note that this parameterization is quite general, in that it permits different capture and survival probabilities for animals with standard tags and radios. However, the probability of moving between core and peripheral areas is assumed to be the same for animals regardless of m a r k type, because model parameters do not appear to be identifiable otherwise. Also note that there is no detection probability for radio relocations that do not involve capture, because the associated probability is assumed to be 1. Models for this situation can be divided into two components. The portion of the likelihood for the recapture data of standard tags is exactly the same as
487
that used in standard multistate modeling (Section 17.3.1). For example, consider the expected value
E(m11] R 1) _ r)1c1(1).1.11~.,1(1) 1\2~
u?2 K 3
"
In order to appear in m23,11animals must be released from location 1 at time 2, survive from time 2 to 3, remain in location 1 (not move between times 2 and 3), and be captured in time 3. The survival and capture parameters are superscripted with "1," indicating animals with standard tags. The multistate capture-recapture portion of the likelihood includes the possibility of m o v e m e n t to location 2 (the peripheral area), but this state is unobservable, so all encounters occur in location 1, the core area. An example expectation for radioed animals that are caught in the core area is
E(a214]A 2) ,
=
A a c a ( 2 ) ~ l , a I r , I(2) ~3~ 't'3 /-'4 9
Animals in the above statistic must be released from the peripheral area at time 3, survive until time 4, move from the peripheral area to the core area between times 3 and 4, and be captured at time 4. The c o m p l e m e n t a r y expectation for radioed animals that have relocated to the core area but have not been caught there is
E(b2!4, [ A2) = ~A2c2(2),1,2111 - P1 ( 2 ) ] ~3~ '4'3
9
The above statistic is observable only because of the radio. Animals m a r k e d with standard tags can only be observed w h e n they are captured. Radioed animals also can be observed (but not caught) in the peripheral area, as in the following expectation:
E(b~,2IAI)
= a l ~ l ( 2 ) , l , 12 ~
'4'1
9
Note that there is no need for the complement of a capture probability in the above expectation, because capture probability is k n o w n to be 0 in the peripheral area. Finally, radioed animals can be located w h e n dead, leading to expectations of the following type:
E(d21 A2) = A2~[1 - $2(2)]. In the above example, neither capture nor m o v e m e n t parameters are needed, because survival is associated with location at the beginning of the interval and capture of dead animals is not possible. We k n o w from Section 17.3.1 that the capture-recapture portion of the likelihood can be written as the product of conditional (on releases) multinomials, with one multinomial for each group of releases. Similarly, the radio portion of the likelihood can also be written as the product of multinomials that are conditional on
488
Chapter 17 Estimating Survival, Movement, and Other State Transitions
the releases A~ and A 2 in each location at each time period. For each release group, we have four possible fates: caught in core area, relocated (i.e., detected by radiotelemetry) but not caught in core area, relocated in peripheral area, and died. All four fates are observable for radioed animals, but only "caught in core area" is observable for tagged animals. 17.5.3.3.
Assumptions
The model of Powell et al. (2000a) requires the standard capture-recapture assumptions (Section 17.3.1) about homogeneity of rate parameters (survival and capture probabilities are similar for all animals with a particular mark type; movement probabilities are similar for all animals regardless of mark type) among individuals, and independence of fates. Standard assumptions also are required for survival and movement rate estimation from telemetry data (Sections 15.4 and 15.5). In addition, the model of Powell et al. (2000a) assumes that (1) movement between times i and i + 1 depends only on location at time i (the Markovian assumption) and (2) emigration from the two-patch system is not possible. In the most general sampling situation, assumption 2 means that the peripheral area must represent "the rest of the world," and satellite tracking would be necessary to sample such a peripheral area. In reality, there may be certain sampling designs (certain kinds of areas at particular seasons of the year) for which a well-defined and easily sampled peripheral area will be reasonable. 17.5.3.4. E s t i m a t i o n
Maximum likelihood estimation was carried out by Powell et al. (2000a) using a modified version of MSSURVIV (Hines, 1994). Estimation was based on the product multinomials for the summary statistics representing numbers of animals observed with the different possible fates. The parameterizations for survival and capture probabilities permit direct estimation of radio effect parameters. For example, define a parameter reflecting a radio effect on survival as
OL-~ sr(2)/S r(1). Then the survival probability for radioed animals can be rewritten as sr(2) __ Otsr(1).
and this parameterization can be used to estimate a radio effect directly, where oL = 1 denotes no radio effect on survival and 0 < oL < 1 indicates a negative
effect of radios (relative to standard tags) on survival probability. 17.5.3.5. Alternative M o d e l i n g
Various reduced-parameter versions of the model of Powell et al. (2000a) are of interest. For example, one can test for radio effects with a model for which survival and capture probabilities of radioed animals are constrained to be equal to those for animals with standard tags. Stationarity in survival and capture probabilities is also of interest. The models also can be made more general with the incorporation of group (e.g., sex) effects, age effects, and possible capturehistory dependence for the component marked with standard tags. Modeling various rate parameters as functions of time-specific or location-specific covariates should be possible as well. We envision many different types of combined-data models that are tailored to specific sampling designs. In some situations, it will not be possible to radiolocate animals in peripheral areas, but only to state with certainty that the radioed animals are no longer in the core area. In this case, emigration from the core area is a possible fate for radioed animals, but there is no group of animals in the noncore area on which to base the estimation of parameters associated with the noncore location. Several approaches to modeling this situation can be taken, depending on a variety of factors, such as whether emigration is viewed as temporary (as in Powell et al., 2000a) or permanent. In considering the joint use of telemetry and capture-recapture data, the investigator must assess the value of the additional information provided by telemetry data. In particular, telemetry data can be useful in permitting separate estimation of survival and emigration probabilities, which frequently are confounded in capture-recapture studies. The ability to locate animals off the study area, to detect all animals on a study area with certainty, and to locate dead animals, at least on the study area, all should contribute in various ways (depending on sampling design) to the separation of mortality and emigration probabilities. In addition, telemetry data should prove to be useful in increasing the precision of parameter estimates. For example, we have seen little use of telemetry data to aid in the estimation of capture probability. In standard capture-recapture modeling, a "0" at the end of a capture history is ambiguous, in that it can correspond to an animal's absence (death or permanent emigration) or to its presence and noncapture. With radios, an animal is always known to be present (or not) in an area exposed to capture efforts, so that the capture or non-
17.6. Study Design capture of an animal known to be in the capture area can be viewed as the outcome of a Bernoulli trial with associated capture probability. 17.5.3.6. Model Selection, Estimator Robustness, and Model Assumptions The approach to model selection described in Section 17.1.8 is applicable to models of both telemetry and standard tag data, and was used by Powell et al. (2000a). We are aware of no specific goodness-of-fit tests developed for such models and tentatively recommend the Pearson chi-square statistic at this time. Discussions of estimator robustness and model assumptions presented in Sections 15.4, 15.5, and 17.1.9 should be relevant to models with combined telemetry and capture-recapture data. The use of radioed animals that can be detected with certainty should reduce problems associated with heterogeneous capture probabilities with standard tags.
17.6. S T U D Y D E S I G N Designs of open population capture-recapture investigations can benefit from the general advice provided for model development in Chapter 3; that is, study design should be tailored to the questions being addressed and the parameters to be estimated. Issues such as replication and spatial and temporal variance components lead to important design recommendations, which are dealt with (at least generally) in Chapters 4-6 of this book. The focus here is on aspects of study design that are especially relevant to the conditional capture-recapture models for open populations. Given a narrow focus on estimation, it is important to tailor study designs to estimation-related study objectives. In the past, such tailoring necessarily was tied to one of a small number of estimation models and sampling methods. The available estimation methods involved closed-form estimators and variances developed by biostatisticians (e.g., Cormack, 1964; Jolly, 1965; Seber, 1965, 1970b). However, the development of flexible software for computing estimates based on user-defined models (e.g., White, 1983; Lebreton et al., 1992; White and Burnham, 1999) has dramatically changed this situation. In the preceding sections, we frequently have focused on specific models in order to illustrate model development, while also stressing alternative models and approaches in an effort to emphasize flexibility. Biologists now can develop a study design and associated model set for a wide range of estimation problems.
489
The general recommendation to tailor a design to the specifics of the biological or management question makes the job of providing general design suggestions more difficult. However, some general design suggestions focus on three questions that are relevant to study designs: (1) what parameters are to be estimated, (2) how can assumption violations be minimized, and (3) how can precise estimates be obtained?
17.6.1. Sampling Designs and Model Parameters It should be clear from material in the previous sections of this chapter that the estimable parameters are determined largely by study design. For example, single-site capture-recapture studies and their associated models (Sections 17.1 and 17.2) can be used to estimate local survival rate, the complement of which includes both permanent emigration and mortality. If primary interest is in separation of these two components of loss, then additional data are needed. In particular, the models making use of various kinds of auxiliary observations (Section 17.5) should be selected for this purpose. The kind of auxiliary data obtained will depend on the specifics of the study, including the location of the primary study area (Does the study area cover most of the population's range? Is it isolated from other potential habitat?), the status of the study organism as a harvested species (Can recoveries of harvested animals be used?), and the ability of the organism to carry a radio transmitter (Can the animal carry a radio without adverse effects?). The robust design (Chapter 19) provides another source of auxiliary data that can be used to estimate parameters not estimable otherwise. Estimation of the probabilities of moving between locations requires sampling at the locations of interest. Simultaneous sampling at all sites is preferred, but if simultaneous sampling is not possible, then the investigator should try to approximate this situation to the degree possible. Consider two possible designs for sampling four sets of mist nets (at four different sites) for birds. In one, the investigator samples one site for 1 day, then moves to a different site and samples the next day, etc., finishing the rotation in 4 days. The next month, this rotation is repeated. In the other design, the investigator samples the first site for 1 day, waits a week, then samples the second site for 1 day, waits another week, etc. Although both designs sample all four locations each month, the first design more closely approximates simultaneous sampling. If there is little movement among sites over the 4 days of sampling,
490
Chapter 17 Estimating Survival, Movement, and Other State Transitions
then multistate models can be used with data from this design. The use of multistate models with physiological or behavioral state variables requires the assignment of every captured animal to a state, though sometimes it is possible to do this for one state but not another. Reproductive activity is a characteristic about which interesting biological questions frequently are addressed. In many sampling designs, it will be simple to assign some animals unambiguously to a "breeder" category based on observations at a nest, e.g., with new young, during copulation, etc. However, observations of animals not engaged in these activities may not necessarily mean that the animal is not also a breeder. A useful research topic involves the use of probabilistic or imperfect state assignments in multistate modeling.
17.6.2. Model Assumptions 17.6.2.1. Homogeneity of Rate Parameters Previous investigations of capture-recapture assumptions have indicated that heterogeneous rate parameters can lead to biased estimates (Carothers, 1973, 1979; Gilbert, 1973; Pollock and Raveling, 1982; Nichols et al., 1982b; Nichols and Pollock, 1983b; Johnson et al., 1986; Rexstad and Anderson, 1992; Burnham and Rexstad, 1993), so it is important to design studies in ways that minimize heterogeneity. One aspect of design that is relevant to virtually all studies involves the information recorded for each animal on capture or resighting. As noted in Section 17.1.2, potentially relevant information includes group variables such as sex, attributes such as age that change nonstochastically, and attributes such as physiological condition and reproductive activity that vary stochastically. If the study area includes different habitat types, then geographic stratification also may be useful. By selecting the appropriate model from among the various possibilities presented in this chapter, it is possible to incorporate such information into an investigation. Model selection a n d / o r formal testing procedures then can be used to decide whether or not the selected variables are relevant to variation in demographic rate parameters (e.g., survival or movement probabilities) or sampling probabilities. If the variables are indeed associated with variation in model parameters, then group-specific or state-specific parameters should be retained in the model structure as a means of reducing problems associated with heterogeneity. If they are not relevant, the group-specific or state-specific parameters can be constrained to be equal across groups or states, with no loss of precision in estimation. Heterogeneous capture probabilities can be induced
by heterogeneity in sampling intensities, which should be at least partially under the control of the investigator. With regard to spatial sampling, it is important to sample all areas of a study area with similar intensity when practicable. When traps, mist nets, or other stationary sampling devices are used, it is important that all animals in the study area be exposed to these devices. This usually is ensured by a uniform spatial placement of devices, as in a trapping grid. In such spatial arrangements, the distance between adjacent devices should be smaller than the average daily movements or average home range radius of the species being sampled. Regardless of the exact nature of the spatial arrangement of sampling devices, the intention should be to have at least one trap or device (preferably more) within an animal's home range, so that animals should encounter at least one device each sampling occasion. If the number of sampling devices is not adequate to cover a study area in this manner, then the devices should be moved to randomly selected locations within the sampled area at each sampling occasion (Pollock et al., 1990). The use of multiple sampling methods is a way to reduce heterogeneity. The basic idea is that if certain animals behave in a manner that renders them especially difficult to encounter with one sampling method, then they may be more vulnerable to being sampled by an alternative sampling method. In this respect the combined use of physical capture and resighting as methods for obtaining samples should be useful. Similarly, the models for which capture-recapture data are augmented by auxiliary observations (Section 17.5) should provide ways of reducing problems associated with heterogeneity. A final component of study design that reduces problems associated with heterogeneous capture or resighting probabilities is sampling intensity. In general, heterogeneous capture probabilities lead to larger biases in parameter estimates when the probabilities are relatively small. For example, a capture-recapture study of a population in which half the animals have capture probabilities of 0.10 and the other half have capture probabilities of 0.35 should produce estimates with larger bias, compared to a study on a population in which half the animals have capture probabilities of 0.70 and the other half capture probabilities of 0.95. Thus, extra effort to increase capture probabilities should reduce problems associated with heterogeneous capture probabilities. Trap response in capture or survival probabilities is usually undesirable. Although models have been developed to deal with capture-history dependence, these models require extra parameters and therefore result in reduced precision in their estimates. Certain
17.6. Study Design forms of trap response cannot be modeled simply and present problems in inducing estimator bias. Traphappy responses are often associated with the use of baited traps. Such responses can be reduced or eliminated by the practice of prebaiting, placing baits either beside closed traps or in traps that are locked open. With prebaiting, animals are expected to become accustomed to traveling to the trap site in order to feed, so that when the traps are initially set, animals should then be caught with high probability. Trap shyness can also occur in some sampling situations. Mist-net studies of birds frequently encounter problems with net avoidance. If net avoidance results from birds learning net locations, then frequent relocation of nets within study areas may be useful. Use of trapping and handling methods that produce minimal stress on animals not only can minimize trap shyness but also possible trap response in survival probabilities. Use of reobservation methods that do not require physical capture of animals (as in studies based on resighting) should be useful in reducing trap responses in both capture and survival probabilities.
17.6.2.2. Tag Retention Selection of a marking method is an important aspect of design of survival studies. Clearly, it is desirable to select a mark that does not influence the animal's survival probability or even behavior, yet is likely to persist with negligible rates of loss. Larger marks, such as those used in resighting studies (neckbands, patagial tags) and radiotelemetry studies, are the most likely to result in changes in animal survival. Pilot studies of captive or semicaptive animals often provide inferences about the potential for marks affecting survival. Actual field studies with two kinds of marks [e.g., radios and legbands, as in the example of Powell et al. (2000a) (see Section 17.5.3)] can provide direct estimates of tag-related reductions in survival probability. The problem of tag loss does not appear to be adequately appreciated in many studies. It is not uncommon to see rather large studies that appear to have nonnegligible tag loss, yet with no means to estimate the magnitude of loss. In studies of animal survival, this problem is critical, because the parameter estimates correspond to tag survival rather than animal survival. Any capture-recapture survival study that shows even minimal potential for tag loss should include at least a sample of double-marked animals by which to estimate tag loss and animal survival (Arnason and Mills, 1981; Nichols et al., 1992a; Nichols and Hines, 1993). We have been involved with studies in which tag types have been changed over time (e.g., Spendelow et al., 1994; Fabrizio et al., 1999). Even with
491
tag loss estimates from double-tagging, the use of multiple tag types in survival studies makes analyses very complicated, at best. Our recommendation is to change tag types as infrequently as possible.
17.6.2.3. Instantaneous Sampling Another assumption that appears to be underappreciated is that of instantaneous sampling. Though this assumption is never met completely, it often is possible to select sampling periods during which animals experience negligible mortality. Indeed, the rule of thumb is to select sampling periods such that the time between sampling periods and, more importantly, the mortality likely to occur between successive sampiing periods, are large relative to the duration of the sampling period and the mortality occurring during this period (see discussion in Section 17.1.2). Selection of an appropriate sampling schedule thus involves both the season of the year and the duration of the sampling period. Whenever possible, it is best to avoid sampling during seasons of the year of suspected high mortality (e.g., harvest seasons; periods of severe weather, such as winters in some areas). Similarly, it is best to try to achieve high capture probabilities with intense sampling over a short period, rather than less intense sampling over a long period. As noted in Section 17.1.9, however, it is sometimes possible to model mortality during the sampling period (e.g., Tavecchia et al., 2002). 17.6.2.4. P e r m a n e n t Emigration
The assumption that all emigration is permanent is commonly listed for capture-recapture estimators for open populations. However, as noted by Burnham (1993), Barker (1995, 1997), and Kendall et al. (1997) (also see Sections 17.5.1 and 19.5.1), random temporary emigration produces no bias in survival estimates, but changes the interpretation of capture probability. However, temporary emigration may sometimes follow a first-order Markov process, such that animals have different capture probabilities depending on whether they were in the area exposed to capture efforts in the previous sampling period. Temporary emigration of this sort can produce biased estimates of parameters of interest (Kendall et al., 1997). Markovian temporary emigration is best handled using the robust design (Chapter 19), although open models with trap dependence in capture probabilities (Sandland and Kirkwood, 1981; Pradel, 1993) (also see Section 17.1.6) sometimes can be used to approximate Markovian temporary emigration. Auxiliary observations permit estimation under a model of Markovian temporary
492
Chapter 17 Estimating Survival, Movement, and Other State Transitions
emigration when the emigration probabilities are constant over time (Barker, 1995) (also see Section 17.5.2).
17.6.3. Estimator Precision Various aspects of study design are relevant to the precision of resulting estimates. Good precision (i.e., small variances and coefficients of variation) is an important determinant of test power and inferential strength, and should be a major consideration in study design. Pollock et al. (1990) presented information about the influence of study design on precision of survival estimates. A general conclusion is that for any set of conditions (e.g., fixed survival probability and population size, and fixed number of sampling periods), cv(q~) decreases as capture probability increases. Certainly, any design modifications that can increase capture probabilities will be useful. These modifications could include increases in the density of capture devices, or the number of observers trying to resight marked animals, or the number of consecutive days on which traps are set or observations are taken. Of course, longer sampling periods deviate more strongly from the instantaneous sampling assumption, so that a tradeoff exists between meeting this assumption and increasing capture probability. Another general inference is that for any set of conditions, estimator precision tends to increase as the number of sampling occasions increases. This is true for time-specific estimates q~i, but the relationship is even stronger when time-invariant parameters are estimated (e.g., q~). For any set of conditions, cv(q~i) decreases as population size increases. Because the study population frequently is defined by the investigator, population size is under investigator control, at least to some extent. A study design involves a tradeoff between size of the sampled population and sampling intensity, in that effort (expressed in terms of numbers of traps, number of person-hours of observations, etc.) can be either spread out over a larger area to sample a larger population, or it can be concentrated in a smaller area to produce a higher capture-resighting probability. In many situations, increases in precision resulting from increased capture probability may be larger than those resulting from increased population size, but the important point is to keep this tradeoff in mind when considering the specific design of a new study. A final determinant of estimator precision is the magnitude of the survival rate. Initially, it might appear that this quantity is not under control of the investigaton However, the quantity of interest is not survival rate scaled to some arbitrary time (e.g., 1 year), but the probability of surviving the interval between succes-
sive sampling periods. Thus, sampling frequency should be tailored to the organism under study. Sampling meadow voles (Microtus pennsylvanicus) at annual intervals, for example, would represent a poor design. Because few marked animals in year i would survive to have a chance of being recaptured in year i + 1, there would be little information for estimating either survival or capture probability. It generally is a good idea to select sampling intervals that provide a relatively high survival probability (e.g., S i > 0.5). However, probability estimators associated with rare events have their own difficulties. For example, the numerical algorithms used in capture-recapture software (e.g., MARK) (White and Burnham, 1999) often have difficulties with parameter estimates near boundaries (e.g., probabilities near 1 or 0). Thus, it is useful to avoid intervals that are so short that deaths are rare and survival probabilities approach 1.
17.7. D I S C U S S I O N Because they allow for mortality, migration, and recruitment, open populations require models that include biological attributes not found in closed population models. The need to include parameters for these attributes means that open population models are considerably more complicated and in consequence the precision of parameter estimates is comparatively lower than for closed populations. This is yet another manifestation of the tradeoff between complexity and precision (see Section 7.1). The modeling approach in this chapter builds on the CJS model, which extends the closed population models in Chapter 14 by incorporating nonstationary survival probabilities. Data for the CJS model consist of marked and unmarked captures at each of a number of sampling occasions, which are summarized in terms of particular capture histories or as summary statistics in an mij -array. Numbers of recaptures in the model are conditioned on the number of releases at each sampling occasion of either unmarked animals (for capturehistory data), or the combination of marked and unmarked animals ( mij-array data). The model therefore represents only a part of the information available in a sample, and a fully efficient use of data requires the modeling of the unmarked captures. The statistical form of the CJS model consists of a conditional product-multinomial distribution of recaptures, from which maximum likelihood estimates and their variances can be derived. Closed forms for the maximum likelihood estimators can be seen as multitemporal analogs of the Lincoln-Petersen estimator (see Section 14.2).
17.7. Discussion A large part of this chapter has dealt with extensions of the CJS model to allow for a cohort structure, which can be incorporated in the model via cohort-specific survival and capture probabilities. Thus, Pollock's (1981b) model (Section 17.2) recognizes age cohorts, and the multistate models of Section 17.3 accommodate both phenotypic and geographic cohorts. The inclusion of both temporal and cohort variation in the probability structure generates a wealth of special cases in model parameterizations, whereby any number of constraints involving stationarity conditions, equality of parameters across cohorts, and other parameter restrictions can be imposed on model parameters. Because the constrained models include fewer parameters than unconstrained models, a collateral benefit is increased estimator precision in the resulting parameter estimates. Though the conditional models in this chapter focus on the estimation of survival and capture probabilities, we used the artifice of reverse-time modeling in Section 17.4 to address recruitment to the population. The approach reverses the direction of time and replaces the focus on survival in the CJS model with a focus on recruitment, building on the recognition of Pollock et al. (1974) that a backward process with recruitment and no mortality is statistically equivalent to a forward process with mortality and no recruitment. Thus, the survival probabilities of a forward process are reinterpreted in a backward process as "seniority parameters"
493
(Pradel, 1996) that inform the recruitment process via Eqs. (17.35) and (17.36). Finally, we explored methods to combine capture-recapture data with information collected from other sources such as band recoveries, radiotelemetry, and resightings of marked animals between capture occasions. In each case we saw that the additional information provides opportunities to improve estimator performance and to estimate new parameters, but at some cost in the mathematical intricacies of data management and analysis. There are numerous opportunities for additional modeling and analysis with these and other combined approaches, as their statistical analysis and application are quite new and yet to be fully explored. The models in this chapter all are based on multinomial distributions of recaptures, conditioned on initial captures. A fully efficient use of capturerecapture data requires the statistical modeling of initial captures as a component, along with the multinomial recapture distributions, of a comprehensive statistical model for the sampling process. In the next chapter we add additional stochastic features to the probability models described here, so as to account for random variation in initial captures. This added feature allows us to focus on a broader suite of biologically informative parameters, including a simultaneous accounting of recruitment and mortality as well as the estimation of population size.
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C H A P T E R
18 Estimating Abundance and Recruitment with Mark-Recapture Methods
18.1. DATA STRUCTURE 18.2. JOLLY-SEBER APPROACH 18.2.1. Model Structure 18.2.2. Model Assumptions 18.2.3. Estimation 18.2.4. Alternative Modeling 18.2.5. Model Selection, Estimator Robustness, and Model Assumptions 18.2.6. Example 18.3. SUPERPOPULATION APPROACH 18.3.1. Model Structure 18.3.2. Model Assumptions 18.3.3. Estimation 18.3.4. Alternative Modeling 18.3.5. Model Selection, Estimator Robustness, and Model Assumptions 18.3.6. Example 18.4. PRADEUS TEMPORAL SYMMETRY APPROACH 18.4.1. Model Structure 18.4.2. Model Assumptions 18.4.3. Estimation 18.4.4. Alternative Modeling 18.4.5. Model Selection, Estimator Robustness, and Model Assumptions 18.4.6. Example 18.5. RELATIONSHIPS AMONG APPROACHES 18.6. STUDY DESIGN 18.6.1. Parameters to Be Estimated 18.6.2. Model Assumptions 18.6.3. Estimator Precision 18.7. DISCUSSION
In this chapter we consider the estimation of population size and recruitment using capture-recapture data for open (to gains and losses between sampling occasions) populations. The relevance of population size and recruitment in this book should be clear, because population size is a state variable of interest in most of the population models that have been discussed, and recruitment of new animals is one of the processes responsible for population change (see Chapters 7 and 8). Methodologically, this chapter can be viewed as an extension of Chapter 17. We consider exactly the same kinds of data on animals that are marked, released, and recaptured at discrete sampling periods throughout the course of a study. In Chapter 17 the modeling of capturehistory data was discussed in terms of survival and recapture or resighting probabilities. The standardtime (as opposed to reverse-time) models of Chapter 17 were developed by first conditioning on animals that are marked and released, and then writing capture probabilities for each capture history in terms of survival and capture parameters. These conditional probability models are incorporated as components of the more comprehensive models presented in this chapter. In Sections 18.2, 18.3, and 18.4, three classes of models representing different parameterizations for the same data are presented and discussed. Each parameterization permits estimation of abundance N i and quantities related to recruitment. In particular, the models in Section 18.4 exploit the temporal symmetry in capture-recapture data that was noted in Section 17.4 on reverse-time modeling. One reverse-time pa-
495
496
Chapter 18 Estimating Abundance and Recruitment
rameterization permits direct estimation of the finite rate of population increase, ~'i -- Ni+l/Ni, as a model parameter.
18.1. DATA S T R U C T U R E For standard capture-recapture sampling, the data collected are identical to those used for the models of Chapter 17. However, for capture-resighting studies there is an important difference between models of this and the previous chapter. Estimation of survival and recapture probabilities using the models of Chapter 17 was shown to depend only on reobservations of marked individuals. On the other hand, the estimation of abundance and recruitment using models in this chapter requires information on the number of unmarked animals that are caught or sighted in sampling efforts. In standard capture-recapture studies, unmarked animals that are captured are given tags permitting individual identification, and the number of these unmarked captures is important in estimating abundance. In studies in which reobservations are obtained primarily by resighting, an effort must be made to count the number of unmarked animals encountered during the resighting sampling efforts. These counts of unmarked animals, which are not needed for survival rate estimation with the models of Chapter 17, play a key role in estimation of population size and recruitment. Model development in this chapter focuses on data for a single age class of animals (e.g., adults) and thus is closely related to the models of Section 17.1. The reason for this focus is that multiple-age models cannot estimate abundance of young animals based on count statistics, such as the number of young caught. This inability is directly related to the fact that capture probabilities cannot be estimated for an initial age class in age-stratified models (see Section 17.2). Most of the formal development for abundance estimation has focused on "adult" animals, or at least on single-age models (e.g., Jolly, 1965; Seber, 1965, 1982; Brownie et al. 1986; Pollock et al., 1990; Schwarz and Arnason, 1996). We note that the estimation of abundance for age classes other than the first is possible based on the models in Section 17.2, as described in Section 18.2.4. In addition, the estimation of abundance for even the initial age class is possible using the robust design described in Chapter 19. The classical conditional model for single-age class data was labeled in Chapter 17 as the CormackJolly-Seber model. Cormack (1964) focused directly on conditional modeling of survival and capture probabilities, whereas the modeling of Jolly (1965) and Seber (1965) also included additional model components for estimation of population size and recruitment. Thus,
it is appropriate to refer to the classical model that includes abundance and recruitment, as well as survival and capture probability, as the Jolly-Seber model. The data structure for models of this chapter is identical in most cases to that presented for single-age conditional models in Chapter 17. We can again think of two kinds of summary statistics, capture-history data and mq-array data. The capture-history data are the numbers of animals exhibiting each observable capture history (Table 17.1). For example, Xl01 denotes the number of animals in a three-period study that exhibited capture-history 101 (caught in periods 1 and 3, but not in period 2). In studies based on resightings, the numbers of unmarked animals observed during resighting efforts are simply treated as animals seen but not released back into the population. For example, the entry "0010 - 3 5 " in a table such as Table 17.1 would indicate that 35 unmarked animals were seen on sampling occasion 3 of a four-period study. The minus sign simply indicates that these 35 animals were not released into the population with marks (see Section 17.1.1). We introduce x~, to accommodate the removal of individuals from the sampled population at some point in the study time frame. Thus in a threeperiod investigation xi-10 is the number of animals exhibiting capture history 110 and not released following the final capture (sampling period 2 in this case). The mq-array data are summarized as in Table 17.2. In addition to information in the mq-array, abundance estimation requires the number of unmarked animals captured or sighted at each occasion (denote this number as ui). Recall from Chapter 17 that mij is the number of animals released at time i (members of R i) that are next caught or resighted at period j. The number of marked animals caught at period j can thus be computed as j-1 mj = ~,
mq.
i=1
Let n i be the total number of animals (marked and unmarked) caught at time i (n i -- m i 4- ui). If there are no losses on capture or, more generally, if all animals encountered at i are released back into the population with tags, then the number of animals caught equals the number released (n i -- Ri). In this case, the number of unmarked animals does not need to be recorded separately in the mq-array, because it can be simply computed as u i = n i - m i, where n i = R i. Two additional statistics (also defined in Section 17.1.2) are required: r i = ~]=i+1 mij is the number of animals released at i ( R i) that subsequently are recaptured; z i is the number of animals caught before sample period i, not caught in i, and recaptured at some period after i. In addition, deaths on capture can be modeled. Thus,
18.2. Jolly-Seber Approach let d i and d~ be the numbers of m i and ui, respectively, that are not released back into the population at i. The numbers of animals released into the population following each sampling period i thus can be written as R1
U 1
--
d{
R i = m i + ui
--
d i - d~
=
and
for i = 2, ..., K. It is worth emphasizing that in this chapter, u i represents the number of previously unmarked individuals that are captured or observed at time i, including those that are released after capture and those that are not. This differs from the meaning of u i in Chapter 17 (see Section 17.1.2), where ui was restricted to individuals that are both captured and released. By allowing for the possibility of different fates (release or removal) following capture, additional stochastic elements are introduced in the models discussed here that do not appear in the Cormack-Jolly-Seber model of Chapter 17.
18.2. JOLLY-SEBER APPROACH 18.2.1. Model Structure Parameters required for the Jolly-Seber model include the capture probability (Pi) and its complement (qi = 1 - Pi), the survival probability (q~i), and the probability of not seeing an animal again following period i [• see Eq. (17.1)], all of which were defined formally in Section 17.1.2. In addition, the parameters "l]i and ~ represent the probabilities of release for marked ( m i) and unmarked (u i) animals caught at i. Thus, E(di]mi) = mi(1
-
,l-]i )
and E(d~]ui)
=
ui(1 - n l ) .
The following parameters are u n k n o w n random variables, the values of which are to be estimated: Ni
Mi
U i =
Bi
N i - M i
The total number of animals in the population exposed to sampling efforts in sampling period i. The number of marked animals in the population just before sampling period i. The number of unmarked animals in the population just before sampling period i. The number of new animals joining the population between samples i and i + 1 and present at i + 1.
497
As with the conditional modeling of the previous chapter (e.g., Section 17.1.2), it is possible to consider models based on capture-history data or on m q-array summary statistics. Consider the modeling of animals caught in the first sampling period, Ul, and their subsequent capture histories for a three-sample study. We use the notation {x~, xoT} to denote the set of possible capture histories (in this example, observable histories for animals caught in period I of a three-period study), where the " - " superscript again indicates animals not released following final capture: P[{x+, x~}] = ul!(ul - ul)! p~',(1 - pl) U.... ul! 1
{
Ul!
(18.1)
X ['II(x-)!(xo; )! (~1{X1)..... (1 - xl{)...... (.q~1P2.q2x2)Xllo[.q{~plp2(1 _ ~i2)]/110 o. x [~1~q~1(1 - p2)q)2P3-q3] ..... [n'l~P1(1 - p2)qo2P3( 1 - n3)] T M X ('q{q~lP2"rl2qo2P3~3) .... [~plP2.q2qo2P3( 1 _ ~3)jxm},
where the index ~o ranges over the capture histories 100, 110, 101, and 111. Equation (18.1) differs from its counterpart under the conditional Cormack-Jolly-Seber approach [Eq. (17.2)] in two important respects. First, Eq. (18.1) does not condition on the new releases in period 1 (the R1), but instead includes an initial binomial term that involves the capture of u I animals from the available population of U1 animals. Second, even the modeling of the subsequent histories of the u I animals caught at time 1 differs from Eq. (17.2) in including the "11iand TI} and thus in modeling the process by which animals are not released (trap deaths, investigator removals, etc.). The complete model for data from the entire threesample capture-recapture study is written as the product of three expressions such as Eq. (18.1), with an expression for each group of unmarked animals that are caught, u 1, u 2, and u 3. We also present the approach of Seber (1982) and Brownie et al. (1986) for modeling such data using the m q-array summary statistics. This approach decomposes the distribution function for the observed variables, {/,/i}, {di, dl}, {m/j}, into three components as P({Ui}, {di, d~}, {mij })
-- {Pl({Ui} l {Ui}, {pi})] {P2({di, d~} ] {m i, ui}, {'1-]i, '1-]~})]
(18.2)
X ~P3({mij}[{ai}, {q0i, pi})].
The first component deals with the capture of unmarked animals and can be written as Pl({Ui}]{Ui}, {Pi}) =
(18.3) 1-I i=1
i! _ pui(1 _ p i ) U i - u i U (U i ui)!
498
Chapter 18 Estimating Abundance and Recruitment
(Seber, 1982). The second component of Eq. (18.2) concerns marked and unmarked animals that are caught but not released back into the population: Pa({di, dl} l {mi, bli}, {~qi, 11~}) __ i.=~ 1 d~l(ui. bli!- d~)W. tlli! z ,~ui-d~ p (1 -- Tlir)d;" --
X =
(18.4)
mi ! d i ! ( m i _ di)! (Tli)mi-di(1 -- ~qi)di .
The third component of the distribution of Eq. (18.2) is simply the conditional probability distribution written for the mq in Eq. (17.6).
18.2.2. Model Assumptions The assumptions for the Cormack-Jolly-Seber model listed in Section 17.1.2 also are required for the Jolly-Seber model. However, assumption (1), that every marked animal in the population at sampling period i has the same probability of being recaptured or resighted, must be modified for application to Jolly-Seber modeling. Under the models presented in Eqs. (18.1), (18.2), and (18.3), the capture probability parameters Pi also apply to unmarked animals. Thus, for application to the Jolly-Seber model, assumption (1) must be modified to state that every animal (marked and unmarked) in the population at sampling period i has the same probability Pi of being captured or sighted. The discussion of model assumptions presented in Section 17.1.2 is relevant to the Jolly-Seber and related models. Much of this discussion was directed at assumptions (1) and (2) involving homogeneity of the rate parameters ~i and Pi. The revision of assumption (1) to include unmarked animals leads to additional possibilities for assumption violations. For example, permanent trap response in capture probability refers to the situation in which different capture probabilities apply, depending on whether the animal is marked or unmarked (Nichols et al., 1984b) (see Section 17.1.2). Because estimation of survival and capture probabilities in conditional models depends only on recaptures of marked animals, it does not matter for these models (Chapter 17) that animals may exhibit increases or decreases in capture probability following initial capture. However, under the Jolly-Seber model [see Eqs. (18.1 )-(18.3)], the capture probabilities estimated using recaptures are assumed to apply also to unmarked animals, and permanent trap response renders this assumption false. The suggestions presented in Section 17.1.2 for dealing with model assumptions all should be relevant to abundance estimation. For example, when variation in capture probability is associated with state variables that are both static and discrete (e.g., sex), stratification
of capture-history data into groups frequently is useful. For deterministically dynamic state variables such as age, special models can be developed as in Section 17.2 (also see Section 18.2.4). Multistate models can again be used to deal with state variables that are discrete, yet stochastically dynamic. Given special attention to homogeneity of rate parameters for marked and unmarked animals under the models of Chapter 18, the discussion of assumptions presented in Section 17.1.2 should be adequate for the models of Chapter 18 as well. However, there are important differences between the models of Chapters 17 and 18 as to the robustness of estimators for the rate parameters identified in Chapter 17 and the additional parameters and unknown random variables of Chapter 18. These robustness issues are discussed in Section 18.2.5.
18.2.3. Estimation Equation (18.1) represents the probability distribution for capture histories of new (unmarked) animals released in period 1. The probability distribution for all capture histories resulting from an entire study is written as the product of K such expressions, one for the unmarked animals caught at each sampling occasion of the study. Equation (18.2) represents the probability distribution for the rely-array summary statistics over an entire study. Viewing the right-hand sides of Eqs. (18.1) and (18.2) as likelihood functions, it is possible to obtain the maximum likelihood estimates of model parameters q~i, Pi, ~qi, and ~ , using the methods of Chapter 4. Specifically, maximum likelihood estimation of these parameters is based on the portions of Eqs. (18.1) and (18.2) that do not include the unknown random variables Ui. For example, the P2 component of Eq. (18.2) specified in Eq. (18.4) can be used to estimate ~i and ~q~ based on the numbers of marked and unmarked animals that are caught and the numbers in these groups that are released. Similarly, the P3 component of Eq. (18.2) is specified in Eq. (17.6) and is used to estimate q~iand Pi based on the capture histories of animals that are caught and released, as described in Section 17.1.2. Closed-form maximum likelihood estimators based on the Cormack-Jolly-Seber model for parameters q~/and Pi were presented in Section 17.1.2 [Eqs. (17.7), (17.8), and (17.10)]. The focus in this chapter is on the unknown random variables N i and B i, estimators for which can be obtained in multiple ways. For example, Seber (1982) notes that the conditional expectation E ( n i ] N i) = N i p i
(18.5)
can be used to obtain a moment estimator for abundance by 1~ i -- Yli/Pi.
(18.6)
18.2. Jolly-Seber Approach Estimator (18.6) corresponds to the canonical estimation approach of Section 12.2, in that it is simply a count statistic divided by the corresponding estimate of detection probability. Using t h e Cormack-JollySeber model estimator Pi = m i / M i for capture probability [Eq. (17.7)] in conjunction with Eq. (18.6) yields (18.7)
1Qi -- l ~ i n i / m i .
This estimator of population size in Eq. (18.7) still requires an estimate M i of an u n k n o w n random variable Mi, and the estimator of Eq. (17.10) is typically used for this purpose. A reduced-bias version of the estimator in Eq. (18.7) is frequently used under the Jolly-Seber model with time-specific survival and capture probabilities (Seber, 1982; Pollock et al., 1990). Closed-form estimators for var (/~i) and cov(/~i,/~j) are presented by Jolly (1965), Seber (1965, 1982), and Pollock et al. (1990). The abundance estimators in the preceding development are presented for the general Jolly-Seber model (e.g., see Seber, 1982; Pollock et al., 1990). However, slightly different approaches sometimes can be used for reduced-parameter models. For example, Jolly (1982) and Brownie et al. (1986) estimated abundance as the sum of the estimated numbers of marked and unmarked animals alive in a given period:
/Qi = Mi q- ~/i.
(18.8)
Jolly (1982) and Brownie et al. (1986) used the relationships E ( m i + zi l M i ) = M i ( 1 - qixi)
and (18.9)
E ( u i l U i) -- Uip i
to develop the estimators 1~ i __
(18.10)
mi + Z i (1 - dtiy(i)
and
499
This expression is similar to Eq. (17.34) in that population size at time i + 1 is viewed as the sum of two components: (1) new recruits not present in the population at i (B i) and (2) survivors from the previous period [ q ~ i ( N i - yl i q- Ri) , where the t e r m - - t l i q- R i simply subtracts the number of animals removed from the population during sampling efforts]. The conditional expectation of Eq. (18.12) leads to the estimator
/~i-- ]Qi+I --
~i(1Qi-
Hi q- Ri)
(18.13)
for the number B i of new recruits. On reflection this estimator is intuitively reasonable. The number of recruits between i and i + 1 is expressed as the difference between estimated abundance at i + 1 and the estimated number of survivors from the previous period. This recruitment estimator is available for sampling periods i = 2, ..., K - 2. Estimators for variances and covariances associated with /3i are presented by Jolly (1965), Seber (1965, 1982), and Pollock et al. (1990). An estimator with reduced bias also is available (Seber, 1982; Pollock et al., 1990).
18.2.4. Alternative Modeling The material presented in Sections 18.2.1-18.2.3 concerns single-age models (e.g., for adult animals) that are parameterized with time-specific capture and survival probabilities. Indeed, most of the work on abundance estimation has involved this specific model (also see Section 18.3). Historically, two classes of alternative models for abundance estimation have received attention: (1) partially open models in which only gains or only losses to the population can occur and (2) timeconstant models in which capture a n d / o r survival parameters are assumed constant over time. 18.2.4.1. P a r t i a l l y O p e n M o d e l s
(18.11)
CIi -- u i / P i ,
where qi -- 1 - Pi. This approach to abundance estimation is used in programs JOLLY and JOLLYAGE (Brownie et al., 1986; Pollock et al., 1990). Note that all of the above abundance estimators require estimates of capture probability ]9i and are thus available for the sampling periods i - 2, ..., K - 1 for which capture probability can be estimated (Section 17.1.2). Under reduced-parameter models (e.g., capture probabilities assumed to be constant over time; Pi -- P), capture probability sometimes can be estimated for additional periods (e.g., periods 1 and K) as well (Section 18.2.4). Estimation of recruitment under the Jolly-Seber model is based on the relationship E ( N i + I [ N i , Bi) = B i + ~ P i ( N i - d i - d l )
= Bi+ ~i(Ni-
n i + Ri).
(18.12)
These models were considered by Darroch (1959), who provided estimators for the case of no losses on capture, and later by Jolly (1965), who viewed these models as special cases of his general open model (also see Seber, 1982). The death-only model can apply to isolated populations not subject to immigration, if sampling is restricted to a sufficiently short time period that new recruits resulting from reproduction are not added to the population. Other sampling situations include those in which timing of the study rules out the possibility of new recruits (see Haramis and Thompson, 1984) or those in which recruits can be identified and excluded from the analysis. The death-only model assumes no recruitment into the population over the course of the K sampling periods (B i -- 0 for i = 1, ..., K - 1). Thus, an animal first captured at any sampling period during the study (a
500
Chapter 18 Estimating Abundance and Recruitment
member of u i) is known (by assumption) to have been alive at all previous periods (- k) animals caught at the breeding colony are assumed to be breeders, regardless of mark status. Thus, if capture probabilities are the same for marked and unmarked breeders [MIk+) and U}/k+), respectively], then abundance of breeders can be estimated as /~ik+) = n !k + ) / ~Ik+ ) ' where nl k+) = m! k+) + ul k+) (i.e., the number of breeding age animals caught at i is the sum of the marked and unmarked breeders caught, respectively). Abundance for animals of age 0 cannot be estimated using the models of Section 17.2.4, although estimation is again possible using the robust design (Chapter 19). We know of no way to estimate abundance of nonbreeding birds of age v > 0 using this class of models, even with the robust design.
18.2.4.8. Multistate Models Use of Eq. (18.6) in the context of multistate modeling (Section 17.3) is straightforward and uses estimators of the same form as used for grouped data [Eqs. (18.19) and (18.20)]. In both the Markovian and memory models considered in Section 17.3, capture probability for sampling period i was assumed to be specific to the state of the animal at that period. Thus, estimation of state-specific abundance is accomplished by _
i/Pi,
where ~ is the estimated number of animals in state s at sampling period i, n s is the number of these animals that are caught at i, and/3 s is the estimated capture probability for this group of animals. Total abundance for animals in all states (denote as N i) is simply estimated as the sum of these state-specific estimates:
s Once again, the critical assumption underlying this approach to estimation is that the marked and unmarked animals present in state s at sampling period i (M~ and ~/, respectively) exhibit the same capture probability.
503
18.2.4.9. Models Utilizing Auxiliary Data The estimation of abundance with Eq. (18.6) also applies to the models described in Section 17.5 that utilize auxiliary data. The key to successful application of this type of estimation is to be sure of a proper "match" between the estimated capture probability and the number of animals captured in the category of interest. Returning to the canonical estimator of Section 12.2, the point here is that the estimated detection probability must correspond to the count statistic (i.e., must estimate the probability that a member of the group of interest appears in the count statistic).
18.2.4.10. Variances and Confidence Intervals Darroch (1959), Jolly (1965), and Seber (1982) derived variance estimators under the partially open models considered above. Similarly, Jolly (1982) and Brownie et al. (1986) provided expressions for computing the information matrices, and thus the variance and covariance estimates, for the reduced-parameter models they considered. For the other models discussed above, there has been little previous work on abundance estimators and their associated variances (but see Arnason and Schwarz, 1999). In cases where Eq. (18.6) is used to estimate abundance, a bootstrap approach to variance and confidence interval estimation is recommended. If a closed-form estimator is needed, then we suggest the estimator va"'r(/Qi)
=
n2~r(pi) /~4
+
ni(1
-- Pi)
132
(18.23)
based on the delta method (Appendix F). In Eq. (18.23) the numbers of animals caught, ni, are statistics that come directly from the sampling, and the estimates and v'a'r(/3i) are computed by the software used to fit the particular model (e.g., MARK) (White and Burnham, 1999). Confidence intervals /Qi can be approximated using the approach of Chao (1989) [see also Rexstad and Burnham (1991) and Section 14.2.4]. The approach is based on the estimated number of animals not captured at sampling period i, foi = 1Qi -- rli" Here ln(f0 i) is treated as an approximately normal random variable, yielding the 95% confidence interval (n i - foil C, ti i q- foiC), where
]9i
for
C=exp
{1.96 [In ( 1 + var(/Qi)~] 1/2}. ~2i ]
(18.24)
The statistical properties of estimators such as those of Eqs. (18.6), (18.23), and (18.24) are not well understood, and additional work on abundance estimation for these models may well provide better estimators. Until then,
504
Chapter 18 Estimating Abundance and Recruitment
we recommend the above as reasonable approaches to inference about abundance under alternative models that cannot be fit using POPAN-5 (Arnason and Schwarz, 1999). 18.2.4.11. Individual Covariates
The one class of alternative models that does not fit nicely into the framework of canonical estimation is that in which capture probability is modeled using individual covariates. If survival is modeled using individual covariates, yet capture probability is modeled as a group or populationqevel parameter, then the above estimation approach [e.g., Eq. (18.6)] can be used to estimate abundance. However, if capture probability is estimated at the individual level based on covariates, then the animals captured on occasion i represent a heterogeneous mixture of capture probabilities. In theory one could use an average of these individual capture probabilities for all animals caught at i, but it is not clear how such an average would be computed in order to yield an unbiased estimate of abundance. A reasonable approach to estimation in the situation in which capture probability is modeled as a function of individual covariates involves an estimator of the type proposed by Horvitz and Thompson (1952). This approach was used by Huggins (1989) and Alho (1990) for estimation of abundance with closed-population capture-recapture models, with capture probability a function of individual covariates (see Section 14.2.2). The approach has been proposed by McDonald and Amstrup (2001) for use with open models, and we follow their recommendations here. Retaining the general notation of Section 17.1.7, let Jjm be an indicator variable that assumes a value of 1 if animalj is captured in sampling period m, and 0 if the animal is not caught during m. Let ]~jm be the estimated capture probability for animal j in period m, based on covariates associated with animal j and on an assumed relationship between capture probability and the relevant covariates. Abundance at period m then can be estimated as
n~ ~m
l~rn ~- ~ Pjm"
(18.25)
j=l
where n m is the number of animals caught at period m and N m is abundance at period m. The estimator in Eq. (18.25) is similar in appearance to the estimator used in the absence of heterogeneity [Eq. (18.6)]. However, because of the heterogeneity in capture probability and the ability to estimate an individual's capture probability as a function of measured covariates, abundance at period m can be estimated by summing the reciprocals of the estimated capture probabilities for animals that are caught at m. Note that if all animals
have the same value of the relevant covariate (i.e., if there is no heterogeneity), then Eq. (18.25) reduces to Eq. (18.6). McDonald and Amstrup (2001) investigated the properties of estimator (18.25) using simulation and concluded that it exhibited little bias. They also proposed an approximate variance estimator for N m, which performed well in simulations for small to moderate levels of heterogeneity, but not for large levels. McDonald and Amstrup (2001) suggested that bootstrap variance estimates might be useful. Although more work on this estimator should prove useful, the important point is that the Horvitz-Thompson estimator provides a reasonable approach to abundance estimation when capture probability is modeled as a function of individual covariates.
18.2.5. Model Selection, Estimator Robustness, and Model Assumptions In practice, model selection for the Jolly-Seber and related models is virtually identical to the process described in Chapter 17. As noted in Section 18.2.1, the conditional models of Chapter 17 (in particular, Section 17.1.2) can be viewed as the third component (P3) of the Jolly-Seber model [see Eq. (18.2)]. This component frequently is written as conditional on either the number of unmarked animals (ui) caught in each period [Eq. (17.5)] or the number of releases (R i) in each time period [Eq. (17.6)]. The subsequent capture-history data on marked animals provide the information needed for testing between competing models and for assessing model appropriateness and fit. Thus, between-model tests and goodness-of-fit tests for the Jolly-Seber and related models are usually based on the P3 component of the likelihood (e.g., Pollock et al., 1985; Brownie et al., 1986). As to the components P1 and P2 of the Jolly-Seber likelihood [Eq. (18.2)], P2 is essentially a binomial model of the number of removals of captured animals (e.g., trap deaths). Historically, Jolly-Seber modeling has not focused on the removal parameters (~1i, ~1~), although it sometimes is assumed that removal probabilities are the same for marked and unmarked animals ('qi = ~i). Models for these parameters could include tests for equality for marked and unmarked animals, tests for absence of temporal variation, and several other possibilities. However, under most capturerecapture sampling designs, removals are not viewed as a part of the natural population dynamics of interest to biologists. Removals are thus modeled in a general way with separate parameters ~i and ~q~ in Eq. (18.4) for marked and unmarked individuals that should not influence inferences about the more interesting processes. By assuming different removal parameters for
18.2. Jolly-Seber Approach each time, P2 effectively removes this model component and associated information from the assessment of Jolly-Seber model fit. The component P1 of the decomposed Jolly-Seber likelihood [Eq. (18.3)] models the number of unmarked animals ui caught at each sampling period conditional on the number Ui of unmarked animals in the population, with the latter treated as unknown random variables. Under this model, likelihood component P1 is used in the estimation of population size [Eqs. (18.8) and (18.11)] but is not useful in assessing model fit. In other parameterizations (Section 18.3) (see Crosbie and Manly, 1985; Cormack, 1989; Schwarz and Arnason, 1996) entry probabilities are used to model the entry of new, unmarked animals into the population. Under these alternative parameterizations, the ui can be useful in selecting appropriate models and assessing model fit. Model selection and goodness-of-fit testing under the Jolly-Seber modeling approach described in this section thus follows the procedures discussed in Section 17.1.8 for the Cormack-Jolly-Seber model, which is component P3 of the Jolly-Seber model. Therefore the goodness-of-fit tests of Pollock et al. (1985), Brownie et al. (1986), and Burnham et al. (1987) are appropriate. Model selection can proceed via likelihood ratio testing with nested models, although we recommend the alternative information-theoretic approach using AIC and its small-sample and quasilikelihood derivatives (Burnham and Anderson, 1998). The variance inflation factors ~ for lack of model fit should be appropriate for computing variance estimates of abundance [in this case the "corrected" variance is computed as ~ v~r (/~/i)]. The discussion of estimator robustness and model assumptions presented in Section 17.1.9 is applicable to the estimators for survival and capture probability under the Jolly-Seber model. As noted in Section 18.2.2 and repeatedly emphasized above, the critical additional assumption underlying abundance estimation with the Jolly-Seber and related models is that marked and unmarked animals exhibit the same capture probabilities. Here, we discuss robustness of open-model abundance estimators to deviations from underlying model assumptions. Equality of capture probabilities for all animals present in the population at any sampling period i is an assumption unlikely to be met exactly in any sampling situation. Heterogeneity of capture probability, in which different animals present at i exhibit different probabilities of being caught at i, can produce substantial bias in abundance estimates (Gilbert, 1973; Carothers, 1973; Nichols and Pollock, 1983b), depending on the form of heterogeneity. In the unlikely situation in which there is heterogeneity in capture probability, yet no covariation between capture probabilities of an indi-
505
vidual at different sampling occasions (i.e., an individual that has a relatively low capture probability at period i could just as easily have a relatively high capture probability at some other period i + j), there should be little bias in the abundance estimator. However, the more likely scenario is that individuals will tend to exhibit relatively low or high capture probabilities throughout the study (e.g., see Gilbert, 1973; Carothers, 1973; Nichols and Pollock, 1983b). In this case, animals in the population with high capture probabilities tend to be caught and become members of the marked component of the population, Mi, whereas animals with low capture probabilities tend to remain in the unmarked component of the population, Ui. As noted throughout this chapter, the estimation of capture probability is based on recaptures of marked animals. In the presence of heterogeneity of capture probabilities, the estimates Pi apply to the marked component (Mi, the animals with higher capture probabilities, on average), but are too high for the unmarked component of the population, Ui. Thus, the ]9i will be positively biased with respect to the average capture probability of the entire population, N i = M i + U i. Because ]~i appears in the denominator of the abundance estimator [Eq. (18.6)], the abundance estimator therefore is negatively biased. The magnitude of the bias in the population size estimator is dependent on characteristics of the heterogeneity (i.e., of the distribution of capture probabilities over individuals). Moderate to large degrees of heterogeneity [often expressed as the coefficient of variation of the distribution of capture probabilities over individuals, cv(p) (Carothers (1973)] tend to produce substantial negative bias in the Jolly-Seber estimator /~i. The exception to this tendency occurs when average capture probability is relatively high (e.g., >0.5), in which case heterogeneity in capture probabilities is relatively unimportant (Gilbert, 1973). The possibility of heterogeneous capture probabilities causing severe negative bias in Jolly-Seber estimates of population size has led to development of methods for bias reduction. Hwang and Chao (1995) used a sample coverage approach to address this problem. They derived estimators for the sample coverage (also see Section 14.2.3) and cv(p) (a statistic reflecting the degree of heterogeneity) and used these estimators to approximate bias of the Jolly-Seber/~i (as well as of the abundance estimators for the partially open models) and to provide new abundance estimators with reduced bias. These estimators have seen little use, though they have performed well in simulation studies. Pledger and Efford (1998) used simulation and inverse prediction (Carothers, 1979) to deal with heterogeneous capture probabilities for survival rate
506
Chapter 18 Estimating Abundance and Recruitment
estimation. Their approach used simulation to establish the functional relationship between the degree of heterogeneity [e.g., the coefficient of variation cv(p), or a metric linearly related to cv(p)] and the bias of the estimator, N i. Then, the cv(p) is estimated from the actual data, as is abundance. The abundance estimate is known to be biased, as it is based on the assumption of homogeneity in the capture probabilities. However, the magnitude of the bias can be estimated using the estimated cv(p) and the simulation-based relationship between cv(p) and bias of/~/i. Inverse prediction then can be used to obtain a new, bias-corrected estimator for abundance. Pledger and Efford (1998) used different estimators [including that of Hwang and Chao (1995)] for cv(p) and metrics related to cv(p) and concluded that the test statistic for heterogeneous capture probabilities developed by Carothers (1971, 1979; also see Leslie, 1958) led to the best estimates of abundance. The approach appeared to perform well in simulation studies that also included the Hwang-Chao estimator. This approach has seen little use because of its recent development. Trap response (a form of capture-history dependence) in capture probabilities also can influence abundance estimation. Temporary trap response can be dealt with via modeling (Section 17.1.6), and abundance can be estimated based on assumptions about the capture probabilities of unmarked animals (Section 18.2.4). Permanent trap response refers to the situation in which unmarked animals exhibit one capture probability and marked animals exhibit another. Such a response cannot be dealt with via modeling, because the information about capture probability comes from marked animals only. A trap-happy response, in which marked animals show higher capture probabilities than do unmarked animals, yields a positive bias in capture probability and produces negative bias in the abundance estimator (Nichols et al., 1984b). A trap-shy response occurs when marked animals exhibit lower capture probabilities than do unmarked animals. This response produces estimates of capture probability that are negatively biased, yielding abundance estimates that are positively biased (Nichols et al., 1984b). The biases in abundance estimates produced by permanent trap response can be substantial and are most severe when the population exhibits substantial turnover and the proportion of marked animals in the population is small (Nichols et al., 1984b). We know of no way to deal adequately with permanent trap response in Jolly-Seber type models other than to use the robust design (Chapter 19) to estimate size of the unmarked component of the population via closed capturerecapture models. Although homogeneity of capture probabilities is of obvious importance to abundance estimation, homoge-
neity of survival probabilities is also relevant. Although some research has been conducted on the robustness of Jolly-Seber survival estimates to heterogeneous survival probabilities (Section 17.1.9), there has been little to no work on effects of such heterogeneity on abundance estimators. Pollock et al. (1990) reported results for the situation in which survival probabilities are positively related within individuals (animals having high survival probability during one interval likely to show high survival probability for other intervals) but are independent of capture probability. They showed that this situation generally produces positively biased abundance estimates (Pollock et al., 1990). Robustness of abundance estimators to violations of the assumption of homogeneous survival probabilities merits further investigation. Tag loss is not as large a problem with abundance estimators as with survival estimators. When probabilities of tag loss do not vary as a function of tag age, Jolly-Seber abundance estimates remain unbiased (Arnason and Mills, 1981). Precision of abundance estimates will be reduced by tag loss, however, because there are fewer marked animals on which to base estimation of capture probability (Arnason and Mills, 1981). Because survival estimates are negatively biased in the presence of tag loss (Section 17.1.9), Jolly-Seber estimates of number of recruits, Bi [Eq. (18.13)], are positively biased by tag loss (Arnason and Mills, 1981). We are aware of no investigations of the effects of agedependent tag loss on abundance estimates, but this problem should cause heterogeneous "survival rates" (in the event of tag loss, open-model survival parameters correspond to tag survival) and thus has the potential to produce bias in abundance estimates. As discussed in Section 17.1.9, violation of the assumption of instantaneous sampling can result in heterogeneous survival probabilities. The problem arises because long sampling periods can result in populations that are open to losses (and gains) during the sampling period. Indeed, if the population size is changing during the sampling period, then it is not even clear what the true quantity of interest is (population size at what point during the sampling interval?). However, we are unaware of any work on the consequences of long sampling intervals for abundance estimates. If the population is thought to be open during the sampling period, then we recommend the robust design approach of Schwarz and Stobo (1997) and Kendall and Bjorkland (2001) in which mortality is modeled during the sampling period as well as between periods (also see Chapter 19). The assumption of no temporary emigration is relevant to abundance estimation and the interpretation of Jolly-Seber abundance estimates. In the presence of random temporary emigration (Section 17.1.9), the
18.2. Jolly-Seber Approach Jolly-Seber abundance estimate is positively biased for N i, the number of animals in the area exposed to sampling efforts during sampling period i (e.g., Kendall et al., 1997). However, Kendall et al. (1997) also considered a "superpopulation" of N~/animals that are associated with the area exposed to sampling efforts during sampling period i, in the sense that they have some nonnegligible a priori probability of being located in the sampled area at period i (see Chapter 19). In the case of random temporary emigration, the Jolly-Seber estimator for capture probability estimates the product of the probabilities of (1) being in the sampled area at i and (2) being caught, given presence in the sampled area. Thus, the Jolly-Seber abundance estimator provides an unbiased estimate of ~ (Kendall et al., 1997). Similarly, the Jolly-Seber estimator for number of recruits is unbiased for recruitment to the superpopulation. Markovian temporary emigration refers to the situation in which an animal's probability of being in the area exposed to sampling at time i depends on whether it was in the sampling area at time i - 1. Under Markovian temporary emigration, the JollySeber abundance estimator is biased with respect to both N i and N ~ and the magnitude of the bias is dependent on the form and nature of Markovian temporary emigration (Kendall et al., 1997). Finally, violation of the assumption of independent fates leads to unbiased abundance estimates, but biased estimates of variance (Section 17.1.9). In the case of dependent fates, with animals in pairs or family groups behaving similarly, variance estimates are negatively biased, but quasilikelihood procedures can be used for variance inflation and for adjusting likelihood ratio tests and AIC model selection statistics (Section 17.1.9). 18.2.6. E x a m p l e
We illustrate the Jolly-Seber model and estimators for abundance and recruitment with the meadow vole data from Section 17.1.10. We focus on abundance and recruitment of adult males over the 6 months of the study. As noted in Section 17.1.10, the CJS model, and hence the Jolly-Seber model, fit the data reasonably well. We thus based estimates on this model, using the bias-adjusted versions (Seber, 1982; Pollock et al., 1990) of the estimators in Eqs. (18.7) and (18.13), as implemented in program JOLLY (Pollock et al., 1990). Abundance estimates are available under the full Jolly-Seber model for all sampling periods except the first and last. For these data the estimates ranged from about 55 to 75 (Table 18.1). Jolly (1965) presented two variance estimators for abundance, and the associated standard errors for both are recorded in Table 18.1. The conditional standard error SE(I~ilN i) reflects variation associated with the
507
TABLE 18.1 Estimates of Meadow Vole Population Size (N i) and Number of Recruits (B i) under the General Jolly-Seber Model a Abundance Sample 9 period
1 2 3 4 5 6
Recruitment
/Vi [S-E(~,IN,),
Sampling dates
~E(/Qi)] b
6 / 27-7 / 1 8/1-8/5 8/29-9/2 10/3-10/7 10/31-11/4 12/4--12/8
74 59 62 55
mc (2.14, 3.89) (3.53, 5.84) (3.01, 5.66) (2.96, 5.61) c
I~i[~'E(Bi)] c 17 (2.56) 21 (2.71) 19 (2.82) me c
a For adult male meadow voles studied at Patuxent Wildlife Research Center, Laurel, Maryland, 1981 (data in Tables 17.5 and 17.6). b ~'E(l~ilNi ) is the conditional standard error including only error of estimation, whereas g'E(/Qi) is the unconditional standard error that also incorporates demographic stochasticity in the death process. c Quantity not estimable under Jolly-Seber model.
fact that all animals are not detected at each sampling period (i.e., Pi < 1). This component of variation is sometimes referred to as "error of estimation" (Jolly, 1965) or "sampling variation." The "unconditional" standard error SE(Ni) reflects both sampling variation and demographic stochasticity associated with the death process. The latter standard error actually is conditional on the number B i of new recruits at each sample period but incorporates stochasticity in the survival of these animals during subsequent time periods. The abundance estimates in Table 18.1 are relatively precise because of the high capture probabilities (see the ]9i in Table 17.7). Numbers of new recruits could be estimated for only three periods, but the estimates were similar, ranging from 17 to 21. Numbers of recruits were estimated with less precision than abundance (e.g., Pollock et al., 1990). In addition to the traditional Jolly-Seber modeling, abundance can be estimated using the group-specific canonical estimator in Eq. (18.19). In the example analyses of Section 17.1.10, the most appropriate model for the meadow vole data of both sexes was model (q~s+t, p). Using this model we estimated sex-specific abundance N~ for each period by =
where i = 1,..., K and s is an indicator of sex. Model comparisons suggested that a single capture parameter was appropriate for both sexes and all sample periods (Table 17.8), so that the single capture probability could be used for all n s. The AICc values of Table 17.8 also suggested that model (%+t, Ps,t) was a reasonable model for the data of Section 17.1.10. Under this model, abundance was estimated as 1Q~ = ?l si / p s
Chapter 18 Estimating Abundance and Recruitment
508
for i = 2, ..., K. Sex-specific recruitment in turn could be estimated based on a sex-specific generalization of Eq. (18.13), with the abundance estimates given by the above canonical estimators and survival estimates (see Fig. 17.2) coming directly from the respective models (q)s+t, P) and (q~s+t, Ps,t). Abundance estimates based on the above canonical estimators with models (q~s+t, P) and (q~s+t, Ps,t) are presented in Table 18.2. The associated standard errors were computed using a bootstrap approach (see Appendix F). The approach involves conditioning on estimated population size at the first sample period for which it can be estimated, and on the estimated numbers of new recruits for subsequent sampling periods. Capture histories for animals in these groups were simulated, with both capture and survival at each sample period treated as Bernoulli trials. The resulting capture histories were used to estimate capture probabilities under models (~s+t, P) and (q)s+t, Ps,t)" The capture probability estimate then was used with the n s for that bootstrap iteration to estimate abundance as above. The standard error of ~ was computed based on 1000 iterations as 1000
SE(/~/)
IEm=
' 2) modeling is similar to that presented in Chapter 17, except that survival probabilities for the sampling process now are incorporated into the model. Thus, every capture event requires both a
\j=l
;)
(18.37) 9
Finally, the conditional probability (conditioned on the total M of animals caught) associated with a particular capture history [denoted as P(h)] can be obtained by dividing the expected number of animals with that history [e.g., as in Eq. (18.36)] by the expected number of total individual animals caught during the study [as in Eq. (18.37)1:
(18.35)
q~i[1 - pi(1 - T]i)]
E(XOllOlO]N1) = N1k~2P2~12q~2pgngq~ 3
capture probability Pi and a probability "l]iof surviving the sampling process. Equation (18.36) does not lead directly to a probability distribution, because the expectation contains the initial population size, N 1, an unknown random variable. Let x h be the number of animals exhibiting capture history h, and M denote the total number of animals caught in the entire study:
(18.34)
q)i/'Yi+l "
= hi(N +/N[-)
513
E(Xh) E(M) "
(18.38)
From Eqs. (18.36) and (18.37), the initial population sizes in the numerator and denominator of Eq. (18.38) cancel, leaving the conditional probabilities of interest expressed in terms of estimable model parameters. Then the likelihood L for the set of animals observed in a study can be written generally as the product of the conditional probabilities associated with all the individual capture histories: L = l-I P(h)Xh.
(18.39)
h
Pradel (1996) described likelihood (18.39) in more detail in terms of the model parameters and sufficient statistics. He suggested three different parameterizations for the above likelihood, each of which might be useful in addressing specific questions, all of which retain capture (Pi) and survival (q)i) probabilities. Of these, perhaps the most natural parameterization incorporates the reverse-time parameters %. Thus, Eq. (18.35) is substituted into the capture history expectations [Eqs. (18.36) and (18.37)], so that all probabilities (Ph) are written in terms of Pi, (Pi, and %.
514
Chapter 18 Estimating Abundance and Recruitment
A second parameterization uses population growth rate h i as a model parameter. Based on the definition in Eq. (18.34), the expression "~i = q~i-1/hi-1
is substituted for the "Yi of the original parameterization. A third parameterization is based on a measure fi of recruitment rate, which denotes the number of recruits to the population at time i + I per animal present in the population at i. This measure of recruitment is used in discrete-time matrix population m o d e l s m f o r example, in the single-age model, Ni+ 1 = Niq~ i + N i f i.
(18.40)
Equation (18.40) can be rearranged to yield Ni+I/Ni
= ~i if- fi
-- ~ i / ~ i + l .
Thus, the parameterization of a model with fi can be obtained by substituting r ~i-1 nt- fi-1
(18.42)
Ki "-- ~Pi/~i+l
= hi
'Yi --
parameters to the interval [0, 1]. For the h i parameterization, Pradel (1996) used a log transform for population growth rate (hi), in order to constrain it to be positive. All three parameterizations ([q~t, Pt, '~t], [q~t, Pt, ht], and [OPt, Pt, ft]) described in Section 18.4.1 have been implemented in program MARK (White and Burnham, 1999). Pradel's temporal symmetry models are relatively new and have seen only limited use. It appears that the numerical optimization algorithms may sometimes perform better (e.g., fewer convergence problems) with the y-parameterization than with the other two parameterizations. If primary interest is in population growth rate, it may be better to fit model (~t, Pt, "Yt) to data and then estimate population growth rate using Eq. (18.34) by
for i = 2, ..., K - 2. This estimator also is computed in program MARK (White and Burnham, 1999). The parameter fi can be estimated in a similar manner, based on estimates from model (q~t, Pt, ~t) and a rearrangement of Eq. (18.41):
(18.41)
for "Yiof the original parameterization. Equation (18.41) is an intuitive expression for the seniority parameter % Recall that this parameter is defined as the probability that an animal alive at period i is a survivor from the previous period, i - 1. All animals alive at i are either survivors from period i - 1 (expectation Xi_lq~i_ 1) or new recruits (expectation X i _ l f i _ l ) , so Eq. (18.41) is natural expression for the proportion of survivors.
fi = ~i(1 -- "Yi+I) "Yi+I
for i = 2, ..., K - 2. Future work on the models of Pradel (1996) should include detailed investigations of the identifiability of parameters under the different model parameterizations. Under the time-specific model with y-parameterization (oPt, Pt, ~/t), the parameters q~l, r
"", q~K-2;
"~3, ~4, "", "~K;
18.4.2. Model Assumptions Because Pradel's (1996) temporal symmetry models simply represent different ways to parameterize the original Jolly-Seber model, the basic assumptions are the same as for the Jolly-Seber and superpopulation approaches (see Sections 18.2.2 and 18.3.2). The general assumption of homogeneity of rate parameters now applies to Pradel's "~i as well as to the usual Pi and q~;.
Maximum likelihood estimates can be obtained for the likelihood of Eq. (18.39) or its analog based on sufficient statistics (Pradel, 1996). In Pradel's (1996) implementation of this model, he used a logit transform for q~i and '~i as a means of constraining these
P2, P3, "", PK-1; "Y2Pl;
q~K-lPK
can be estimated. Note that the list includes K - 2 survival parameters, K - 2 capture probabilities, K - 2 seniority parameters, and two product parameters with components not separately identifiable, yielding a total of 3(K - 2) + 2 = 3K - 4 parameters. Under the time-specific model with h-parameterization (OPt, Pt, ht), the parameters q~l, r
18.4.3. Estimation
(18.43)
"",
h2, h3, "", hK-2;
q~K-2; hl/Pl;
P2, P3, "", PK-1; ~K-lPK;
hK-lPK
can be estimated. This parameter list includes K - 2 survival parameters, K - 2 capture probabilities, K - 3 population growth rates, and three product parameters, again yielding a total of 2(K - 2) + (K - 3) + 3 = 3K 4 parameters.
18.4. Pradel's Temporal Symmetry Approach
18.4.4. Alternative Modeling Various types of alternative modeling should be possible using the basic models of Pradel (1996). For example, models with parameters constrained to be constant over time can be used to incorporate various hypotheses of potential biological interest. As noted in Section 17.4.1, models incorporating constancy of the ~/i (~/i = ~ for all i) reflect temporal similarity in the relative contributions of new recruits and old survivors to population growth. Models with stationarity of h i also are potentially useful for investigating population regulation and for testing the assumptions that underlie matrix population modeling (Chapter 8) based on stationary growth rates. One topic meriting consideration in reduced-parameter models that utilize these parameterizations involves the manner in which the h i and fi parameters are defined as functions of the parameters ~i that also appear in the model [e.g., see the estimators of Eqs. (18.42) and (18.43)]. Thus, modeling one set of parameters as temporally constant (e.g., q~i -- q~) may impose unintended constraints on the parameters h i and fi. Because of the lack of work on this topic, we simply recommend caution at this time. In cases where interest is focused on a parameter such as h i, a conservative approach might be to allow full time specificity in capture and especially survival probabilities when evaluating alternative models for the hi. However, whether this approach performs better than others is yet to be determined. The potential to describe parameters as functions of time-specific covariates offers interesting possibilities with these models. For example, it may be useful to model recruitment-related parameters (~/i and fi) as functions of environmental variables thought to influence either reproduction or immigration or both. The ability to model population growth rate as a function of environmental covariates also should prove useful. It often is of interest to investigate time trends in the hi, which is accomplished by modeling h i with time as a covariate. Though ecologists have long been interested in time trends, a focus on trends in the trend parameters (h is usually the quantity selected to express "trend" in population size) is relatively new (see Franklin et al., 1999). There may be large potential in using the h-parameterization in conjunction with data from other sorts of surveys (other than capture-recapture) in which count data are collected for the purpose of estimating trends in population size. For example, assume that we conduct capture-recapture studies on the same area where line transect counts also are collected (see Section 13.2) to estimate abundance. It should be possible to develop
515
joint likelihoods for the separate data types that share h i, thus combining information to better estimate population growth rate [e.g., see Alpizar-Jara and Pollock (1996, 1999) for a similar approach to a different problem]. In other situations, it may be possible to use simple count statistics for which no effort is made to estimate detectability. If count statistics have detection probabilities that are constant over time (see Chapter 12), then the ratio of counts Ci and Ci+ 1 in two successive years provides an estimate of population growth rate (i.e., Ci+I//Ci should estimate hi). One way to assess the reasonableness of this index assumption would be to use Pradel's (1996) models to model population growth rate using the counts as covariates (e.g.,)k i = ~Ci+ 1//Ci). If this model performs well (if it describes the variation in the data nearly as well as a model with no covariate model for h i, and if is near 1.0), then this can be taken as some evidence that the counts provide reasonable indices, at least over the period of study. In that case the model with count statistics as covariates should provide more precise estimates of population growth rate. It may be useful to consider using the h-parameterization in conjunction with individual covariate modeling. In studies of closed populations, or at least populations for which emigration and immigration are not important, the individual h values can perhaps be viewed as fitness estimates associated with individuals characterized by those covariates. It is possible to use a variance components approach (see Burnham et al., 1987; Skalski and Robson, 1992; Link and Nichols, 1994; Gould and Nichols, 1998) based on the conceptual framework of random effects modeling, to estimate the true temporal variance of h i . This variance is relevant to extinction probability (e.g., Lewontin and Cohen, 1969; Leigh, 1981; Goodman, 1987a) and emphasizes the potential utility of the direct estimation and modeling of h i for population viability analyses (see White et al., 2001) (also see Section 11.2.1). Finally, we note that most of the alternative modeling described above can be implemented using MARK (White and Burnham, 1999). In particular, MARK includes model parameterizations that incorporate ~/i, h i, and fi.
18.4.5. Model Selection, Estimator Robustness, and Model Assumptions Model selection should follow the same basic approach discussed for the conditional models of Chapter 17 and the other classes of models described above (Sections 18.2.5 and 18.3.5). As with the superpopulation models of Schwarz and Arnason (1996), the models of Pradel (1996) contain an extra set of parameters
516
Chapter 18 Estimating Abundance and Recruitment
(either seniority, population growth rate, or recruitment rate), providing additional flexibility in modeling. The recommendations of Chapter 17 apply here for the use of AIC, likelihood ratio testing, and quasilikelihood procedures. The goodness-of-fit tests recommended for conditional models (e.g., Pollock et al., 1985a; Brownie et al., 1986; Burnham et al., 1987) apply to these models, because the entries of new unmarked animals into the sample provide little additional information for assessing model fit, especially in the case of the models with time-specific parameters [e.g., model (q)t, Pt, 'Yt)]" Because they were only recently developed, these models have seen only limited use, and topics such as estimator robustness are yet to be investigated extensively. Hines and Nichols (2002) focused on the k-parameterization and investigated three possible sources of bias. The investigation was tailored to a particular set of analyses for the spotted owl, Strix occidentalis caurina (Franklin et al., 1999). However, some of the specific findings are likely to be relevant to other studies. The first potential problem investigated by Hines and Nichols (2002) involved expansion of the study area. Because this issue is not associated specifically with a model assumption, it has not been mentioned previously. Basically, capture-recapture estimates apply to a particular area under investigation, and if this area changes in size between sampling periods i and i + 1, the relevant rate parameters (e.g., q~i, ki) can be expected to reflect these changes. For example, we can envision certain study situations where there is a tendency to detect animals just beyond the periphery of the original study area and to target their capture and addition to the study. This tendency would be expected to result in increases in the size of the study area over time, with estimates of )ki that are larger than if there had been no expansion of study area. The estimates would not be biased, in the sense that they would reflect changes in numbers of animals on the expanding study area. However, if the interest is in growth or decline of a biological population, then an effort should be made to restrict attention to areas of similar size, so that inferences apply to biological processes and not changes in sampling area. If N;+ 1 is the number of animals exposed to sampling efforts in period i + 1 that were not exposed to efforts in period i, then the approximate bias (with respect to the original sampling area of period i) i n ~i is given by Bias(Ki) = E(K i) - )k i N;+ 1/Ni. Thus, expansion of the study area in sampling period i + 1 will result in positive bias in Ki with respect to
the population growth rate )ki o n the area sampled in period i. Another potential assumption violation considered by Hines and Nichols (2002) is permanent trap response, a violation known to produce biased estimates of abundance (Section 18.2.5) and seniority parameters (Section 17.4.1), but not survival rate (Section 17.1.9) (see Nichols et al., 1984b). Trap-happy response (higher capture probabilities for marked animals than for unmarked animals) produced a positive bias in K and Ki, whereas a trap-shy response led to a negative bias. The intuition underlying this result is based on the way of expressing [Eq. (18.34)] and estimating [Eq. (18.42)] population growth rate as a function of survival and seniority parameters, i.e., )ki-- q)i/'~i+l"
Survival rate estimates are not biased by permanent trap response because they are based on recaptures of marked animals only (Section 17.1.9). Estimation of seniority parameters is based on captures of marked and unmarked animals in previous periods, whereas estimation of capture probabilities is based on marked animals. In the case of trap-happy response, the estimated capture probability based on marked animals will be too high for unmarked animals, leading to seniority parameter estimates that are negatively biased (the estimated number of animals in i + 1 that were unmarked prior to i will be too small; see Section 17.4.1). If the ~i+1 a r e too small, then the ~i of Eq. (18.42) will be too large. Similarly, trap-shy response produces positive bias in seniority parameter estimates and negative bias in estimates of population growth rate. As expected, the magnitude of the bias in population growth rate was largest for the largest differences between capture probabilities of marked and unmarked animals. The bias also varied as a function of sampling period (Hines and Nichols, 2002). Under a trap-happy response with true )ki constant over time, for example, the )ki exhibited a negative time trend, with the largest positive biases occurring i n ~2 and the smallest biases occurring in )KK_ 2. O n reflection, this trend in estimator bias makes sense, because it involves the larger numbers of unmarked animals in the early sampling periods. The key point here is that time trends in )ki should be considered with caution in sampling situations when there is a possibility for trap response in capture probabilities. Heterogeneity in capture probabilities also was investigated as a potential source of bias in )~i. Although heterogeneous capture probabilities are known to cause serious bias in abundance estimates (Section 18.2.5), they appear not to cause problems for estimates of population growth rate (Hines and Nichols, 2002). If we consider estimating population growth rate as
18.4. Pradel's Temporal Symmetry Approach the ratio of abundance estimates in two successive sampling periods, then both the numerator and denominator will be negatively biased by heterogeneity. If the relative bias is similar for the two estimates, the estimate of their ratio (ki) can be expected not to show substantial bias, an expectation confirmed in the study of Hines and Nichols (2002). In the case of model (q~t, Pt, ~kt) with true population growth constant over time, the time-specific estimates k i showed a slight negative time trend, with positively biased estimates in the early time periods and negatively biased estimates in the later time periods. The combined effects of a trap-happy response and heterogeneity of capture probabilities also were examined by Hines and Nichols (2002). Results were similar to those obtained under trap response, the more important of the two assumption violations with respect to bias. Effects of tag loss and of sampling that is not instantaneous have not been investigated for the temporal symmetry models. Random temporary emigration (Burnham, 1993) should result in parameter estimates that are unbiased with respect to the superpopulation (Kendall et al., 1997), but not with respect to the population available for capture in the specific sampling periods. Markovian temporary emigration is expected to produce biased estimates of most parameters. Clearly, the topic of bias under the various parameterizations of Pradel's (1996) temporal symmetry models merits careful investigation. 18.4.6. E x a m p l e
We again use the m e a d o w vole data of Tables 17.5 and 17.6, in this case to illustrate the temporal symmetry approach of Pradel (1996). We applied program MARK (White and Burnham, 1999) to estimate parameters under all three parameterizations of the temporal symmetry approach. For each parameterization, we fit the full, time-specific model, as well as a reducedparameter model in which the parameters other than survival and capture probabilities (either seniority, population growth rate, or recruitment rate) are assumed constant over time. The natural parameterization with seniority parameters under model (oPt, Pt, "Yt) yielded estimates of seniority "Yi ranging from 0.60 to 0.71, indicating that about 60-70% of the adults in the population over the course of the investigation consisted of survivors from the same population in the previous month (Table 18.4). Monthly population growth rates were estimated as derived parameters from the r and ~/i, and ranged from 0.83 to 1.07 (Table 18.4). AICc favored the reducedparameter model (q~t, Pt, "Y), which produced an estimate for the proportion of survivors of 65% (Table
517
TABLE 18.4 Estimated Seniority Parameters (~i) and Population Growth Rates (h i) for Meadow Voles under Two Temporal Symmetry Models a Model (~t, Pt, ~t )b
Sample period
~/i
S"E(~i) ~i a
1
me
me
2 3 4 5 6
me 0.71 0.67 0.65 0.60
0.83 1.07 0.90 me e
0.068 0.070 0.075 0.063
Model (~ot, Pt, ~l)c
~(~i )
~f
0.105 0.138 0.123
0.65
S"E(~t) ~i a ~ ( ~ i )
me 0.031
1.31
0.094
0.87 1.10 0.92 me me
0.090 0.106 0.094
aBased on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). b Model (q~t, Pt, ~t ) AICc = 989.9. CModel (q~t,Pt, "Y) AICc = 984.8. e Estimated as the derived parameter Ki = ~i/~i+1" e Quantity not estimable under the model. fSingle estimate q corresponds to sample periods 2-6.
18.4). The derived estimates of population growth rate based on the estimates from the reduced-parameter model were similar to those under the time-specific model for the periods for which they could be estimated. The constant-parameter model (q~t,Pt, "Y) permitted estimation of an additional population growth parameter, ~1, and also yielded estimates with smaller standard errors than the time-specific model (q~t,Pt, "Yt). Estimates under the two )~-parameterization models are presented in Table 18.5. Estimates under the timespecific model (oPt, Pt, kt) are identical to those derived from model (q~t, Pt, "Yt) and displayed in Table 18.4. The estimates also can be compared with the derived estimates
Ki -- 1Qi+l / (1Qi -- di)
TABLE 18.5 Estimated Population Growth Rates Meadow Voles under Two Models a Sample
period
(~ki) for
Model (~t, Pt, ]kt)b
Model (%, Pt, k)c
~i
~e
~(~k)
1.04
0.042 e
1
d
2 3 4 5
0.83 1.07 0.90 d
S"E(~ki) d
0.105 0.138 0.123 d
a Based on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). bModel (q~t,Pt, ~'t) AICc = 989.9. CModel (oPt,Pt, k) AICc = 993.4. dparameters not estimable under the model. eSingle estimate )~applies to periods 1-5.
Chapter 18 Estimating Abundance and Recruitment
518
from the Jolly-Seber model (see Section 18.5), where the abundance e s t i m a t e s / ~ i are obtained using the biascorrected Jolly-Seber estimator as in Table 18.1 and the d i refer to trap deaths or losses on capture. The subtraction of trap deaths is intended to restrict inference about population growth to ecological (as opposed to investigator-related) processes. Estimates of population growth computed in this way are effectively identical to the estimates produced directly by the Pradel (1996) model (~t, Pt, )kt)" The reduced-parameter model (q~t,,,Pt, X) yields an estimated population growth rate of ~ = 1.04. However, a comparison of the AICc values for the two models suggests that time specificity in ]ki is needed to model the data adequately (Table 18.5). This result is logically consistent with the result that model (~t, Pt, ~/) was appropriate for modeling the same data. If survival is time specific and the proportion of new animals is time invariant, it then follows that population growth rate should vary over time. Finally, we note that use of model selection (AICc) or test (LR) statistics comparing models with timespecific vs. time-invariant population growth rate should be relevant to decisions about whether to use asymptotic rates of change from matrix population models as descriptions of population growth. The time-specific recruitment model (q~t, Pt, ft) yielded estimates of new recruits at i + 1 per animal at i ranging from 0.24 to 0.35 (Table 18.6). These estimates can be compared with the derived estimates
fi-- B i / ( l ~ i - di) obtained under the JollyzSeber model (see Section 18.5), where the /~i and N i are Jolly-Seber estimates and d i represents trap deaths. As was the case with population growth rate, the derived recruitment esti-
TABLE 18.6 Estimated Per Capita Recruitment Rates for Meadow Voles under Two Models a
(fi)
Sample period/
Model (%, Pt, ft )b
Model (%, Pt, f)c
fi
S'E(fi )
~e
S"E(f)
1 2 3 4 5
d 0.24 0.35 0.31 a
md 0.076 0.109 0.098 a
0.35
0.039
aBased on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). bModel (%, Pt, ft) AICc = 989.9. CModel (%, Pt, f) AICc = 987.6. aQuantities not estimable under the model. eSingle estimate f applies to periods 1-5.
mates are effectively identical to the estimates under model (q~t, Pt, ft). The AICc was slightly lower for the constant-f model, which yielded an estimate of f = 0.35 (Table 18.6). Note that the AICc values under the general models with all parameters time specific are identical for the three parameterizations of Tables 18.4-18.6. This is expected, because they are statistically equivalent ways of representing the same data. However, the AICc values for the reduced-parameter models with parameters constant over time are not identical for the different parameterizations. This again is expected, because the parameterizations in these models yield statistically different representations, with different consequences as to model fit.
18.5. R E L A T I O N S H I P S AMONG APPROACHES
The Jolly-Seber (Jolly, 1965; Seber, 1965), superpopulation (Crosbie and Manly, 1985; Schwarz and Arnason, 1996), and temporal symmetry (Pradel, 1996) approaches described above are simply three different ways of modeling the same data. In this section we attempt to clarify some of the relationships among the three approaches, as an aid to understanding the information contained in open-model capture-recapture data. In order to facilitate understanding, we consider the case of no losses on capture, though the arguments in this section remain intact absent this assumption (e.g., see Section 18.4.1). All three approaches include survival and capture probabilities, though capture probability parameters are viewed slightly differently under reverse time with losses on capture. However, the same Pi are used for all three approaches in the case of no losses on capture. In the discussion below, we first consider the Jolly-Seber quantities N i and B i and write them in terms of the superpopulation and temporal symmetry approaches. We then focus on the three parameterizations of Pradel (1996) and consider estimation of the associated parameters under the Jolly-Seber and superpopulation approaches. The Jolly-Seber approach treats population size ( N i) and number of recruits (B i) as unknown random variables to be estimated. Under the Schwarz-Arnason superpopulation approach, the expected number of recruits in any sampling period is simply given by the product of superpopulation size and the appropriate entry probability: E(BilN) = N ~ i ,
(18.44)
leading to the estimator in Eq. (18.31). Under the Schwarz-Arnason superpopulation ap-
18.5. Relationships among Approaches proach, the expected value of abundance in period i can be written as
E(Nil N) = N(~oq~lq~2 "'" q~i-1 if- ~1q~2q~3 "'"
(18.45)
X q~i-1 q- "'" q- ~i-1)
(see Schwarz, 2001). In Eq. (18.45), expected abundance in period i is written as the sum of the expected number of animals that first entered at each previous sampling occasion and survived until i. The recursive abundance estimator in Eq. (18.32) is based on Eq. (18.45). Under the temporal symmetry models of Pradel (1996), the expectation for B i can be written as
E(BilNi+I) = Ni+I(1 - '~i+1)-
(18.46)
Thus, the expected number of recruits at i + 1 is simply the product of population size at i + 1 and the proportion of these animals that are recruits. Equation (18.46) suggests the estimator Bi = /~/i+1( 1 - ~i+1)
(18.47)
for recruits between period i and i + 1. One way of writing the expected value for abundance in period i under the models of Pradel (1996) is to condition on abundance in the first sampling period and to multiply this abundance by the appropriate population growth rate parameters:
of the seniority parameter (the probability that an animal present at i+1 is "new" in the sense that it was not present at i) as the ratio of estimated new animals at i+1 to abundance at i+1. Now consider estimation of the three parameters (~i, Ki, fi) used by Pradel (1996) under the superpopulation approach of Schwarz and Arnason. Begin by writing the expected population size as a function of Pradel's (1996) per capita survival and recruitment parameters:
E(Ni+I[Ni) = Ni(q~i + fi)"
j=l
Ki-- ~i q- fi.
fi =
Pradel's (1996) per capita recruitment rate fi is defined as the expected number of animals in the population at time i+1 per animal in the population at time i. The natural estimator for this quantity under the Jolly-Seber approach is simply
fi = Bi/l~i.
(18.48)
Finally, the seniority parameter of Pradel (1996) can be estimated under the Jolly-Seber approach either as ~i+1 -" l~i~Pi/l~i+l
(18.49)
or as 1 - "Yi+I = Bi/l~i+l.
1[
~j=0
~i = /~i+l/l~i"
(18.50)
Estimator (18.49) is the ratio of estimated survivors from period i still present at i+1 to the estimated abundance at i+1. Estimator (18.50) shows the complement
(18.52)
Equation (18.52) expresses population growth rate intuitively, as the sum of survival and recruitment rates (see Section 8.1). Under the superpopulation approach of Schwarz and Arnason (1996), recruitment rate can be estimated as
~.j.
Estimation of population growth rate under the Jolly-Seber approach relies on the definition of ~ki as the ratio of two abundances and substitution of the appropriate estimates:
(18.51)
Equation (18.51) simply defines the expected population size in sampling period i + 1 as the sum of expected survivors and new recruits, written as the product of abundance and the sum of survival probability and recruitment rate. By rearranging Eq. (18.51), population growth rate written as
i-1
E(Ni[NI) = Nil-- [
519
~i
]
(18.53)
~j I-II=j+l i-1 ~Pl
(Schwarz, 2001). Estimator (18.53) is obtained by writing fi as a function of Jolly-Seber approach estimators (18.48) and then substituting the corresponding superpopulation estimators from Eqs. (18.44) and (18.45). The superpopulation estimator for population growth rate is then given by
~i = ~Pi if- fi' where the estimator for recruitment rate is based on Eq. (18.53). Finally, it is possible to estimate Pradel's (1996) seniority parameter by substituting the appropriate superpopulation estimators into Eq. (18.50), to obtain 1
-
qi+l
=
~i q- ~j=0 i-1 q~l] /-11 ~j II1=j+l
Last, we note that the superpopulation of Schwarz and Arnason (1996) can be written as the sum of the numbers of new recruits to the population over all sampling periods [e.g., see Eq. (18.26)]. Because the numbers of recruits can be estimated using both the Jolly-Seber [Eq. (18.13)] and temporal symmetry [Eq.
520
Chapter 18 Estimating Abundance and Recruitment
(18.47)] models, estimation of N under these approaches can be based on Eq. (18.26) (e.g., see Shealer and Kress, 1994). Thus, any quantity estimated using one of the approaches considered in this chapter can be estimated (although perhaps indirectly) via the other two approaches. At present, there is little basis for recommending one estimation approach over another based solely on estimator properties. The superpopulation and temporal symmetry approaches are sufficiently new that there are yet to be comprehensive investigations of estimator properties. However, if investigator interest is on a particular abundance or recruitmentrelated parameter, it seems reasonable to use the approach that permits direct modeling of that parameter.
18.6. S T U D Y D E S I G N As we have emphasized throughout this book, study design should always be tailored to the questions under investigation. In this section we follow the approach of Section 17.6 and focus on aspects of study design that are especially relevant to the estimation of abundance, recruitment, and related parameters. The models of Chapter 18 are flexible enough to allow one to focus on specific questions involving these parameters. Following Section 17.6, we discuss three considerations that are relevant to study design: what parameters are to be estimated, how assumption violations can be minimized, and how precise estimates can be obtained. Because the models of this chapter are obtained by adding components to the likelihoods of Chapter 17, virtually all of the design considerations discussed in Section 17.6 also are relevant to Chapter 18. The following discussion focuses on aspects of study design not covered in Section 17.6.
18.6.1. Parameters to Be Estimated One aspect of study design that is important to the estimators of this chapter but not to those of Chapter 17 involves the reobservation process. The model parameters of Chapter 17 can be estimated based solely on reobservations of marked animals. However, if abundance or recruitment is of interest then every sampiing occasion for which an abundance or recruitment estimate is desired must include sampling of unmarked animals. As discussed in Section 18.1, unmarked animals need not be captured and marked, but it is necessary to record the number of unmarked animals observed during the process of recording the identities of marked animals. Thus, one must be able to distinguish different unmarked individuals during the sampling process, so that an unambiguous count
of them can be made (see Hestbeck and Malecki, 1989b; Kautz and Malecki, 1990; Dreitz et al., 2002). Many studies of open populations using capturerecapture are designed to investigate population dynamics for relatively long periods of time, over which births and deaths contribute a substantial fraction of the population gains and losses. For example, studies of small mammals might span several years and multiple generations of animals. On the other hand, some studies use open models to investigate dynamics over very short periods in which movement would account for most gains and losses. For example, ornithologists sometimes are interested in estimating the number of migratory birds using migration stopover sites (e.g., Nichols, 1996; Nichols and Kaiser, 1999; Kaiser 1999), and fishery biologists are interested in the numbers of fish migrating seaward from spawning streams (e.g., Schwarz et al., 1993b; Schwarz and Dempson, 1994). In these short-term studies, the study areas are viewed as "flow-through" systems, and one objective is to estimate the number of animals going through the system over the course of the study. In such studies, the superpopulation size is the parameter of primary interest, and the estimation approach of Schwarz and Arnason (1996; also see Crosbie and Manly, 1985) is especially appropriate. An important design aspect of such studies is to be sure that the sampling occasions cover the entire period of interest. For example, if capture and recapture do not begin until after the arrival of birds at a migration stopover location, then the estimated superpopulation size N will not include birds that arrived and departed before the sampling began. Although this chapter has focused on single-age, single-stratum models, we noted in Section 18.2.4 that abundance and recruitment also can be estimated for multiple ages or locations or physiological states. For studies with geographic stratification, the design must include approximately simultaneous sampling at multiple locations, as discussed in Section 17.6.1. Similarly, animals must be assigned to age class or physiological state at each reobservation if age- or state-specific estimates are desired. In the case of multiple-age classes, it is possible to decompose recruitment into components associated with immigration versus in situ reproduction (Nichols and Pollock, 1990) (see Section 19.4). Because sampling design considerations necessary for application of this approach include the robust design, we defer its discussion until Chapter 19.
18.6.2. Model Assumptions 18.6.2.1. Homogeneity of Rate Parameters The discussion in Section 17.6.2 included various suggestions for dealing with heterogeneity of rate pa-
18.6. Study Design rameters (also see Pollock et al., 1990), and these suggestions should be just as applicable to the new parameters introduced in Chapter 18 (e.g., the entry parameters of the superpopulation approach; the seniority parameters of the temporal symmetry approach) as they are to capture and survival probabilities. Stratification by factors such as location, age, sex, size, reproductive state, and physiological state can be useful whenever rate parameters are thought to differ among strata. The key design issue is then to record data on the stratification factors (i.e., factors likely to be associated with variation in rate parameters). In addition to stratification, the use of multiple sampling methods is also a good approach for reducing the likelihood of heterogeneous observation probabilities. A design issue relevant to estimation of abundance and recruitment involves the use of resighting as a means of "recapturing" animals. As noted in Section 18.1, the use of resighting requires that the investigator record the number of unmarked animals encountered while resighting marked animals. The important element of such sampling is that marked and unmarked animals must have the same probabilities of being observed. Assume, for example, that Canada geese are being sampled, and that some birds are marked with neckbands. Assume further that a large group of birds is under observation. In this case, an effort should be made to scan the group for neck collars, recording the number of individuals whose necks are observed without collars as well as the identities of birds with collars. If a bird is seen to have a collar, but flies off before the collar can be read, one approach is to not record this bird (i.e., the bird does not add to either the marked or the unmarked group). Under this approach, it is important not to record birds seen to be unmarked that fly off before their band numbers could have been read had they been marked. Such a bird could be assigned unambiguously to the unmarked group, but such an assignment will lead to higher sighting probabilities for unmarked birds (because a marked bird seen for a similar length of time could not be identified). Thus, it is best to not record these birds in the sample. Similarly, any birds in the group whose necks are not examined would not be recorded in the sample at all. This example is just one of many possible field situations, but it illustrates the point that every effort should be made to ensure that marked ( M i) and unmarked (U i) animals in the sampled population have similar probabilities of entering the sample and being included in the associated count statistic (i.e., of appearing as a member of m i or u i, respectively). Spatial sampling with devices such as traps should be conducted in a way that ensures each animal in the area of interest is likely to encounter at least one sampling device (Section 17.6.2). As noted previously,
521
high sampling intensities [e.g., capture probabilities :>0.5; see Gilbert (1973)] can reduce the effects of heterogeneous capture probabilities on abundance estimates. Finally, the various methods for reducing traphappy or trap-shy responses (Section 17.6.2) should also be considered. The above recommendations concern the design of a study so as to minimize violations of the assumption of homogeneous rate parameters. An alternative approach for studies focused on abundance estimation is to implement the robust design (Chapter 19). The original motivation for this design involved the ability to use closed population models (Chapter 14) (Otis et al., 1978) for estimation of abundance (Pollock, 1982). Closed population models and estimators have been developed for situations in which capture probabilities vary among individuals (heterogeneity) and between marked and unmarked individuals (behavioral response models). Indeed, the "robust design" was so named because it provides the ability to obtain estimates of abundance (as well as survival and recruitment) in the presence of nonhomogeneous capture probabilities (Pollock, 1982).
18.6.2.2. Tag Retention As noted in Section 18.2.5, tag loss does not produce bias in Jolly-Seber abundance estimates but does lead to reduced precision of these estimates. Jolly-Seber recruitment estimates are biased by tag loss. The best design advice regarding tag loss is to use doubletagging (at least for a subset of animals) in cases where it is suspected (Section 17.6.2), because this permits estimation of loss rates and thus provides some ability to deal with any resulting problems of estimator bias (e.g., with respect to survival and recruitment estimators).
18.6.2.3. Instantaneous Sampling Violation of this assumption causes problems with estimation of abundance and recruitment. Indeed, if the population is open to gains and losses during the sampling period, then it is not even clear what is meant by "abundance during period i." The appropriate design recommendation is to select the seasonal timing and duration of the sampling periods in an attempt to reduce the possibility of nonnegligible mortality, immigration, and emigration (Section 17.6.2).
18.6.2.4. Temporary Emigration Markovian temporary emigration can result in biased estimates of abundance and recruitment (Kendall et al., 1997). Perhaps the best way to deal with this
522
Chapter 18 Estimating Abundance and Recruitment
possibility is to include in the study design a way to either (1) estimate the time-specific conditional capture probabilities for animals in the study area and exposed to sampling efforts or (2) estimate rates of migration to and from areas surrounding the primary sample area. The robust design (Chapter 19) has been proposed as a means of estimating conditional capture probabilities for animals in the study area (Kendall and Nichols, 1995; Kendall et al., 1997; Schwarz and Stobo, 1997). The direct estimation of movement rates can be accomplished by establishing another stratum (e.g., the area surrounding the principal study area) to be sampled via capture-recapture methods (e.g., using multistraturn models; Section 17.3). An alternative approach is to mark a subset of animals with radios and use telemetry to estimate directly rates of temporary emigration (e.g., Pollock et al., 1995; Powell et al., 2000a).
18.6.3. Estimator Precision Under the Jolly-Seber approach, abundance and especially recruitment tend to be estimated relatively less precisely (e.g., larger coefficients of variation) than survival probability (see Pollock et al., 1990). Precision is thus an especially important consideration for studies directed at estimation of abundance and recruitment. All of the design recommendations provided in Section 17.6.3 are relevant for abundance and recruitment estimation as well. As with survival rate estimation, increases in capture probability lead to increases in precision. The sample size figures of Pollock et al. (1990) plot cv(Ni) and cv(Bi) as functions of capture probability, and knowledge of this relationship often is useful in study design. Basically, all of the tradeoffs and considerations previously discussed (Section 17.6.3) are even more important for abundance and recruitment estimation because of the inherent tendency of these estimates (especially recruitment) to be relatively imprecise.
18.7. D I S C U S S I O N The basic approach for the models of Chapter 17 was to condition on release of a marked animal in a specific sampling period and then to model the remainder of its capture history as a function of capture and survival probability parameters. In the models of Chapter 18, we relaxed the conditioning in Chapter 17 by adding components that account for the entry of unmarked animals into the population. The primary motivation for adding these new components is to esti-
mate additional quantities such as population size, recruitment, and related parameters. The Jolly-Seber, superpopulation, and temporal symmetry approaches described in this chapter are simply three different ways of parameterizing the extra model components. The equivalence of the three approaches is emphasized in Section 18.5, where we show that any quantity estimated using one approach can be estimated (although perhaps indirectly) via the other two approaches. For example, the Jolly-Seber approach focuses on the direct estimation of numbers of animals (N i) and numbers of recruits (Bi). These quantities are treated as unknown random variables to be estimated after the modeling of survival and capture parameters, absent model parameters for abundance or recruitment. On the other hand, the superpopulation approach considers the total number of animals found in the study area during at least one sampling period of the entire study and the probabilities that a member of the superpopulation entered the sampled population at each of the sampling periods during the study. The temporal symmetry approach incorporates simultaneous backward and forward models for capture history data and utilizes seniority parameters (probability that a member of the population at sampling period i + 1 is "old" in the sense of having been in the population the previous period). Alternative parameterizations for the temporal symmetry models use either population growth rate or recruitment rate. It is remarkable that a simple capture history matrix (a vector of ls and 0s for every animal caught during a study) provides the information needed to estimate all these quantities. Time-specific estimates of abundance, survival, recruitment, and the various derivative parameters provide a very detailed description of the dynamics of the studied population. Of course, certain questions require auxiliary data (e.g., decomposition of losses into deaths and movement; decomposition of gains into recruits resulting from in situ reproduction and immigration), but the basic demographic bookkeeping associated with changes in numbers of animals on a predefined study area can be accomplished using the data from a simple capture history matrix. In Chapter 19 we deal with capture-recapture data obtained at two different temporal scales, thereby permitting simultaneous use of both closed (Chapter 14) and open (Chapters 17 and 18) models. We will see that the use of this "robust design" permits not only the estimation of quantities that could be estimated using either closed or open models separately, but also the estimation of quantities that could not be estimated without both in combination.
C H A P T E R
19 Combining Closed and Open Mark-Recapture Models: The Robust Design 19.1. DATA STRUCTURE 19.2. A D HOC APPROACH 19.2.1. Combining Open and Closed Models 19.2.2. Estimation Based Solely on Closed Models 19.3. LIKELIHOOD-BASED APPROACH 19.3.1. Models 19.3.2. Model Assumptions 19.3.3. Estimation 19.3.4. Alternative Modeling 19.3.5. Model Selection, Estimator Robustness, and Model Assumptions 19.4. SPECIAL ESTIMATION PROBLEMS 19.4.1. Temporary Emigration 19.4.2. Multiple Ages and Recruitment Components 19.4.3. Catch-Effort Studies 19.4.4. Potential for Future Work 19.5. STUDY DESIGN 19.6. DISCUSSION
dynamics (population size), the rate of change in that state variable, and the vital rates responsible for that change. This chapter represents a synthesis of capturerecapture approaches to the estimation of population size and vital rates, by combining in a single model the advantages of both open- and closed-population methods. Here we view the long-term study of an open population as a sequence of short-term studies of closed populations. Several advantages accrue to population sampling at two distinct temporal scales, including more robust estimation of the parameters considered previously and estimation of certain parameters not otherwise estimable with either open or closed models when considered separately. Both advantages are a direct consequence of the additional information provided by the short-term capture-history data. In one sense, the robust design can be considered to be a special case of using auxiliary data (Section 17.5) produced from short-term sampling. The original motivation for the robust design was a concern about estimator robustness, especially as relates to heterogeneity in capture probabilities. Previous to his formulation of the robust model, Pollock (1975) extended the work of D. S. Robson (1969) by incorporating certain kinds of capture-history dependence (see Section 17.1.6) in the context of Jolly-Seber models. However, other kinds of variation in capture probability, notably heterogeneity among individuals and permanent trap response, could not be dealt with adequately in an open-model setting. Although the survival estimators for the Jolly-Seber model are relatively robust to these sources of variation (e.g., see Carothers 1973, 1979), its abundance estimators are not
In Chapters 14 and 16-18 we focused on the estimation of population parameters based on studies of marked animals. In Chapter 14 we saw that capturerecapture models can be used to estimate population abundance over short periods of time during which the population is assumed to remain unchanged in size and composition. In contrast, open-population models (Chapters 16-18) allow one to include population gains and losses between sampling periods and thus to estimate population size, population rate of change between successive sampling periods, and rates of survival, recruitment, and movement between sampling periods. In terms of system dynamics, these quantities include the principal state variable for population
523
Chapter 19 The Robust Design
524
(e.g., Gilbert, 1973; Carothers, 1973). On the other hand, capture-recapture models for closed populations were developed to deal with trap response and heterogeneity in capture probabilities, leading to robust estimates of abundance under these conditions (Pollock, 1974; Burnham and Overton, 1978; Otis et al., 1978). Building on both approaches, Pollock (1981a, 1982) suggested sampling at two temporal scales, with periods of short-term sampling over which the population is assumed to be closed and longer term sampling over which gains and losses are expected to occur (also see Lefebvre et al., 1982). In particular he recommended that closed models be used to estimate abundance, with data arising from each short-term sampling episode (see Chapter 14). These data then can be pooled (with each animal recorded as caught if it was observed at least once during the closed population sampling) to estimate survival based on the Cormack-Jolly-Seber estimators (see Chapter 17). With the abundance estimates from the closed models and survival estimates from the open models, recruitment in turn can be estimated as in Eq. (18.13). Pollock (1982) suggested that such a sampling design should provide estimators that are robust to various sources of variation in capture probabilities.
stricted to include a fixed number of secondary occasions over all the primary occasions (l i = l for all i = 1, ..., K). As an example, a small mammal population might be trapped for five consecutive days every 2 months. Capture-recapture data from the robust design can be summarized in several ways. Perhaps the most basic summary is analogous to the X matrix of Section 14.2.1, with Xgij an indicator variable reflecting either capture (Xgij = 1) or no capture (Xgij = O) for individual g in secondary sampling period j of primary sampling period i. For example, a study with K = 4 primary sampling periods and l = 5 secondary sampling periods within each primary period would correspond to an X matrix with 20 columns. A row vector corresponding to a particular animal might be 01101 00000
A schematic representation of the robust sampling design is presented in Fig. 19.1. The design consists of K primary sampling occasions, between which the population is likely to be open to gains and losses. At each primary sampling occasion, a short-term study is conducted, with the population sampled over l i secondary sampling periods, during which it is assumed to be closed [although this assumption can be relaxed; see Schwarz and Stobo (1997)]. Though one can have a different number of secondary sampling occasions for each primary occasion, the design also can be re-
Secondary Periods
1
1
/ I N...11 2
10111,
consisting of four groups of five capture values. The first group of five numbers gives the capture history over the five secondary periods of primary period 1, showing that the animal was captured on occasions 2, 3, and 5 of primary period 1. The second group of numbers indicates that the animal was not captured at all during primary period 2. In primary period 3, it was captured on the third secondary occasion, and in primary period 4, it was captured on secondary occasions 1, 3, 4, and 5. The X matrix consists of all such capture history vectors for all animals caught at least once during the study. For example, the X matrix for the male meadow voles used in the example analyses of Sections 17.1.10, 18.2.6, 18.3.6, and 18.4.6 is presented in tabular form in Table 19.1. Note that a " - " designation in the final column of Table 19.1 indicates that the animal was not released back into the population following the last capture (the last "1") in the record. The individual capture history data also can be collapsed into various kinds of summary statistics. Here we follow the general notation of Kendall et al. (1995),
19.1. D A T A S T R U C T U R E
Primary Periods
00100
2
1
.
/ I N 12 2...
.
.
K
1
2...1K
FIGURE 19.1 Schematicrepresentation of Pollock's (1982) robust design for capture-recapture sampling. Primary sampling periods i = 1..... K are separated by relatively long time intervals over which the population is likely to be open to gains and losses. At each primary period i, sampling is conducted at l i secondary sampling periods. Secondary periods are separated by relatively short time intervals over which the population may be closed to gains and losses. Models for open populations are used for the capture history data summarized at the level of primary periods, whereas either closed or open models (as appropriate) are used for data summarized at the level of secondary periods within each primary period.
19.1. TABLE 19.1
525
Data Structure
C a p t u r e - R e c a p t u r e Data for A d u l t M a l e M e a d o w Voles a Primary sampling period b
Identification number
1
2
3
4
4321
00100
00000
00000
5311
00000
00000
00010
7701
11011
11100
7720
11110
7725
11111
7736
5
6
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00110
11111
11111
11111
11111
11111
11101
11111
11111
11011
11111
7745
11101
01100
00000
00000
00000
00000
7752
10101
11011
00000
00000
00000
00000
7762
00000
00100
00000
00000
00000
00000
7764
11111
11100
11000
11111
11111
11111
7772
10110
00000
00000
00000
00000
00000
7773
10101
01110
00000
00000
00000
00000
7775
11111
11100
00000
00000
00000
00000
7782
00000
00000
00010
00000
00000
00000
7785
11100
01110
00000
00000
00000
00000
7786
01100
11000
11111
00000
00000
00000
7792
01101
01100
01011
11111
11101
11101
7796
11111
11100
00000
00000
00000
00000
7811
11111
11110
11101
11111
11111
01001
7824
11111
11110
00000
00000
00000
00000
7828
11011
00000
00000
00000
00000
00000
7832
00001
11000
00000
00000
00000
00000
7840
01100
11100
00000
00000
00000
00000
7846
11101
10000
00000
00000
00000
00000
7847
11000
01100
00000
00000
00000
00000
7853
11011
10100
00000
00000
00000
00000
7855
10100
00000
11001
11110
00000
00000
7856
10000
01100
00000
00000
00000
00000
7857
11111
01100
11001
10111
11110
00000
7858
10101
11100
00010
10100
00000
00000
7860
01010
00000
00000
00000
00000
00000
7863
01011
01000
10010
11111
00000
00000
7865
01000
01000
10000
00000
00000
00000
7866
01000
00000
00000
00000
00000
00000
7867
01111
00000
00000
00000
00000
00000
7868
01110
01100
00000
00000
00000
00000
7869
01100
11100
11110
11111
11110
11010
7871
01111
00000
00000
00000
00000
00000
7872
00000
01110
00000
00000
00000
00000
7874
01011
00000
00000
00000
00000
00000
7875
01111
11100
00000
00000
00000
00000
7879
00101
00010
01000
11110
00010
01010
7882
00100
11110
11101
11111
00000
00000
7887
00100
11110
01000
00000
00000
00000
Not released ( - )
(continues)
526
C h a p t e r 19
The Robust Design
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
5
6
7890
00010
00000
00000
00000
00000
00000
7891
00010
11100
00000
10000
10100
10000
7892
00000
10100
00000
00000
00000
00000
7894
00010
11000
00000
00000
00000
00000
7895
00010
11100
00000
00000
00000
00000
7896
00001
11100
10000
11110
11111
10100
7901
00000
11100
00000
00000
00000
00000
7904
00000
11100
00000
00000
00000
00000
7905
00000
11101
01010
11000
00000
00000
7906
00000
11000
01000
00000
00000
00000
7907
00000
11100
11110
00000
00000
00000
7910
00000
11100
00010
10001
00000
00000
7912
00000
01100
00000
00000
00000
00000
7913
00000
01000
00100
00000
00011
01010
7918
00000
00000
11100
11001
00000
00000
7919
00000
01010
01100
00000
00000
00000
7920
00000
01000
10110
00001
11100
11011
7921
00000
01010
00000
00000
00000
00100
7922
00000
01000
00000
00000
00000
00000
7925
00000
00100
00000
00000
00000
00000
7930
00000
00100
00000
11000
00000
00000
7932
00000
00100
00001
01000
00000
00010
7935
00000
00010
00000
00000
00000
00000
7936
00000
00010
00000
00000
00000
00000
7937
00000
00010
00000
00000
00000
00000 00000
7938
00000
00010
00010
10010
00000
7940
00000
00010
00000
01011
01001
01111
7941
00000
00010
01000
01001
00000
10000
7944
00000
00000
10001
00000
00000
00000
7945
00000
10100
11111
11111
11111
11110
7946
00000
00000
11111
11111
00000
00000
7948
00000
00000
01011
00000
00000
00000
7949
00000
00000
01010
00111
11111
11111
7953
00000
00000
01000
00000
00000
00000
7954
00000
00000
00000
01011
00000
00000
7957
00000
00000
01000
00000
00000
00000
7958
00000
00000
00100
00000
00000
00000
7964
00000
00000
00010
10110
00000
00000
7967
00000
00000
00001
00001
11011
11101
7969
00000
00000
00000
10000
01111
11111
7970
00000
00000
00000
11111
00000
00000
7974
00000
00000
01000
10011
11111
11110
7975
00000
00000
00000
10001
00000
00000
7976
00000
00000
00000
00000
11101
11111
Not released ( - )
(continues)
19.1.
527
Data Structure
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
7978
00000
00000
00000
00000
01101
00110
7980
00000
00000
00000
10000
00000
00000
7983
00000
00000
00000
01110
00000
00000
7986
00000
00000
00000
01000
00000
00000
7990
00000
00000
00000
00000
00000
01000
7992
00000
00000
00000
00000
00000
00111
7995
00000
00000
00000
01000
11111
11011
7999
00000
00000
00000
01100
11110
11111
8002
00000
00000
00000
00110
00000
00000
8003
00000
00000
00000
00100
10100
10100
8007
00000
00000
00000
00100
00000
00000
8008
00000
00000
00000
00100
00000
00000
8009
00000
00000
00000
00100
00000
00000
8010
00000
00000
00000
00011
10111
00000
5
6
8014
00000
00000
00000
00000
01010
11011
8016
00000
00000
00000
00010
01101
00000
8017
00000
00000
00000
00000
01111
00010
8019
00000
00000
00000
00000
00010
00010
8022
00000
00000
00000
00001
00000
00000
8027
00000
00000
00000
00001
00000
00000
8028
00000
00000
00000
00001
00000
00000
8029
00000
00000
00000
00001
00111
00000
8032
00000
00000
00000
00000
00000
11111
8033
00000
00000
00000
00000
00000
10011
8034
00000
00000
00000
00000
00000
10000
8036
00000
00000
00000
00000
10111
11111
8038
00000
00000
00000
00000
11000
10110
8040
00000
00000
00000
00000
00000
01110
8044
00000
00000
00000
00000
10100
00000
8045
00000
00000
00000
00000
10001
00110
8046
00000
00000
00000
00000
00000
01011
8048
00000
00000
00000
00000
00000
01111 01010
8050
00000
00000
00000
00000
00000
8051
00000
00000
00000
00000
00000
11100
8052
00000
00000
00000
00000
01010
00000
8055
00000
00000
00000
00000
01010
11111
8056
00000
00000
00000
00000
00000
11110
8058
00000
00000
00000
00000
00000
11111
8061
00000
00000
00000
00000
01110
00100
8062
00000
00000
00000
00000
00000
00110
8064
00000
00000
00000
00000
01000
11011
8069
00000
00000
00000
00000
00100
00010
8070
00000
00000
00000
00000
00000
00001
8074
00000
00000
00000
00000
00100
00000
Not released ( - )
(continues)
528
C h a p t e r 19
The Robust Design
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
5
6
8080
00000
00000
00000
00000
00010
00000
8087
00000
00000
00000
00000
00001
01000
8090
00000
00000
00000
00000
00000
11100
8092
00000
00000
00000
00000
00000
10000
8093
00000
00000
00000
00000
00000
10100
8095
00000
00000
00000
00000
00000
10110
8097
00000
00000
00000
00000
00000
11000
8099
00000
00000
00000
00000
00000
10110
8100
00000
00000
00000
00000
00000
10110
8225
00000
01000
00000
01001
11111
11111
8421
00000
00000
00000
00000
00000
11101
8601
00000
00000
00000
00000
00000
01000
8602
00000
00000
00000
00000
00000
01000
8604
00000
00000
00000
00000
00000
01000
8606
00000
00000
00000
00000
00000
01001
8608
00000
00000
00000
00000
00000
01000
8610
00000
00000
00000
00000
00000
00100
8613
00000
00000
00000
00000
00000
00110
8616
00000
00000
00000
00000
00000
00010
8619
00000
00000
00000
01000
00000
00011
8620
00000
00000
00000
00000
00000
00001
8621
00000
00000
00000
00000
00000
00001
8624
00000
00000
00000
00000
00000
00001
8633
00000
11000
10011
00101
01001
11111
8645
00000
00000
00000
00000
00000
00010
8652
00000
00000
11111
11101
11101
11110
9321
10000
00000
00000
00000
00000
00000
9322
11111
11100
11101
11111
11011
10010
9334
11111
01100
00000
00000
00000
00000
9343
11111
11101
00000
00000
00000
00000
9345
11110
11100
10011
00000
00000
00000
9350
11111
10100
00000
00000
00000
00000
9359
11101
11000
00000
00000
00000
00000
9362
11110
11100
11011
11110
00000
00000
9381
11111
11110
11111
11101
11111
01110
FIB5
00000
00100
10100
00001
00000
00111
TCB1
00000
00000
00100
00000
00000
00000
TCB2
11111
01110
11110
00000
00000
00000
TCF7
11111
10100
00000
00000
00000
00000
Not released ( - )
a At Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981. Data follow Pollock's robust design, with the columns under each primary period representing the five consecutive days of trapping each month. bInitial date in 1981 of each primary period: 1, 6/27; 2, 8/1; 3, 8/29; 4, 10/3; 5, 10/31; 6, 12/4.
529
19.2. Ad Hoc Approach who used the notation of Chapters 17 and 18 for the open-model portion of the capture-history data: ui
mi
Yli ~ U i -}- m i
Ri
mhi
The number of unmarked animals caught on at least one secondary occasion within primary period i. The number of animals marked previous to primary period i that are caught on at least one secondary occasion within primary period i. The total number of animals caught on at least one secondary occasion within primary period i. The number of n i that are released back into the population following primary period i. The number of R i that are recaptured at some primary period following i. The number of animals caught in primary period i that were last caught in primary period h.
In addition, the following statistics are associated only with the robust design: X~i
X)~i
x.~
The number of animals from u i that exhibit capture history 00 e f~ over the l i secondary periods of primary period i (where f~ is the set of all possible sequences of 0s and ls over the l i secondary periods). The number of animals from mhi that exhibit capture history co e f~ over the l i secondary periods of primary period i. The total number of animals caught in primary period i that exhibit capture history 00 over the l i secondary periods: i-1
X~ :
E
X~ i"
h=0
This notation allows us to partition the individuals captured at time i into (1) those previously captured at primary period h = 0, 1..... i - 1, and (2) those with secondary capture history oJ. We again designate animals not released following the final capture of the secondary capture history with a " - " preceding the number of animals exhibiting the history.
19.2. AD H O C APPROACH An ad hoc approach to the robust design typically involves a combination of open and closed models. However, it also is possible to develop ad hoc estimators
for abundance, survival probability, and recruitment using only capture-recapture models for closed populations. In what follows we describe both approaches, but emphasize that the former approach is by far the more commonly used.
19.2.1. Combining Open and Closed Models 19.2.1.1. M o d e l s
In his pioneering work, Pollock's (1981a, 1982) robust capture-recapture design involved three different approaches to estimation: (1) estimation of abundance with closed models using secondary capture-history data, (2) estimation of survival rates using standard open models with capture-history data reflecting captures in each primary period, and (3) estimation of the number of new recruits using the closed-model abundance estimates and open-model survival estimates, in conjunction with Eq. (18.13). Thus, the modeling proceeds via independent selection of an open model that incorporates survival and capture probabilities for the primary periods, and a closed model that incorporates abundances and capture probabilities for the secondary periods. The independent modeling of data from the primary and secondary periods distinguishes the ad hoc approach from a likelihood-based approach, in which both types of data are modeled simultaneously within a single likelihood (see Section 19.3). Under the most general ad hoc approach, model selection can be carried out independently for each of the K closed-model data sets (one for each primary period). This can lead to different closed models for different primary periods within a single analysis. Unless there are a priori reasons for expecting different closed models, we recommend the use of a single closed-population model for all K data sets. One reason for this recommendation concerns the magnitudes and directions of biases associated with the abundance estimators of the different models. Any biases are likely to be of similar direction and magnitude if the same model is used for estimation with all K data sets (e.g., see Skalski and Robson, 1992), and this similarity will yield more reasonable estimates of recruitment when the closed- and open-model estimates are combined [see Eq. (19.3)]. A second reason for recommending the use of a common closed-population model involves the imperfect nature of all model selection algorithms, including that of Otis et al. (1978) and Rexstad and Burnham (1991). It often is reasonable to expect similar processes (environmental variation, genetic variation)
530
Chapter 19 The Robust Design
to affect capture probability throughout a study, and it is appropriate under these circumstances to use model selection results from all K data sets to select a single model for the study. For example, if we have K = 8 primary periods and the model selection algorithm of CAPTURE (Rexstad and Burnham, 1991) indicates selection of model Mh for six periods, model M 0 for one period, and model Mth for one period, it may be reasonable to select model Mh for use with all eight data sets. Of course, this approach is ad hoc and without a sound theoretical basis, but it nonetheless seems reasonable. Note that the models for the robust design must account for two kinds of capture probabilities, corresponding to the two different temporal scales. The capture probabilities Pij associated with secondary sampling periods refer to the probability that an animal is captured on secondary occasion j of primary occasion i, given that it is in the population on that occasion. On the other hand, the capture probabilities p* for primary sampling periods refer to the probability that an animal is caught at least once in primary occasion i (i.e., on at least one of the l i secondary occasions), given that the animal is in the population during that sampling period. The latter probabilities are the same parameters used in Chapters 17 and 18, where the "," superscript is used here to avoid confusion with capture parameters for the secondary periods. Here we follow the approach of Kendall et al. (1995), who considered different models for these two different kinds of capture probability. In particular, they allowed for multiple sources of variability over the secondary periods, including temporal variability, behavioral response (i.e., dependence of capture probability on previous capture within the primary period), and heterogeneity (i.e., different capture probabilities among the animals in the population in a primary sample). Models and estimators for these sources of variation in capture probability were discussed in Section 14.2.2 (also see Otis et al., 1978). Recall from Chapter 17 that open models for capture probability also can include time and, in a limited sense, behavioral response, but not individual heterogeneity. Permanent trap response can be included in open models (see Section 17.1.9) because survival estimates are based on marked animals only and exhibit no bias in the face of permanent trap response (Nichols et al., 1984b). Note, however, that permanent trap response at the primary level imposes requirements on models for the secondary sampling period data. Permanent trap response indicates that animals that are unmarked just before primary sampling period i have one capture probability and marked animals have a different capture probability. In order to deal with this response in the closed-population modeling of the secondary periods, animals that were and were not marked prior to primary period i are
placed in two groups and analyzed separately with closed models. Thus, models incorporating behavioral response at the level of primary periods require the fitting of a single open model and (2K - 1) closed models, one closed model each for the marked and unmarked animals in each primary period 2, ..., K, and a single closed model for primary period I (when all animals are unmarked). Kendall et al. (1995) designated models for the robust design as M~, where 13 specifies the model for the primary period capture probabilities and oL specifies the model for the secondary period capture probabilities. A "0" in the subscript or superscript indicates no variation in the specified capture probabilities. Kendall et al. (1995) considered models in which the data within all primary periods exhibit the same sources of variation in capture probability. Thus, they did not consider independent model selection for all K closed models, but instead required a single closed model for use with all K data sets. Note that this formulation leaves unspecified the modeling of survival probabilities between the primary periods. Possible robust design models representing combinations of open (from Chapter 17) and closed (from Chapter 14) models are shown in Table 19.2. Because the model for the primary periods is fitted to the data independently of the fitting of closed models to the secondary period data, the models underlying the ad hoc approach are nothing more than simple combinations of open and closed models. However, given that there is no temporary emigration and all assumptions underlying both models are met, there is a mathematical relationship between capture probabilities at the primary and secondary sampling periods: li
1 - p~ = I-[(1 - P ij) j=l or
xPi =
li
1 - l-I(1 -Pij)
(19.1)
j=l
for i = 1, ..., K. In words, these expressions essentially say that the probability of noncapture on the primary time scale is given by the product of noncapture probabilities on the secondary time scale. Under the simplest model, M ~ the closed model M 0 (Section 14.2.2) (see Otis et al., 1978) is fitted to each of the secondary period data sets, and a model with constant capture probability [e.g., model (q~t,P)] is fitted to the primary period data. Note that model M ~ imposes the constraint that Pij = Pi., i.e., secondary capture probabilities are equal within each primary period.
19.2. A d Hoc Approach TABLE 19.2
Possible Models for Capture Probability under the Robust Design a Source(s) of variation in capture probability
Model
Secondary periods
Primary periods
M~
None
None
Mt
None
Time
M~'
Time
None
M{
Time
Time
M~
Behavior
None
M~
Behavior
Time
Mb
Behavior
Behavior
M~b
Behavior
Time, behavior
M~
Heterogeneity
None
M{~
Heterogeneity
Time
Mt~
Time, behavior
None
M~b
Time, behavior
Time
Mbb
Time, behavior
Behavior
Mttb
Time, behavior
Time, behavior
Mt~
Time, heterogeneity
None
Time, heterogeneity
Time
M~h
Behavior, heterogeneity
None
M~h Mbh
Behavior, heterogeneity
Time
Behavior, heterogeneity
Behavior Time, behavior
M{~b
Behavior, heterogeneity
o Mtb h M~bh Mbbh
All
None
All
Time
All
Behavior
tb
All
Time, behavior
a Following Kendall et al. (1995).
If all primary periods contain the same number of secondary capture periods (l i - l for all i), expression (19.1) then becomes p* = 1 - (1 - p)~
havioral response and heterogeneity for secondary periods. Behavioral response at the secondary sampiing level but not the primary level indicates that behavioral response is temporary, in that marking in previous primary periods is not relevant to capture probability, but marking in a previous secondary period within the same primary period does confer a different capture probability. 19.2.1.2. Model Assumptions
M~h
Mtbh
531
(19.2)
for all periods i = 1, ..., K. Thus, model M ~ in theory links capture probability on the secondary time scale with capture probability on the primary time scale (assuming the absence of temporary emigration; see Section 19.4.1). However, the independent model fitting of the ad hoc approach fails to impose the constraints in Eqs. (19.1) or (19.2). This inadequacy is addressed with the likelihoodbased models of Section 19.3. Other models include model M~, which denotes the robust design model with variable capture probabilities over both secondary and primary periods, and model Mtbh, which indicates time-specific capture probabilities at the level of primary periods, and both be-
The assumptions underlying the above models include those for the respective closed and open models. For example, the assumptions for modeling secondary samples within each primary period are that (1) the population is closed to gains and losses during the period [though this assumption can be relaxed; see Schwarz and Stobo (1997)], (2) marks are neither lost nor incorrectly recorded, (3) capture probability over the secondary periods varies according to specifications dictated by the structure of the selected model, and (4) the fate of each animal is independent with respect to capture probability. These assumptions are discussed in more detail in Section 14.2.3 and need not be revisited here. The assumptions of the ad hoc robust design also include those underlying the open modeling of primary period data: (1) the conditional probability of surviving from primary period i to i + 1 is the same for all animals, (2) the conditional probability of being caught at each primary period is the same for each marked animal in the population at that time, and (3) the fates of animals with respect to both survival and capture are independent. In addition, the closed model assumptions of marks retained and correctly recorded, and closure during the primary period [the equivalent of the instantaneous sampling assumption (4) of Section 17.1.2], are also required by the open models. These assumptions, and ways of dealing with their violation, are discussed in Section 17.1.2. 19.2.1.3. E s t i m a t i o n Estimation under Pollock's (1981a, 1982) ad hoc robust design uses open models for survival rates (~i), closed models for abundance (/~/i), and a combination of both for the recruitment estimator /~i = /~i+1 -- ~Pi(l~i-
Yli -}- Ri),
(19.3)
with (n i - R i) the number of animals caught during the primary period but not released back into the population. An approximate variance estimator for B i is presented by Pollock (1982) and Pollock et al. (1990), on assumption that the survival and abundance estimators are independent. Note that Eq. (19.3) has the
532
Chapter 19 The Robust Design
same appearance as expression (18.13), the only difference being the derivation of the abundance estimates. Estimation of ourvival rates was described in detail in Chapter 17, and abundance estimation with closed models was discussed in Chapter 14. The combined use of open and closed models under the robust design permits estimation of some parameters that are not identifiable using the standard, openmodel approach to estimation. For example, under the Jolly-Seber model, abundance can be estimated only for periods 2 through K - 1, because of an inability to estimate p~ and p~. However, under the robust design, the information needed to estimate these capture probabilities comes from the secondary samples, so capture probabilities and abundance can be estimated for periods I and K. The ability to estimate N 1 in turn permits estimation of B1 with Eq. (19.3). The combined approach also allows one to estimate separately the final capture and survival probabilities, which otherwise can be estimated only as the product q~K-1 P~ in the Cormack-Jolly-Seber model (see Section 17.1.2). There are several ways to estimate q~K-1 using the robust design. One such approach depends on the ability to estimate p~: using data from the secondary periods of primary period K based on Eq. (19.1 ). Thus, under the assumption of no temporary emigration (see Section 19.4.1), p~ can be estimated as
IK /~ = 1 - 1 - [ ( 1 - PKj).
(19.4)
j=l
Given this estimate of p~, a natural estimator for ~K-1 is formed by dividing the estimate of q~K-1 P~ by the estimate of p~:: ~K-lP~ q~K-1 = ~ ,
(19.5)
where the estimate of the product parameter is obtained via maximum likelihood in the same manner as the other survival estimates (e.g., see Section 17.1.2). Two additional approaches to estimation of ~K-1 are presented in Section 19.2.2. The availability of estimates of q~K-1 and N K under the robust design also permits estimation with Eq. (19.3) of recruitment BK_ 1 for the final primary period. Thus, the robust design permits estimation of the quantities of interest for all primary sampling periods: /~1, "",
I~K;
]91, "",
FK;
~1, "",
~K-1;
B1 ..... BK-1.
Kendall and Pollock (1992) present a good discussion of alternative estimators using the ad hoc approach under the robust design.
19.2.1.4. Alternative Modeling The ad hoc approach to estimation under the robust design can accommodate virtually any of the models described for open (Chapter 17) and closed (Chapter 14) populations. Because the models for primary and secondary periods are independent, the discussions of alternative modeling in these previous chapters are directly relevant to the robust design as well. For example, reduced-parameter models, covariate models, and models with capture-history dependence all can be used in conjunction with the robust design. In the case of temporary emigration, where an animal in the population of interest is not present in the area exposed to sampling efforts during a particular primary sampling period i, the capture probabilities for conditional modeling of open populations (e.g., Chapter 17) reflect the product of the probabilities of being present in the area exposed to sampling efforts and of being caught given presence in this area (Kendall et al., 1997). However, capture probabilities based on the secondary samples within a primary period reflect only the conditional probability of capture, but not temporary emigration. Under some forms of temporary emigration, this difference in the interpretations of the two kinds of capture probabilities can be used to estimate the probability of an animal being a temporary emigrant. This topic is sufficiently important that it merits separate discussion in Section 19.4.1. The discussion thus far has been in terms of singleage models, but multiple-age modeling is possible as well (see Pollock and Mann, 1983; Nichols and Pollock, 1990; Nichols and Coffman, 1999). The robust design also can be used for reverse-time modeling, and in fact, age-specific modeling in reverse time actually requires the robust design (Nichols et al., 2000a). Because the robust design with age specificity (using both standard-time and reverse-time approaches) permits estimation of quantities of special biological interest, these models are discussed separately in Section 19.4.2. Multistate modeling of data from the primary periods also can be used in the robust design. In multistate modeling, the closed modeling of the secondary period data should be stratified by animals in the different observable states (Nichols et al., 1992b; Nichols and Coffman, 1999). Finally, the robust design is useful in studies that include auxiliary data (e.g., band recoveries), a topic that is discussed briefly in Section 19.4.4.
19.2.1.5. Model Selection, Estimator Robustness, and Model Assumptions Model selection follows the principles discussed in Chapters 14 and 17 for closed and open models, respec-
19.2. Ad Hoc Approach tively. If permanent trap response is believed to operate at the level of primary sampling periods, then the secondary data within each primary period should be stratified into animals caught and not caught in previous primary periods. However, within each stratum, model selection can proceed as usual. Estimator robustness in the face of variable capture probabilities for individuals was the primary motivation for development of the robust design (Pollock, 1981a, 1982). As discussed in Section 17.1.9, survival estimates based on open models tend to be quite robust to variation among individuals in capture probability (also see Carothers, 1973, 1979). Abundance can be estimated using the suite of closed-population models developed specifically to deal with individual heterogeneity in capture probabilities (Chapter 14) (Otis et al., 1978; Pledger, 2001). Under certain assumptions (no temporary emigration, equivalent behavioral responses in capture probabilities at the primary and secondary levels, etc.), robust design models may impose additional logical constraints on the capture probability parameters of the closed and open models [e.g., Eqs. (19.1) and (19.2)]. These constraints typically are expressed in terms of a relationship between the two kinds of capture probability parameters [e.g., Eq. (19.1)]. However, the independent fitting of models under the ad hoc approach was not designed to impose these constraints, and they are best handled via likelihood-based estimation (see Section 19.3).
19.2.2. Estimation Based Solely on Closed M o d e l s For an open-model treatment of primary period data, Kendall et al. (1995) excluded from their list of models those with heterogeneous capture probabilities (see Table 19.2). However, it often is reasonable to think of animals as having innate tendencies to exhibit relatively high or low capture probabilities, with these tendencies extending over the duration of a long-term study and perhaps for the life of the individual. There are no open models that permit heterogeneity in capture probabilities, so survival estimation has proceeded by assuming similar capture probabilities for all animals within the group being investigated. In this section we note that it is possible to estimate survival probabilities using ad hoc estimators based entirely on closed-population models. If models including heterogeneity are used, it then becomes possible to estimate survival rate in the presence of heterogeneous capture probabilities. Here we consider approaches to the estimation of survival probability that are based entirely on the use
533
of closed models. These estimators can be used as alternatives to Eqs. (19.4) and (19.5) to estimate ~PK-1 (or if desired, all the survival parameters). One approach is based on the closed-form estimator
/~i+1 ~i = 1 ~ i _ mi + Ri
(19.6)
from the Cormack-Jolly-Seber (CJS) open model [Eq. (17.8)]. A slightly simpler estimator was presented by Nichols et al. (1992b): ~Pi "- ~'IR-~I / a i ,
(19.7)
where/VI,R_~I denotes the number of individuals in R i that are estimated to be alive at i + 1. Estimator (19.7) simply conditions on the animals released at i and estimates the number still present at i + 1. Note that Eq. (19.7) is the standard CJS estimator [Eq. (19.6)] for sample period 1, because there are no previously marked animals at that time. The estimator in Eq. (19.7) should be less efficient than that of Eq. (19.6), and we present it only because it is easily computed, with bootstrap confidence intervals, using program COMDYN, developed by Hines et al. (1999) for community-level analyses (Chapter 20). Under the CJS approach, the estimates/~i in Eq. (19.6) are based on the open-model estimator of Eq. (17.10). However, under certain conditions Mi also can be estimated using a closed-model estimator for the probability p* of capture in primary period i. Thus, an estimator for the number of marked animals in the population just before sampling in primary period i is 1Vii = m i l ~ * .
(19.8)
Equation (19.8) is simply the canonical estimator in expression (18.6) applied to a special subset of animals (in this case, marked animals that are caught in period i). If/~I~ ;-1 is desired [e.g., for Eq. (19.7)], then the m i in Eq. (19.8) is simply replaced by m Ri-1, the number of animals caught at i that also were caught at i - 1. An estimator for p* in Eq. (19.8) is obtained in a manner similar to that used for p~ (Eq. 19.4). Let Pij be the time-specific capture probability for secondary periods under model M t, or the time-specific capture probability for an animal not previously caught in primary period i under model Mtb. Then p* can be estimated as li
fi* = 1 -
l-I(1 -fiij).
(19.9)
j=l
On the other hand, if a heterogeneity model (e.g., M h o r Mth) is used for the secondary-period data, the
Chapter 19 The Robust Design
534
p* can be estimated as the average probability of being caught at least once during primary period i:
scribed above may not be particularly useful with standard capture-recapture data. However, in special cases in which heterogeneity is believed to be extreme (e.g., in community studies; see Chapter 20), we do recommend this approach. Finally, we note that the mixture models (see Section 14.2.3) developed by Norris and Pollock (1995,1996) and Pledger (2000) to deal with heterogeneity may prove to be especially useful for both ad hoc and likelihood approaches to the robust design.
(19.10)
~* = ni/lxl i,
where/~i is based on the selected heterogeneity model (e.g., Mh or Mth) and n i is the number of animals caught at least once during primary period i. We note that the use of Eq. (19.8) to estimate the number ( M i or M Ri-1) of marked individuals in the set of interest involves estimation of p* using all animals caught during primary period i [Eqs. (19.9) and (19.10)]. The approach thus assumes equal capture probabilities for animals that were marked before primary period i and those that were not. An alternative approach that does not require the assumption of equal capture probabilities for previously marked and unmarked animals focuses only on animals that were caught before primary period i (the m i o r mRi-1). The capture histories in primary period i for this subset of animals can be used directly with a closed model estimator (e.g., program CAPTURE) to estimate M i (or M~;-1). This approach to estimation is somewhat more conservative than that of Eqs. (19.8)-(19.10) and is probably most reasonable when heterogeneity models are being used (especially when high or low capture probabilities are thought to extend over the duration of the study). In any case, estimation of M i o r M Ri-1 using any of the above methods provides the estimates needed to estimate survival [Eqs. (19.6) and (19.7)]. Thus, it is possible to estimate all of the quantities of interest (q~i, Pi, Ni, Bi) using only closed models in conjunction with capture histories over the secondary periods. Because open-model survival estimators are robust to heterogeneity of capture probabilities, the ad hoc approach de-
TABLE 19.3
Example
The robust design capture-history data presented in Table 19.1 for adult male meadow voles at Patuxent Wildlife Research Center are used to illustrate the robust design. Use of these data permits comparison with the open-model approaches of Sections 17.1.10, 18.2.6, 18.3.6, and 18.4.6. Here we present results using a robust design that combines results of modeling with both closed- and open-population models. Under the original robust design approach (Pollock, 1981a; Pollock, 1982), we used the CJS survival estimates from Table 17.7. Abundance was estimated using program CAPTURE (Otis et al., 1978; Rexstad and Burnham, 1991) with the capture-recapture data from the five secondary periods within each primary period. The discriminant function model selection algorithm of CAPTURE indicated that model Mh was appropriate for primary periods 1 and 3-6, but not for period 2. The data from primary period 2 provided strong evidence of temporal variation and behavioral response. During primary period 2, a raccoon (later caught and removed) disrupted traps on the last 2 days of sampling, leading to very small numbers of captures. We thus reanalyzed truncated capture histories from the first 3 days of trapping during primary period 2, and
R e s u l t s of S e l e c t e d Tests f r o m Program C A P T U R E for A d u l t M a l e M e a d o w Voles a Test for heterogeneity (M o vs. M h)
Closure test Primary sample period (i)
1 2b 3 4 5 6
M h goodness of fit
z
P
X2
df
P
X2
df
P
-1.30 - 1.18 0.97 0.15 -0.84 -1.78
0.10 0.12 0.84 0.56 0.20 0.04
43.31 pc 22.12 67.05 19.39 50.04
2 _ 1 2 2 2
0.10). The richness estimate for 1990 was smaller than that for 1970, although the 95% confidence intervals for the two estimates overlapped substantially (Table 20.4). The estimated rate of change based on these richness estimates was 0.88, but the confidence interval for rate of change included values >1.0. The average detection probability of 0.83 for 1970 and 0.79 for 1990 provided little evidence of a difference (X2 = 5.86, P = 0.11), justifying the use of Eq. (20.9) for estimating rate of change. The resulting rate estimate of 0.85 was similar to that based on Eq. (20.8) but was substantially more precise. The confidence interval for the estimate was (0.73, 1.0) (Table 20.4), providing evidence that avian richness declined on the route between 1970 and 1990.
Species Detection Statistics for Wisconsin BBS Route 1, 1970 and 1990
Species group
Species detected
Total species detected (1970), R70 Total species detected (1990), R90 Members of R70 detected in 1990, (m90 I R70) Members of R90 detected in 1970, (m70 I R90)
66 80 57 57
Species detected on exactly h of the five groups of stops (fh) fl
f2
f3
f4
f5
17 23 9 10
19 15 10 17
9 9 6 9
10 12 11 10
21 21 11
11
568
Chapter 20 Estimation of Community Parameters TABLE 20.4
Estimates of Quantities Associated with Community Dynamics Based on Avian Species Seen on Maryland BBS Route 25, 1970 and 1990 Naive
Quantity (0)
"estimates "a
Estimator
Species richness (1970)
65.0
Species richness (1990)
55.0
95% c o n f i d e n c e interval
0
S-E(0)
N70
78.5
10.4
66.9-104.1
/~/90
69.4
11.1
55.6-94.3
^
Members of R70 present in 1990 b
48.0
(/~90 I R70)
54.7
13.2
35.0-86.0
Members of R90 present in 1970 b
48.0
(/~70 I R90)
51.1
6.9
38.5-67.1
Complement of extinction probability
0.74
q~70,90
0.84
0.15
0.54-1.00
Complement of turnover
0.87
q~90,70
0.93
0.09
0.70-1.00
Rate of change in richness [Eq. (20.8)]
0.85
K70, 90
0.88
0.18
0.61-1.28
Rate of change in richness [Eq. (20.9)]
0.85
K70, 90
0.85
0.07
0.73-1.00
Number of colonizing species [Eq. (20.14)]
870, 90 -~ P70
3.3
Average detection probability (1970)
7.0 1.0 c
0.83
0.10
0.62-0.97
Average detection probability (1990)
1.0 c
-~ P90
0.79
0.11
0.58-0.99
12.2
a "Estimates" based on the assumption that all species are detected. b Confidence intervals for (l~/ljla i) can include values 0, or minimizing a + bF(x) for b ~ 0. Thus, it is possible to transform a problem of the form minimize
F(x)
subject to xeX
into the equivalent problem maximize
-F(x)
subject to xeX. This transformation does not affect the optimizing solution, because both problems produce the same optimal value x*. In what follows we occasionally use this fact to transform problems involving, e.g., cost minimization, into maximization problems.
22.1. THE GEOMETRY OF OPTIMIZATION The identification of optimizing values involves a search among alternative values of x that lie in the opportunity set X. The opportunity set typically is defined by nonnegativity constraints x -> O, along with
and hi(x) ~ b1 9
o
hs(x) 0. In Fig. 22.1a the opportunity set includes all nonnegative numbers, and the objective function displays both a local maximum at x** and a global maximum at x*. Imposing the inequality constraint x -> b restricts the range of allowable values for x, as shown in Fig. 22.1b. In this case only one of the maximizing values is retained in the opportunity set. Unconstrained optimization involving two decision variables and a nonlinear objective function is displayed in Fig. 22.2. The nonlinear nature of the objective function is indicated by contours, i.e., curves defined by {x I F(x)_ = constant}, and by the gradient vector cgF/Ox = (3F/OXl,0F/3x 2) indicating the direction of steepest ascent in the objective function. Starting at some point x in the opportunity set, an efficient search procedure for locating the optimal value x* would move in the direction of 3F/Ox. Figure 22.3 displays a nonlinear objective function and nonlinear equality constraint g(x) = 0. In this case the feasible values for x* consist of points on the curve corresponding to the constraint. An optimal solution is found at the point of tangency between the constraint and a contour of the objective function (the point at which the gradients 0F/0x and 0g/0x are coincident). m
22.2. Unconstrained Optimization
585
In w h a t follows we make liberal use of differential methods, involving differentiation of both the objective function and the system constraints. As a matter of notational convenience we adopt the convention that differentiation of F(x) with respect to the column vector x produces a row vector OF/Ox = (OF/Ox 1, 3F/Ox2, ..., OF~ 0xn). Conversely, differentiation of G(_K)with respect to the row vector )~ = ()~1, ~2, ..., )~n) produces a column vector OG/O)~ = (0G/0)~ 1, 3G/OK2, ..., 3G/O~m)'.
22.1.1. Convexity Requirements
FIGURE 22.1 An optimization problem involving a single decision variable x. (a) Over the range of all positive values for x, local maxima can be identified at x* and x**.(b) Over the more restrictive range of values x - b, a single maximum can be identified at x*.
At that point the value of the objective function is greater than at any other point on the constraint curve. In Fig. 22.4 a nonlinear objective function is constrained by a set of linear inequality constraints. Figure 22.4a shows an optimizing value at a b o u n d a r y of the opportunity set. Under these conditions a change in the active constraints w o u l d lead to a different optimizing value x*. Figure 22.4b shows an optimizing value x* that is interior to the opportunity set X. In this case a change in the boundaries of the opportunity set w o u l d not affect the optimal choice of x*. Finally, Fig. 22.5 exhibits a linear objective function along with linear inequality constraints. Either a unique value x* maximizing F(x) is found at a vertex of the opportunity set X (Fig. 22.5a), or else optimal values for _x are found at any point along a b o u n d i n g surface of X (Fig. 22.5b). In either case a change in the active constraints w o u l d affect the choice of an optimizing value for x.
It is useful to describe the property of convexity as it applies to sets and functions, preparatory to developing the optimization procedures. If certain geometric properties involving convexity obtain for the objective function and opportunity set, then an optimization problem is assured of having a solution. Furthermore, u n d e r certain convexity conditions a local solution is guaranteed to be global over the opportunity set. Set convexity captures the idea that if x I and x 2 are two points in X, then any point on the line joining them also is in X. The property is expressed mathematically as follows: the set _X is convex if axl + (1 - a)x 2 _X for all x I E Xr X2 (~ X~ and a ~ [0,1]. Thus, there can be no "depressions" along the surface of a convex set, for then it w o u l d be possible to exit the set in moving along a line from one side of the depression to the other (Fig. 22.6). Function convexity also concerns the geometry of lines joining two points in X (Fig. 22.7). Thus, the function F(x) is convex if its value at points along a line segment in X is less than the corresponding average of values at the segment endpoints. In mathematical parlance, F(x) is convex on X if F[ax I + (1 - a)x 2] aF(x 1) + (1 - a)F(x 2) for all x__1 E Xr X2 E Xr and a e [0,11 (Fig 22.7a). By analogy, a concave function is defined by simply reversing the inequality sign: G(x) is concave on X if G[ax I + (1 - a)x 2] -~ aG(x 1) + (1 - a)G(x 2) for all X1 E----XIX2 (~ Xl and a e [0,11 (Fig. 22.7b). In mathematical p r o g r a m m i n g problems, concavity (convexity) in the objective function over convex X is sufficient to guarantee that a local m a x i m u m (minimum) is also global.
22.2. UNCONSTRAINED OPTIMIZATION In this section we describe optimization procedures for problems in which a vector of decision variables is chosen to maximize an objective function F(x), given that allowable values for the decision variables are not constrained. We assume in w h a t follows that the
586
Chapter 22
Traditional Approaches to Optimal Decision Analysis
FIGURE 22.2 An optimization problem involving two decision variables x 1 and x2 and a nonlinear objective function F(x). The nonlinear nature of F(x) is indicated by contours of the form F(x) = c. m
o b j e c t i v e f u n c t i o n is t w i c e d i f f e r e n t i a b l e w i t h c o n t i n u ous derivatives. \ \ \
22.2.1. Univariate D e c i s i o n Problem
\ \
\ \
W e b e g i n w i t h t h e o p t i m i z a t i o n p r o b l e m of c h o o s i n g a v a l u e of a s i n g l e d e c i s i o n v a r i a b l e x to m a x i m i z e F ( x ) . A s s u m i n g t h e a b s e n c e of c o n s t r a i n t s o n x, t h e firstorder stationary condition
X X\
\ \ \
m
\
\
~ L..
~ --- -._ ~ x_.)=o. . . .
~- -"- - F(x)=cI F(x)--c3
x_*
FIGURE 22.3 An optimization problem involving two decision variables xI and x2, a nonlinear objective function F(x), and a nonlinear equality constraint g(x) = c. The optimizing value of x on the constraint is located at a point of tangency between the constraint and a contour of F(x).
d~ (x*) = 0 dx m u s t b e s a t i s f i e d for x* to b e m a x i m i z i n g ( A p p e n d i x H). H o w e v e r , f i r s t - o r d e r s t a t i o n a r i t y is n o t b y itself s u f f i c i e n t to e n s u r e a m a x i m u m , b e c a u s e t h e d e r i v a t i v e a l s o v a n i s h e s for l o c a l m i n i m a a n d i n f l e c t i o n p o i n t s (Fig. 22.8). A d d i n g t h e s e c o n d - o r d e r c o n d i t i o n d2F d x 2 (x*) < 0
(22.1)
22.2.
Unconstrained Optimization
587
X2
x2
a
\
\
\
x
\
\ \
x
\
a
x \
\
\
x \
\
\ x
\
\
N \
\
"\
\
N
: X\\
"'.i
\ \
\
\
XX x
\
\
\
,
---..
\
\\
"'F(x)=c'
X''"
\
\
.
X
f(x)=c2
\
\
! \\ i: \ \\ \
\
\
-i..,, \ " \, , i \
""""-.-... F(x)=c3
i
\\
\
\\
,, \
\
\
\\
\
_x
_X
\,
x2
b
\
x2
\
b \
\
\
---------_______
\
\ \
\
\ \
\
\
\
\
\
\
\
\
\
\
"\\~NN N \\\
_LL-_LL-. _," /
..------~
/ /
..---...
\\\ \
\
\
\
,,
X
\\
"',iN "q \\ ", \\ ? \\X"
\
.
._L
...``/ / /
k x*
F I G U R E 22.4 An optimization problem involving two decision variables x I and x 2, a nonlinear objective function F(x), and a set of linear inequality constraints {g(x) < c}. (a) An optimizing solution at a boundary of the opportunity set. (b) An optimizing solution in the interior of the opportunity set.
F I G U R E 22.5 An optimization problem with linear objective function and linear inequality constraints. (a) The objective function is maximized at a point of intersection of the boundary constraints. (b) F(x) is maximized by any point along the edge of a boundary constraint.
Example
eliminates the possibility of a minimizing value or inflection point, so that together the first- and secondorder conditions guarantee that x* is a local maximum. Note that a local m a x i m u m identified by first- and second-order optimality conditions is not guaranteed to be a global m a x i m u m (Fig. 22.1). However, if (d2F/dx2)(x) 0 and x 2 >_ 0. Partial differentiation of the objective function yields 3 F / O x I = - 4 x I + x2 + 6
and
constraints, with modifications to account for the nonnegativity constraints as in conditions (22.13)-(22.15). After some algebra (see Appendix H) the conditions for optimality can be written as OF (x*)
0x -
k* Og - 3x(X*)---0' b - g ( x * ) ~ O,
3F ag ] 7x (x*)_ - _x* ox-(X*) _x* = 0,
(22.16)
m
x*Eb - g(x*)] = 0, 3F/Ox 2 = xI
6X2 -- 13.
--
X* --> 0, m
The conditions (22.13)-(22.15) above present four possibilities, d e p e n d i n g on whether the optimal stocking levels x~ and x~ are nonzero: 1. If x~ = x~ = 0, then F(x*) = 4. 2. If x~ = 0 but x~ > 0, then condition (22.14) requires c]F/Ox 2 = O, so that Xl-6X 2 = 13 or x 2 = - 1 3 / 6 , violating the nonnegativity condition (22.15) for x 2. 3. If x~ > 0 but x~ = 0, then condition (22.14) requires c]F/Ox I = O, so that 4x 1 - x2 = 6 or x I = 1.5 with F(0, 1.5) = 8.5. 4. Finally, if x~ and x~ both are nonzero, then 3 F / O x 1 = 3 F / 3 x 2 = 0 and x2 = - 4 6 / 2 3 . Again, this violates the nonnegativity condition (22.15). E
It follows that the optimal stocking regime will involve the stocking of only one species, at a level of 1.5. The total cost for stocking at this level will be g(x) = $11,500, leaving a funding residual of $8500 for other uses.
22.4.2. Nonlinear Programming with General Inequality Constraints In this case the optimization problem is maximize
F(x)
subject to g ( x ) 0. Conversely, x~ must be nonnegative and is necessarily zero if O F / 3 x j - ~i )kiOgi/ c~xj < 0 atx*._ 9 Similarly, either k~ = 0 or gj(x*) = bj (or both) for each j = 1.... , m. The Kuhn-Tucker conditions require that gj(x*) 0. Conversely, k~ must be nonnegative and is necessarily zero if gj(x*) < bj. 9 On assumption that there are no inequality constraints, the Kuhn-Tucker conditions reduce to conditions (22.13)-(22.15) for optimization with nonnegative conditions only. This confirms that classical programming can be s u b s u m e d as a special case of general nonlinear programming. 9 If the constraints define a convex opportunity set and the objective function is concave, the Kuhn-Tucker conditions are sufficient to guarantee a global maximum. 9 As before, the optimal Lagrangian multipliers can be interpreted in terms of a marginal change in the
22.4. Nonlinear Programming objective function with respect to the constraint coefficients: OF (x*) = )~* 0b-"
(22.17)
599
at (x*, _k*) for each constraint j. It follows that the Kuhn-Tucker conditions for this problem are 300
-
2x I -
3
500 -- 4x 2 -- 5
m
Example
-
)kI = 0,
-- k I = 0,
- 2 3 + kI - k2 - 0 ,
Consider a situation in which two logistic populations are to be m a n a g e d in an animal caretaker facility. The annual population growth for each population is AX 1 -- 3x1(1 -- X l / 3 0 0 ) a n d AX 2 = 5X2(1 - X2/250), where x I and x 2 represent population sizes. The populations are to be maintained at constant size, and the growth increment is to be sold at $100 per individual each year. Each individual in population 1 requires about $3 a year for maintenance, and each in population 2 requires about $5 per year. Facility x 3 capacity can be e x p a n d e d to accommodate up to 400 individuals and rent and other annual facility costs are expected to total about $23 per unit capacity. M a n a g e m e n t needs to k n o w h o w large a facility to develop and w h a t the population sizes ought to be, in order to minimize net costs. The problem can be expressed as maximize
F(x)
= X 1 ( 3 0 0 -- X1) + X 2 ( 5 0 0 -
3x I
-- 5X 2
2X2)
- 23x 3
subject to X3 ~> X 1 + X2r X3 ~
0, X2 ~
0,
X3 ~ > 0 ,
-
X1) + X 2 ( 5 0 0 -
2X2) -- 3X1
2 3 x 3 + ~.l(X 3 -- x I -- x2) + k 2 ( 4 0 0 -
-- 5X 2 x3).
A s s u m i n g for now that the nonnegativity conditions on x are met by an optimum, the equalities in expression (22.16) require that
ag}
aF
COXi
E J
k2(400
-- X3) = 0, KI-->0, k2>--0.
An examination of possibilities for K 1 and k 2 s h o w s that they both cannot be zero, because that would violate the third condition above. Nor can we have k 1 -0 and k 2 > 0, because the third condition would then require that ~-2 -- --10. Consider the case in which K 2 = 0 and ~1 > 0. From the third condition we have K 1 = 23, so that x I = 142, x 2 = 103, and x 3 = 245 from the first, second, and fourth conditions, respectively. Thus, an optimal decision is to limit capacity below allowable limits and maintain populations at levels of 142 and 103 individuals. This example is unusual, in that it is possible to identify an optimal solution directly from the K u h n Tucker conditions. In general, the Kuhn-Tucker conditions tell us about the mathematical nature of an optimal solution, but by themselves are not particularly useful in helping to find one. It usually is necessary to take advantage of the differential properties of the objective function and constraints in a procedure that accounts explicitly for the constraints and leads in sequential steps to a local optimum.
22.4.3. Solution Algorithms for Constrained Nonlinear Optimization
and the Lagrangian is
L(x, k) = x 1 ( 3 0 0 -
K I ( - - X 1 -- X2 + X3) = 0,
)kJ~xi -" 0
at (x*, _k*) for each state variable xi, and Xj[bj - gj(x)]
= o
As with unconstrained optimization, the methods for constrained problems almost always involve an iterative search for an o p t i m u m x*, each step of which consists of choosing the "best" direction, and determining the length of the step to be taken in that direction. Though a mathematical description can be difficult, in concept the specification of such an iterative algorithm is straightforward. To illustrate, in w h a t follows we focus on gradient or "gradient-like" searches for constrained optimization, recognizing that these are but a few of m a n y approaches that are available (See Appendix H). At each iteration of a gradient-based procedure, the direction of search is initially chosen as the gradient
600
Chapter 22 Traditional Approaches to Optimal Decision Analysis
VF(xk) = OF(Xk)/Ox. A generic algorithm includes the following steps: 1. Identify a feasible starting value x 0 (this may or may not be a simple task, depending on the constraints). For opportunity sets that include the origin, a possible starting value (though not necessarily a good one) is 0. 2. Move in the direction of the gradient VF(x 0) for a distance determined by a selected step size 80 and thereby locate a new feasible point X.1 with F(Xl)>F(x0). 3. At iteration k, move in the direction of the gradient VF(xk) for a distance determined by step size 8k and identify a new feasible point Xk+1 with F(Xk+l)>F(Xk). 4. Repeat until a stopping criterion is satisfied. Key issues for such a search algorithm are (1) the choice of an appropriate step size at each iteration, and (2) the choice of a search direction that remains in the opportunity set. Difficulties arise when a step of size 8k in the direction of the gradient leaves one outside the opportunity set, or when movement along the gradient takes one immediately outside the set (e.g., x k is on the boundary of X and VF(xk) points away from X). The added complexity attendant to searching under constrained optimization is directly related to these situations. Three common approaches to them are the gradient projection method, the method of feasible directions, and the Lagrangian differential gradient method. Gradient projection is based on a suitable modification of gradient search (see Appendix H.1.4) to account for the constraints. It starts with an initial value x 0 in the opportunity set X_ and moves at each step in the direction of the gradient of F, provided that direction remains in _X. If at some step in the iteration, the gradient direction is infeasible [i.e., if x k is on a boundary of X and VF(xk) points away from X], the direction of movement is altered to follow the projection of the gradient vector on the tangent to the boundary of X. The corresponding step size is chosen to increase the value of the objective function while remaining in the opportunity set. Iterative application of the algorithm can be shown to converge to x*, provided the objective function is concave and the opportunity set is convex. An alternative approach is the method of feasible directions, which involves choosing a direction D k that deviates as little from VF(xk) as possible, while ensuring that at least some movement in that direction is possible. If the operative constraints are linear, under some rather mild conditions on the normalization of candidate directions D k, a feasible direction can be found at each step via linear programming (Luenberger, 1989). The corresponding step size in the direction of D k typically is determined by the nearer of (1) the point where
the direction vector leaves the opportunity set X, or (2) the point at which F(x) reaches a maximum in the direction of D k. Yet another approach to constrained optimization is the Lagrangian differential gradient method, which uses gradient search with the Lagrangian L(x, ~) rather than the objective function F(x)._ In this case the algorithm begins at an arbitrary initial value x 0 and moves from that point according to the gradient components OL/Ox and OL/3K of the Lagrangian; hence the name "Lagrangian differential gradient method." If the objective function is concave and the inequality constraints are convex, the procedure converges to optimizing values of x and _Kstarting at arbitrary values of these variables. Algorithms such as gradient projection, the method of feasible directions, and the Lagrangian differential gradient method require evaluation of partial derivatives of both the objective function and the inequality constraints each time a new search direction is determined. The corresponding computational requirements increase quickly with increasing numbers of decision variables and constraints and with increasing mathematical complexity. A further challenge is to ensure that a value x* thus identified is in fact a global maximum rather than a local maximum. Recognizing global maxima becomes much more difficult as the number of decision variables and constraints increases, and especially as the mathematical complexity of the problem increases. There is a wide variety of different approaches to constrained optimization, in addition to the methods mentioned above. Frequently cited procedures include (1) primal methods, in which the problem constraints are used to reduce the dimensionality of a search for an optimal value x*; (2) penalty and barrier methods, involving the approximation of a constrained optimization problem by an unconstrained problem, which then can be solved with procedures for unconstrained optimization; (3) dual methods that focus on the Lagrangian multipliers as the fundamental variables to be optimized, with the idea that determining optimizing values for the Lagrangian multipliers is tantamount (at least in some cases) to finding the optimal solution x*; and (4) Lagrangian methods that focus on simultaneously solving for the optimizing values of the decision variables and Lagrangian multipliers in the Lagrangian function. Many of these procedures are adapted from procedures for unconstrained problems (see Appendix H.1.4). In general, their rates of convergence are controlled by the structure of the Hessian matrix of the Lagrangian, much as convergence rates for unconstrained problems are controlled by the Hessian of the objective function (Luenberger, 1989). m
m
22.5. Linear Programming
601
22.4.4. Summary In this section we have described procedures for nonlinear programming, extending the classical programming problem of the previous section by including inequality constraints on allowable values of the decision variables. Thus, nonlinear programming procedures are used to identify a vector of decision variables that maximizes an objective function of them in the presence of inequality constraints and nonnegativity conditions. The objective function and constraints must be continuously differentiable in the decision variables. Solution approaches involve the following considerations: 9 Lagrangian multipliers _~ are introduced to account for the inequality constraints and are included along with the original objective function in a Lagrangian function. 9 Differentiation of the Lagrangian function with respect to x and _h leads to derivation of the Kuhn-Tucker conditions (22.16). 9 The optimal Lagrangian multipliers ~* describe sensitivities of the optimal value F(x*) of the objective function to changes in the constraint constants of the inequality constraints [Eq. (22.17)]. 9 On condition that the objective function is strictly concave and the constraints describe a convex opportunity set, a local optimum is also global. 9 Many approaches are available for finding an optimizing value x*, depending on the mathematical structure of the problem, the dimensionality of the decision space, and the nature of constraints defining the opportunity set. In most cases, computer-based search procedures must be used to identify values (x*, h_*) satisfying the Kuhn-Tucker conditions. m
22.5. LINEAR PROGRAMMING Linear programming is a special case of nonlinear programming, in which both the objective function and the constraints are linear combinations of the decision variables. A statement of the problem involving n decision variables and m constraints is maximize
a
x_>0, where c = (c 1, c2, Cn)' is a vector of constants in a linear objective function, b = (bl, b2, ..., bin)' is a vector of constraint constants, and
...
aln
a21
a22
...
a2n
am1
am2
...
amn
__
As before, the nonnegativity constraints _x -> _0 restrict feasible solutions to the nonnegative orthant of En. Additional restrictions are imposed by the linear constraints a i l x I + ai2x2 + ... + ainXn ~ b i
in A x ,, I/d/---"
II
//
tl / / . / Xo~ ~ /
-%\ \
\.
\
\
\
\
FIGURE 23.2 Multiplestate trajectories from (0,1) to the terminal curve x(t) = 2 - t. The trajectory of minimum length is a straight line that is perpendicular to the terminal curve.
23.1.
[0, 2]. Because the optimality index is of the form I(x, ic), Eq. (23.3) is operative. We then have OI/Oic = Yc + x + 1, and c = I
613
C a l c u l u s of V a r i a t i o n s
ic = - t + U(t) that the optimal control is linear over time: U(t) = 2 t + 1. 9 I = I(x, t). In this case the objective functional d e p e n d s on the s y s t e m state, but not on the change in s y s t e m state. Then Oi/Oic = 0 so that Euler's equation simplifies to
0I. a-x
= ( i c 2 / 2 4- xic 4- ic 4- x) -- ic(ic 4- x 4- 1)
OI/Ox = O.
= - - i c 2 / 2 4- x.
Differentiation of this expression with respect to x yields 5/= 1, so that x(t) = (t2/2) 4- clt 4- c 2. The initial and terminal conditions can be used to d e t e r m i n e c 1 a n d ca, by x(0) = ca = x 0, and x(2) - xf = 2 + 2c 1 4x 0 or c I = (xf - x0)/2 - 1. For the particular case in which, e.g., (x 0, xf) = (1, 3), this gives x(t) = t 2 / 2 + 1 as an o p t i m i z i n g p o p u l a t i o n trajectory for the problem. From the transition equation ic = [ - x ( t ) - 1] + U(t) the c o r r e s p o n d i n g optimal control strategy is given by U(t) = ic + x + 1 = t 2 / 2 + t + 2.
9 I = l(ic, t). In this case the objective functional d e p e n d s on the time rate of change in s y s t e m state, but not on the s y s t e m state. Then 0 I / O x = 0, and Euler's equation becomes o
(23.5)
But this is s i m p l y an u n c o n s t r a i n e d o p t i m i z a t i o n problem, involving the choice of x to o p t i m i z e the value of I. Thus, an optimality index of the form I(x, t) allows one to solve the calculus of variations p r o b l e m by solving a series of traditional o p t i m i z a t i o n problems, one for each time in the time frame. Example. M a n a g e m e n t seeks to m i n i m i z e the deviations from a target p o p u l a t i o n trajectory a(t), while also m i n i m i z i n g time-specific costs c(t)x(t) associated w i t h p o p u l a t i o n size. An optimality index for this p r o b l e m is I = Ix(t) - a(t)] 2 + c(t)x(t), a n d because it is of the form I(x, t), w e use Eq. (23.5) to get OI/Ox = 2[x - a(t)] + c(t) = 0 or x(t) = a(t) - c ( t ) / 2 . This suggests that the optimal trajectory tracks the target a(t), with modifications at each point in time based on per capita costs
c(t).
or
23.1.4. General Multivariate Problem OI
Oic
= c,
(23.4)
In its classical multivariate form, the calculus of variations p r o b l e m is
w h e r e c is a constant that is d e t e r m i n e d from the initial and terminal conditions.
f'
maximize Ix(t)}
Example. Consider a p o p u l a t i o n that declines in the absence of control as a linear function of time: ic = - t + U(t). A control trajectory is desired that will minimize costs over [0, 2], according to objective functional
J =
f
I(x, 2, t) dt
to
subject to x(t 0) = x0,
2 U2
x(t ) - x ,
- ~ dt. o
w h e r e x' = ( X l , . . . , Xk). Note that the m u l t i v a r i a t e nature of the p r o b l e m allows considerable flexibility in the form of the objective functional. For example, the optimality index can be a function of some, all, or none of the state variables a n d / o r their time rates of change. O p t i m a l i t y conditions for the multivariate p r o b l e m are completely analogous to those for the univariate problem. In particular, the multivariate version of Euler's equation is w
Substituting U(t) = ic + t into the optimality index produces I = ic2/2 4- tic + t 2 / 2 , w h i c h is of the form I(ic, t). We therefore use Eq. (23.4) to get OI/Oic = ic + t = Cl, or x(t) = - t 2 / 2 + clt 4- c 2. As before, initial and terminal conditions can be used to d e t e r m i n e c I a n d c2: x(0) = c2 = x 0 and x(2) = xf = - 2 4- 2c 1 4- x 0 or c I = 1 + ( x f - x0)/2. For (x 0, xf) = (1, 3), this gives x(t) = - t 2 / 2 + 2t + 1 as an o p t i m i z i n g p o p u l a t i o n trajectory for the problem. It follows from
Ox
dt
_
-"
614
Chapter 23 Modern Approaches to Decision Analysis
involving k equations, one for each of the state variables. The corresponding transversality conditions for initial and terminal times are
which can be combined with the constraints to characterize a solution. Example
0
(23.7)
for t = t o, tf. A state variable trajectory {x(t)} that satisfies Euler's equation and the initial and terminal conditions x(t 0) = x 0 and x(tf) = xf is called an extremal, and the optimal solution for a calculus of variations problem with specified b o u n d a r y conditions is necessarily extremal. Note that an extremal trajectory in the calculus of variations plays a role analogous to that of a stationary point satisfying aF/ax = 0 in mathematical programming.
23.1.5. Constraints It is possible to incorporate certain kinds of constraints in the calculus of variations problem. In particular, equality, inequality, and integral constraints can be handled by straightforward extensions of Euler's equation.
A population with linear transitions ~ = 2x + U / 2 is to be m a n a g e d so as to minimize l =
-~- dt, o
while ensuring that the population grows from x 0 to xf over 1 year. One approach is to use the transition equation to transform this problem into the standard calculus of variations format, as described above. Another is to treat the control variable U as another state variable, with x = x I and U = x 2. Then the system transition equation is 21 = 2x I 4- x 2 / 2 , which can be handled as an equality constraint and incorporated into the objective functional with a time-varying Lagrangian multiplier: L = x 2 / 2 + h(2x I 4- x 2 / 2 -
21).
Euler's equation for the problem then becomes
23.1.5.1. Equality Constraints
ax
dt
~_x
A statement of the optimization problem that includes equality constraints is
=
k x 2 4-
h/2
=0, which gives
maximize {x(t)}
I(x, 2, t) dt ~( = -2X
to
subject to
X 2 -"
-X/2
21 = 2X 1 + X 2 / 2 .
g(x, 2, t) = a, x(t 0) = x0,
x(t;~) = x~. As in mathematical programming, a solution approach involves a set of Lagrangian multipliers K_ = (X1, ..., kin), one for each of the constraints in g(x, 2, t) = a. The a u g m e n t e d optimality index is
The first equation yields X - Cl e-2t, SO that x 2 = -- c l e - 2 t / 2 from the second equation and 21 -- 2X 1 -- c l e - 2 t / 4 from the third. The latter equation is solved by x(t) = [16c2e2t - cle-2t]/16, with the parameters c 1 and c2 determined by the initial and terminal conditions: x 0 = c2 - Cl/16 and xf = c2e 2 - c l e - 2 / 1 6 .
m
L(x, 2, X, t) = I(x, 2, t) - )~[a - g(x, 2, t)],
Example Consider the optimal control of the linear system_x = A x + B U to minimize a quadratic objective functional
and a solution is obtained by maximizing J' =
f'
J = 1/2 g(x, 2, )t, t) dt.
to
As before, this leads to Euler's equation,
Ox
d-t\~_~/
-'
[ U ' R U + x ' Q x ] dt, to
subject to the constraints x(0) = x 0 and x(1) = x f (assume without loss of generality that R and Q are symmetric matrices). The Lagrangian for this problem is L = (U' R U + x ' Q x ) / 2
+ h(Ax + BU-
2),
23.1. Calculus of Variations and Euler's equations are o, Ox
dt\O2_J
x,
615
and augmenting the objective functional by means of time-varying Lagrangian multipliers gives Q + h A + ~_ =
o,
1
f0 [x2/2
+ ~.1(x2- 21) 4- ~.2(x3- 22) ] dt.
and Euler's equation then is o, OU
a_(oL dt\OU]
= u'R
+
= o' -"
OL'
OX
The optimal solution is therefore characterized by the system of linear differential equations (see Appendix C)
dt\02]
~'1 4- ~2 X 3 4 - K2
--
=0,
2=Ax+BU,
)~A,
~. = - Q x -
or ~.1 =
0,
with the control trajectory for _U in this system given in terms of the Lagrangian multipliers:
J~2 = --~1,
U = - R -1 BK'.
X3 = -- h 2.
A minimizing solution depends on the existence of an inverse for _Rand also requires that _Rand Q be positive definite. On condition that an initial value x(t o) is specified but x(tf) is not, the transversality condition specifies that (OL/O2)(tf) = K(tf) = 0', and identification of an optimizing control requires the solution of a twopoint b o u n d a r y value problem.
From the first of these equations ~'1 -- Cl, from the second equation ~'2 -- --clt + c2, and therefore the third equation gives x 3 = clt - c 2. From the transition equations we then have x 2 = c l t 2 / 2 - c2t 4- c 3 and x 1 = c l t 3 / 6 - c2t2/2 + c3t + c 4. The initial and terminal conditions can be used to solve for the constants c 1, c2, c3, and c4 in the equation for x 1, producing xl(t) = 3t 3 - 5t 2 + t + 1. The first derivative of x I gives the instantaneous rate of growth as x2(t) = r(t) = 9t 2 10t + 1, and the second derivative of x I gives the optimal control as x 3 = U = 18t - 10. From r(t) = (9t - 1)(t - 1) it is easy to see that the instantaneous rate of growth decreases from r = 1 to r = 0 at t = 1/9, declines yet further to r = - 1 4 / 9 at t = 5/9, and then increases to zero at t = 1 (Fig. 23.3). In response, the population increases for t ~ [0, 1/9] and then decreases to zero at t = 1. At first glance it may seem counterintuitive that an optimal strategy to eliminate the population w o u l d allow it to increase over part of the time frame. Recall, however, that the population was ass u m e d to be increasing initially, with 2(0) = 1. T h o u g h the optimal population growth rate begins immediately to decline from unity, a small increment of time is necessary before the growth rate becomes negative and the population begins to decline.
Example
Consider an exponential population for which initial and terminal growth rates are 1 and 0, respectively. The population rate of growth is to be controlled directly, with an objective of eliminating the population in 1 year. Thus, effort U(t) is to be applied over the interval [0,1] to influence the rate of change r according to d 2 x / d t 2 = d r / d t = U. Note that this is a s o m e w h a t different formulation of the control problem, in that the instantaneous rate of growth parameter is controlled rather than the population. The objective is to minimize l =
flu2
-~- dt
0
subject to initial and terminal conditions on both x and 2, as specified by x(0) = 2(0) = 1 and x(1) = 2(1) = 0. The problem can be formulated in terms of the calculus of variations by changing notation to x I = x and x 3 = U and introducing another variable x 2 = r such that 21 --- X2 and 22 = x 3. It is easy to see that (d/dt)(21) = X2 = X3 or x1 -- X3, which is equivalent to d2x/dt 2 = U in the original problem statement. System dynamics are expressed in matrix form by
[~:] = [~
10][;:] 4- [~] x3,
23.1.5.2. Inequality Constraints A Lagrangian approach can accommodate inequality constraints of the form g(x, 2, t) O.
and =
U 1
Because the controls fluctuate between the maximum and minimum allowable values for U(t), depending on the sign of the switching function a + K(t)b, a solution is termed bang-bang control. Example
Consider a predator-prey system in which prey numbers are to be controlled through the use of a pesticide. Control is expressed as a percentage of the prey population that is targeted for removal, with adjustments for effectiveness. Population transitions are described by a modified version of the Lotka-Volterra equations X1 =
X1(1
- x 2)
9C2 - - X 2 ( X 1 - -
-
bx lu,
1),
where the control term indicates that pesticide application is linear in its effect on prey. In the absence of pesticide application, the population fluctuates about the equilibrium point (x 1, x2) = (1, 1), with amplitudes determined by the initial system state (see Appendix C). An objective of management is to bring the populations to equilibrium with minimum application of the pesticide, i.e., to minimize J =
f,,
to
23.2.3.3. Singular Controls u dt
with 0 -< u -< U m a x over the time frame. The Hamiltonian for this problem is H = u + ~1[x1(1
- - X2) - -
Because the control u does not appear in OH/au = 1 - - K l b X l , setting the latter to zero does not provide a solution to the control problem. However, the maxim u m principle indicates that the optimal control must maximize H at each point in the time frame, which in turn means that u takes a value of either 0 or U m a x , depending on the sign of 1 - K l b X 1. Determination of the actual switching strategy depends on the trajectories of Mt) and x(t), which are difficult to derive based on the canonical equations. In this particular instance, however, it is possible to deduce the appropriate strategy from knowledge about the behavior of the Lotka-Volterra system. When forced by maximum pesticide application, the system oscillates indefinitely about a new equilibrium point that is defined by the parameter b (see Appendix C). The pattern of these oscillations of course depends on the state of the system at the time when pesticide application begins. It is easy to see that only one oscillation pattern for the forced system will include the equilibrium point (1,1) of the unforced system (Fig. 23.4). Provided Umax is sufficiently large, that oscillation pattern will coincide with oscillations of the unforced system in at least one point. Assuming the initial system state is not a point of intersection between the forced and unforced trajectories, the optimal strategy is to leave the system unforced until it evolves to a point of intersection and then apply the pesticide at Umax until equilibrium (1,1) is attained. Because of the oscillatory nature of the system, optimal application must continue at the level U r e a x u p to the time at which equilibrium is attained; otherwise, predator and prey numbers will immediately begin a new pattern of indefinite oscillations and never attain equilibrium.
bXlU ] +
Ka[X2(Xl
--
1)],
In the previous example of a predator-prey system, the switching function is nonzero almost everywhere in T, so that a bang-bang control strategy is optimal over the entire time frame. However, in many cases the switching function can vanish over a nonzero interval
23.2. Pontryagin's Maximum Principle
forced system
250
\f ~,
_[
623
N--~
o "~ 200 r i.. El.
~-~...___._____.~
150
0
,
200
4;0
600 prey
8;0
10'00
,
1200
,,
N
1
FIGURE 23.4 Oscillations for forced and unforced Lotka-Volterra predat o r - p r e y systems. Starting at N 0, the optimal strategy for a m i n i m u m effort objective is to leave the system unforced until it evolves to N 1, and then switch to m a x i m u m control until the equilibrium N* is attained.
of time. Under these circumstances the control problem is said to be singular, and extremal control values are insufficient to describe an optimal control strategy over the time frame. Solution approaches to singular control problems typically require the interpretation of the state a n d / o r costate equations (23.11) to identify the pattern of optimal controls.
Example Previous investigations suggest that the population carrying capacity of a habitat undergoes periodic fluctuations, which can be approximated by the trajectory a(t). Management wishes to control population changes so as to minimize deviations from this trajectory, while recognizing limits on the level of available control. The optimal control problem can be stated as maximize
- ~ T [ x ( t ) - a(t)] 2 dt d o
U m i n -< U -< U m a x
subject to ~=U. The Hamiltonian for this problem is H = - I x - a] 2 + KU, with the state and costate equations given by ~ = OH/3K = U and ~, = - 3 H / O x = 2(x - a), respectively. The Hamiltonian is maximized by
U(t)
= Umi n
if
~,(t) < 0
U(t)
=
if
k(t) > 0.
and Uma x
However, if Mt) = 0, the Hamiltonian becomes H = - [x - a] 2, and the appropriate level of control sustains the equality x(t) = a(t) and thereby causes the Hamiltonian to vanish. Under these conditions the transition equation ~? = U yields U(t) = k = li. An optimal control trajectory therefore consists of intervals of maximum and minimum controls, along with intervals of nonextremal control to track the population carrying capacity exactly. Interval lengths in the optimal control trajectory are determined by the trajectory of K(t). From the costate equation ~, = 2(x - a) we have t
k(t) = 2 / Ix(s) - a(s)] ds, 0
with M0)
-
X(T) =
0
[endpoint conditions on Mt) follow from the transversality condition above, given that endpoint values for x(t) are not specified]. The optimal control strategy therefore adjusts the population rate of change to ensure coincidence of the population and carrying capacity, over intervals in which the necessary change does not exceed allowable limits of control. From the above equation, these intervals are characterized by K(t) = 0. If control limits are exceeded by the changes needed to track the carrying capacity exactly, the optimal strategy calls for maximum allowable controls for K(t) > 0 and minimum allowable controls for K(t) < 0. The locations of interval endpoints for maximum, minimum, and
624
Chapter 23 Modern Approaches to Decision Analysis
target control rates can be seen to be uniquely determined, as a result of the condition
f 0TEx(t) -- a(t)] dt = O. 23.2.4. Sensitivity Analysis As with the Lagrangian multipliers in nonlinear programming, the costate variables can be interpreted in terms of the sensitivities of the objective functional to certain parameter changes. In particular, it can be shown that the initial value K_*(t0) of the optimal costate trajectory expresses the sensitivity of the objective functional to a change in the system initial state:
As with nondynamic optimization, these sensitivities can be given an economic interpretation. Thus, the objective functional sometimes measures total economic value, in terms of price times quantity accumulated over time. If the system state represents resource availability, then the costate variables at t o can be interpreted as the marginal change in economic value with respect to a change in the resource inputs in x(t0). Hence the use of the term shadow price in reference to the optimal costate values.
23.2.5. Discrete-Time M a x i m u m Principle It is possible to derive a version of the canonical equations for problems in which the time frame is discrete. The relevant optimization problem in discrete time is tf - 1
{u(t)} ~ u
~ I[x(t), U(t), tl t=to
+
Fl[X__(tf)]
subject to x(t + 1) = x(t) + ~(x, U, t), x(t0)
x 0.
=
As before, Lagrangian multipliers can be used to incorporate the transition equations and initial conditions into the objective functional, producing a discrete-time version H[x(t), U(t), X(t + 1), t] = m
~
First we consider a situation in which there are no constraints on the control trajectory, i.e., the vector U(t) of control variables can be any point in E k. After some rather complicated mathematics it can be shown (see Appendix H.6) that an optimal trajectory must satisfy aH/OU(t) = 0', k(t) = O H / a x ( t ) ,
and x(t) = OH/OMt)
at each point in the time frame, along with the transversality condition
oJ*lox(to) = _~*(t0).
maximize
23.2.5.1. U n c o n s t r a i n e d O p t i m i z a t i o n w i t h Discrete Time
3F1 -- k(tf)]T](tf) = 0 x(tf)
-
-
and initial condition x(to) = Xo.
Note that these optimality conditions are analogous to the Euler-Lagrange Eqs. (23.9) and (23.10) for continuous problems. In both cases the optimization problem reduces to a two-point boundary value problem, typically requiring the solution of a system of nonlinear transition equations in state and costate variables. In general, both discrete-time and continuous-time problems must be solved by iterative techniques. In most instances the solution of a discrete-time problem converges to its continuous-time analog as the partitioning of the time frame becomes increasingly fine. Example
Management desires to eradicate a pest population over a 10-day period, while minimizing the cost of removal. A unit U(t) of effort on day t results in the removal of o~U(t) individuals, at a cost of U(t)2/2. Based on survey data the population size is estimated to be 100 individuals, so the optimization problem is 9
maximize
- ~ , U(t)2/2
{u(t)}
t=0
subject to x(t + 1) = x(t) - o~U(t),
m
I[x(t), U(t), t] + _~(t + 1)[[x(t), U(t), t]
x(0) = 100, x(10) = 0.
of the Hamiltonian. Optimal controls then can be found by appropriate choice of U(t) to maximize the Hamiltonian at each point in the time frame.
The Hamiltonian for this problem is
/
H = - U ( t ) 2 / 2 + )~(t + 1)Ix(t)- oLU(t)],
23.2. Pontryagin's Maximum Principle so that
625
so that ~U (t) = - U(t) - )t(t + 1)
~U (t) = - U(t) - ~)~(t + 1)
=0,
=0,
)t(t) = -~xH(t)
)~(t) = ~ ( t )
= )~(t + 1)Jr + x(t)/K],
= )~(t + 1), and
and
x(t + 1) = aH/a)t(t + 1) = rx(t) - x ( t ) 2 / 2 K -
x(t + 1 ) = - ~all. ( t + 11
U(t).
From the optimality condition on H we have U(t) = -K(t + 1), so that
= x(t) - oLU(t).
From the costate equation we have )~(t) = c, so that U(t) = -oLc from the optimality condition on H. Substituting this expression for U(t) into the state transition equation then gives x(t + 1) = x(t) + oL2c. It follows from x(0) = 100 that x(t) = 100 + tc(x 2, and in particular x(10) = 0 = 100 + 10col2. Then c = -10/oL 2, and the optimal control and state trajectories are given by U(t) = 10/c~ and x(t) = 100 - 10t, respectively. Thus, the m i n i m u m - c o s t control strategy calls for a u n i f o r m effort over the time frame, which results in a linear decline in the p o p u l a t i o n to extinction at time t = 10.
x(t + 1) = rx(t) - x ( t ) a / 2 K + K(t + 1).
B o u n d a r y conditions for the problem are the initial population value x(0) = 10 and ~(3) = 1 from the transversality condition. Utilizing the costate equation, we can step b a c k w a r d from the terminal time to obtain )~(2) = )~(3)[r- x ( 2 ) / K ] = r -
x(2)/K,
)~(1) = M2)[r - x ( 1 ) / K ] = [ r -
x(2)/K][r-
x(1)/K],
M0) = )~(1)[r- x(O)/K] = [ r -
x(2)/K][r-
x(1)/K]
• [r - x(0)/K]. Example
Consider the optimal harvest of a logistic population x(t + 1) = x(t) + r'[x(t) - x ( t ) 2 / K '] - U(t) = rx(t)
x(t) 2
2K
x(2) = x ( 1 ) [ r - x(1)/2K] + h(2) = x ( 1 ) [ r - x(1)/2K]
2 {U(t)}
-~
x(1) = x ( 0 ) [ r - x(O)/2K] + )~(1) = 1 0 [ r - 5 / K ] + [ r - x(2)/K] [ r - x(1)/K],
U(t)
over T = {0, 1, 2, 3}, where harvest cost at each point in time is a quadratic function of effort, and a terminal value is ascribed to the population size at the end of the time frame. A s s u m i n g an initial population size of 10 individuals, the optimization p r o b l e m is maximize
On the other hand, we can utilize the state transition equation to step forward from the initial time, to obtain
U(t)2/2 + x(3)
t=0
+ [r - x(2)/K],
x(3) = x ( 2 ) [ r - x(2)/2K] + K(3) = x ( 2 ) [ r - x(2)/2K] + 1. The latter system of equations can be solved for x(t), and the resulting values then can be used in the costate equations to identify )~(t). The costate values in turn can be used to identify the optimal control sequence, by U(t) = h(t+l).
subject to x(t + 1) = rx(t) - x ( t ) 2 / 2 K -
23.2.5.2. Discrete-Time Optimization with Constraints on Controls
U(t),
x(O) = 10.
In this case there are constraints on the vector U(t), i.e., U(t) e f~t. As before, the Hamiltonian
The H a m i l t o n i a n for this problem is
H[x(t), U(t), )~(t + 1), t] = _
H = -U(t)2/2
+ )~(t + 1) rx(t) - x ( t ) 2 / 2 K -
U(t)],
_
l
I[x(t), U(t), t] + _K(t + 1)[[x(t), U(t), t]
626
Chapter 23 Modern Approaches to Decision Analysis
is to be maximized by choosing the appropriate value U(t) in f~t: maximize
H(x, U, t)
U(t) ~ ~t
for all t e T. An optimal value is obtained either at an interior point of f~t, in which case 3 H / O U vanishes, or at a b o u n d a r y point of f~t. A general solution of the optimal control problem consists of trajectories {x(t)}, {U(t)}, and {K(t)} for which the Hamiltonian is maximized o v e r ~t at each point in the time frame, and the discrete canonical equations are satisfied:
x(t) = OH/OK_(t),
x(t 0) = x0;
K_(t) = OH/Ox(t),
K_(tf) = cgF1/cgX f.
Substitution of the optimality condition on u(t) into the costate equation produces Mt) = UmaxE1 -- K(t + 1)(1 + r)] + X(t + 1)(1 + r) (1 + r)Mt + 1)
u(t) = Umax, u(t) = O.
Because neither a terminal state nor a terminal value is specified, the transversality condition is X(T)
OF1 / 3 X ( T ) = 0,
=
and u(T - 1) = Umax from (1 + r)MT) < 1. The costate equation then gives K ( T - 1)
=
Umax[1
-
K(T)(1
+
r)~ +
K(T)(1
+ /')
and, from the optimality condition,
Example Previous population surveys indicate that within certain limits, a h u n t e d population exhibits postharvest exponential growth over time. If u(t)x(t) is the harvest in year t and y(t) = [1-u(t)]x(t) is the postharvest population size, then population change over time is given by
x(t + 1) = (1 + r)y(t) = (1 + r)[1 - u(t)~x(t), where r > 0 is the population instantaneous rate of growth. M a n a g e m e n t seeks to maximize the total harvest over T time periods, assuming an initial population of size x(0) = x 0 and a range of harvest rates between 0 and Ureax. The optimization problem is
u(T-
2) =
Umax,
if (1 + r)K(T - 1) < 1. This backward stepping process using Umax and X(t + 1) to identify k(t) continues until (1 + r)X(t*) >1, at which time the optimal harvest rate goes to 0. The harvest rate then remains at 0 for t < t*, because X(t - 1) = (1 + r)X(t) > 1 over that time. Thus, the optimal harvest strategy calls for abstention from harvest early in the time frame to allow the population to increase in size, followed by m a x i m u m harvest throughout the remainder of the time frame. The appropriate time to switch from u = 0 to u = l/max is determined by the intrinsic growth rate r and the value Umax, according to the difference equation X(t)
=
Umax[1
-
X(t + 1)(1 + r)] + K(t + 1)(1 + r).
T
maximize
~
u(t)x(t)
u(t), [0, Umax] t = 0
subject to
x(t + 1) = (1 + r)E1 - u(t)]x(t), x(0) = x0. The Hamiltonian for this problem is
H = u(t)x(t) + k(t + 1)(1 + r)[1 - u(t)]x(t), which is maximized by
u(t) =
Ureax 0
(1 + r)X(t + 1) < 1, (1 + r)K(t + 1 ) > 1.
The Euler-Lagrange equations are
K(t) = ~x(t) = u(t) + K(t + 1)(1 + r)[1 - u(t)] and
x(t + 1) = - ~OH. ( t + 1) = (1 + r)E1 - u(t)lx(t).
23.2.6. Summary The m a x i m u m principle extends beyond the calculus of variations, to include complex constraints on the control variables in U(t). The problem formulation includes possible initial and terminal value functions, and also allows for constraints other than the system transition equations. A solution approach involves the following: 9 Time-varying costate variables are included as weighting factors for the state transfer functions in an augmentation of the optimality index k n o w n as the Hamiltonian. 9 An optimal control strategy is obtained by maximizing the Hamiltonian with respect to the controls at each time over the time frame of the control problem. Pointwise maximization defines the " m a x i m u m principle." 9 Differentiation of the Hamiltonian with respect to the state variables produces the time rate of change
23.3. Dynamic Programming of the costate variables. Differentiation with respect to the costate variables reproduces the state transfer functions. 9 Differentiation of the Hamiltonian with respect to the controls often can be used to identify the form of optimal controls as a function of the state and costate variables. Incorporating the control function into the state and costate transition equations then defines a system of equations in the state and costate variables, absent any direct reference to the controls. 9 Optimal state and costate trajectories are derived as solutions of this system of equations, subject to initial boundary conditions on the state variables and terminal conditions on the costate variables. 9 Solving the canonical equations usually involves backward integration of the costate equations and forward integration of the state equations. The optimal control strategy subsequently is identified as a function of the optimal state and costate trajectories. 9 In some cases, most notably when the Hamiltonian is linear in controls, other methods besides the use of stationarity conditions must be used to identify the optimal control strategy. 9 Initial values of the optimal costate variables express the sensitivities of the optimal value of the objective functional to marginal changes in the initial state variable values. 9 A maximum principle can be formulated for discrete-time problems, with optimal controls derived from Euler-Lagrange equations that are essentially identical to those for continuous systems.
23.3. D Y N A M I C P R O G R A M M I N G As with variational mathematics, the general control problem for dynamic programming is to choose a control strategy {U(t)} from some constrained set ___Uthat maximizes an objective functional of system states, controls, and possibly time. For deterministic systems with continuous time frames the control problem is identical to that of the maximum principle: maximize {U(t)} ~ U B
f t~ I(x, U, t) dt + Fl[X(tf)] to
subject to
2_ = [(x, U, t), x(t0) = x0, x(t~) = x~.
627
An analogous statement for discrete-time systems replaces the integral with a summation over the time frame, i.e., ti l = ~,, I(x, U, t) + Fl[X(tf)], (23.12) t=to
and utilizes difference equations of the form x(t + 1) = x(t) + f(f, x, U, t)
m
m
to express system transitions. Dynamic programming also is amenable to stochastic problems, although the objective functional is stated in terms of expected values"
J = g
[t0
}
I(x, U, t) + fl[x(tf)] ,
(23.13)
where the expectation is with respect to random elements Z(t) that influence system behaviors by x(t + 1) = x(t) + [(x, U, Z, t), with {Z(t)} a time series stochastic process (see Chapter 10). Though dynamic programming and the maximum principle share a common statement of the control problem for deterministic systems, dynamic programming constitutes a substantially different approach to its solution. Rather than incorporating the state transition equations into the objective functional by means of costate variables, the approach here is to use the "Principle of Optimality" (Bellman, 1957) to derive a partial differential equation or difference equation, the solution of which solves the dynamic programming problem. The Principle of Optimality is stated as follows: An optimal policy has the property that, whatever the initial state and decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
The meaning of this principle is illustrated in Fig. 23.5, which describes the decomposition of an optimal state trajectory into two parts: an initial trajectory that starts at an initial condition x(t o) and evolves to x(tl), and a subsequent trajectory that starts at x(t 1) and evolves to the terminal point x(tf). Essentially the Principle of Optimality says that if the overall trajectory is optimal with respect to the system initial condition, then the second trajectory is optimal in its own right, relative to its initial condition. Thus, the optimal behavior of the second trajectory is independent of how the system came to be at the second starting point.
628
Chapter 23 Modern Approaches to Decision Analysis
x(t) x(t~)
x(t,)
J
X(to)
I I I I
to
t,
tr
F I G U R E 23.5 B e l l m a n ' s Principle of Optimality. For o p t i m a l trajectory {x(t)} o v e r [t 0, tf], the trajectory {x(t)} over [t 0, tf] m u s t be o p t i m a l relative to the initial c o n d i t i o n x(t 1) at t = t 1. After Intriligator (1971).
In both its discrete and continuous forms, dynamic programming can be seen to apply to systems for which the past behavior of the system is unimportant in assessing optimal controls. Utilizing a term from stochastic processes, we characterize systems for which future behavior is independent of past trajectories as Markovian (see Sections 10.3-10.6). The future history of a Markovian system at any point in time is influenced only by the system state (and controls) at that time. Indeed, the Principle of Optimality is an expression of the Markovian nature of control systems with no time lags. In order to apply dynamic programming, a Markovian system description is necessary. Of course, the statement of the optimal control problem used here includes system transitions that are Markovian.
23.3.1. Deterministic Dynamic Programming We consider here the application of dynamic programming to problems with nonstochastic behaviors. We may write the optimal value of the objective function (23.12) as J*[x(t), t] to emphasize its dependence on time and the starting point of the state trajectory. Application of the Principle of Optimality yields the fundamental recurrence relation J*[x(t), t] =
max
[I(x, u, t)At
{u(t)} ~ ut m
+ J*(x + Ax, t + At)],
(23.14)
a form that is appropriate for solution of control problems with discrete states and time frames. Additional smoothing assumptions ensuring the continuous differentiability of J[x(t)] yield the Hamilton-Jacobi-Bellman (HJB) equation -ol*/ot =
max [I(x, U, t) + (ol*/ox)~(x, u, t)], {U(t)}, m
ut
(23.15)
which, along with the boundary condition
l,[x(tj), t~] =
Fl[X(tf)],
provides the analytic framework for solving the optimal control problem for continuous systems. From the above discussions, it is clear that the application of the principle of optimality requires one to account for all possible states of the system at each point in the time frame. Thus, dynamic programming describes a field of values for the objective functional, one for every state at every time in the time frame, with each value produced by a state-specific and timespecific optimal control strategy. The net effect is to generate a feedback control rule, wherein the state and time can be fed back as arguments in an optimal control function U*[x(t), t]. This function identifies an optimal action for state x(t) at time t, on assumption that future state and control histories will follow optimal trajectories over the remainder of the time frame. 23.3.1.1. Applications in Continuous Time Few applications of dynamic programming in renewable natural resources use a continuous-time formulation, primarily because of the formidable difficulties in analyzing the HJB Eq. (23.15) in continu-
23.3. Dynamic Programming ous time. The equation involves partial derivatives for k + 1 variables (k state variables in x and time t) and is extremely difficult to solve, even with the aid of highspeed computers, for all but a handful of problems. D
Example
Consider the example discussed earlier, in which direct control of population change is to be applied in an effort to eliminate a pest population over [0, 2], while also minimizing control costs. The problem is to 2
maximize
f
IU(t)}
( x + U2) dt
Jo
subject to
~=u, x(0) = 3, x(2) = 0.
629
trajectory identified previously by the calculus of variations and the maximum principle. 23.3.1.2. L i n e a r - Q u a d r a t i c C o n t r o l in C o n t i n u o u s T i m e
A useful application for which the HJB equation can be solved involves a quadratic objective functional and linear system transitions. To illustrate, consider the expenditure of resources to control a community of pests. Pest population sizes x(t) and the effort U(t) directed to pest control both are assumed to influence population changes in a linear fashion, though pest control efforts have differential impacts, depending on the affected species. Pest damage is measured by x ' Q x , and the cost of pest control through time is U ' R U. The objective of management is to minimize accumulated costs as measured by a quadratic function of these factors, or, equivalently, to maximize the negative of this accumulation. A formal statement of the problem is ti
The HJB equation for this problem is
minimize {U(t)} m
1/2
f to
[ x ' Q x + U ' R U] dt
subject to Yc=Ax
with minimization of the term in brackets given by differentiation with respect to U:
0[
s
(y -}-
U2) q-
J
~r
x(tp = x~,
03"~"
with Q and _R negative-definite matrices. If the time frame is unlimited, a potential solution of the HJB equation has the form ]* = x'P x/2, with P a symmetric matrix that solves the system of equations
or
U
x(to) = Xo,
O]*U] = 2U + 0]*= O, OY
_ _ _
+ BU,
lal* 2 Ox
Q + PA + A'P - PBR-IB'P
Substitution of the expression for U* into the HJB equation produces o3j, : X q- l ( ~ at
-4\ Ox )
2 -- l ( ~
(see Appendix H.7). The optimal control strategy is expressed by
2 U*(t) = - R - 1 B ' P x ( t ) ,
2 \ Ox ) '
which describes a linear feedback strategy in x(t). Substituting U*(t) back into the transition equations then produces
or l(a]*~ 2
4\-O-~xl+
= 0
Ol___~*a=t 0. Yc = A x + B U *
This equation is solved by J* = - x t / 2 - x 2 / t + t3/48, as seen by substitution of the partial derivatives of J* back into the equation. From oJ*/ox = - t / 2 - 2 x / t we obtain the optimal control U* = - ( o J * / O x ) / 2 = t / 2 + x/t, and also the transition equation ~ = t / 2 + x/t. This in turn gives x*(t) = t 2 / 4 + clt + c 2. The constants Cl and c2 are of course determined by the initial and terminal conditions: x(0) = 3 = cI and x(2) = 0 = 22/4 + 2c I + c2, from which we have c2 = - 2 and thus x*(t) = t 2 / 4 + 3t - 2. Note that this is the same optimal
= [A - B R - 1 B ' P ] x
with a solution k x(t) = ~ , ci(vie.Xit), i=1
where k 1, ..., k k are the eigenvalues of A - B R -1 B ' P and __v1, ..., __vk are the corresponding eigenvectors (see Appendix B).
630
Chapter 23 Modern Approaches to Decision Analysis
Example Consider an earlier example in which the growth rate of an exponential population x(t) is to be controlled directly: f = U(t). Recall that the population dynamics for this situation were handled by introducing a second state variable to account for the control of growth, with x(t) r = x2(t), and ~1 = x 2 = U(t). Population dynamics then are given by
Substituting this expression back into the transition equations produces
['~.~2] 1 = {I 00 10]--[~]l-O 1] [42 ~]} [X12]
=i0 1][,]
= xl(t),xI =
E12]= Ea 101[x:] Assume for this example that the objective is to maximize
l = -fto
E2x2+ 2j2],t
to account for both population and control costs. For this problem R = r = 1, B = (0,- 1)', m
Q=
-2
-2
x2
"
It is straightforward to show that the eigenvalues for this system are complex, so that the optimal state trajectory is oscillatory (see Appendix C). Thus, the optimal control strategy is linear in both the population state and its growth rate, and optimal control induces oscillatory population levels. Example Assume for the control problem of the previous example that there are separate controls for the two state variables, with Ul(t) representing direct control of the population and U2(t) representing control of the population rate of growth:
0'
0]ix:] 1
and
_A=
Id:2 =
[0 ~] 0
with an objective functional
"
, = St,~0Ix2 + ~2+ ~2],t
For p=[Pl --
P2] P2 P3
As above, a solution for this problem is given in terms of the matrix P satisfying
we have
Q + PA + A'P - P B R - 1 B ' p
Q + PA + A ' P --
Pl - P2P3] PBB'P/r = [ 4-p2 LPl - P2P3 2p2 - p32J
= [
1_p21_ p2 Pl -- PlP2 -- P2P3
/
Pl -- PlP2- P2P3| 2p2 _ p 2 _ p2 J
=0.
-- Or m
from which it follows that
It is straightforward to show that
22]
P=
and
I 1/2 X/3/2
X/'3/2] 1/2 J
satisfies this equation, and substitution of P into U*(t) = - B ' P x ( t ) / r
= --[xl(t),x2(t)][ 4
= 2xl(t ) + 2x2(t).
U*(t) =
22] I_~]
-R-1B'Px(t)
defines an optimal linear feedback strategy with U'((t) = [V3xl(t) + x2(t)]/2
23.3. Dynamic Programming and
U~(t) = [xl(t) + x2(t)]/2. 23.3.1.3. Applications in Discrete Time Most applications of dynamic p r o g r a m m i n g in renewable natural resources use a discrete-time formulation. The operative form of the HJB Eq. (23.14) is J*[x(t), t] =
[I(x, U, t)&t + J*(x + Ax, t + &t)~,
max {U(t)} m
~
Ut
in which the time increment for system change is At. As a matter of convenience, the time interval in most applications is taken to be 1, so that the HJB equation is expressed by /*[x(t), t] = max {I(x, U, t) +/*[x(t + 1), t + 1]}. (23.16) {U}, Ut m
Several points are worthy of note. First, this formulation displays clearly the stagewise character of the optimization process, in which the problem of selecting an optimal control trajectory over an entire time frame is decomposed into a series of single-stage optimization problems. Thus, the maximization in Eq. (23.16) is with respect to potential actions taken at time t only. Of course, one must identify a maximizing action for every possible state x(t) of the system at time t. Second, maximization at time t requires one to project forward to time t + 1 the consequences of actions at time t. Thus, an action taken at time t with the system in state x(t) engenders a transition to state x(t + 1) at time t + 1. The optimization must account not only for the utility of the action at time t, but also the effect of that action on system dynamics (and therefore on future utilities) from that time forward. Third, maximization at time t requires optimal values at time t + 1 in order to select a maximizing action at time t. When the system is in state x(t), one must k n o w the optimal values associated with state x(t + 1) to which transfer is made from x(t). And this in turn requires the optimal control strategies for states at time t + 1. The optimization at time t simply augments the field of state-specific control strategies and values for time t + 1, by adding additional actions for time t to create a new field of state-specific strategies and values for time t. Fourth, time-specific and state-specific optimal values and controls can be identified by means of a sequence of stagewise optimizations that progress backward from the end of the time frame. The following algorithm can be used for stagewise dynamic prog r a m m i n g problems: 9 Beginning at tf, an action is chosen that maximizes the terminal v a l u e Fl(Xf) , given that these values are
631
influenced by the choice of an action. Otherwise, the value J*[x(tf), tf] for state x__fat time tf is J*[x(tf), tf] = Fl(Xf). 9 The second step utilizes the state-specific values J*[x(tf), tf] for tf to determine state-specific actions at time tf - 1 according to Eq. (23.16). The optimization produces state-specific values l*[x(tf - 1), tf - 1] for a two-stage problem. 9 The third step utilizes the values J*[x(tf- 1), t f 1], to determine state-specific actions at time tf - 2. The optimization produces state-specific values J*[x(tf - 2), tf - 2l for a three-stage problem. 9 The algorithm continues to step backward through the time frame until the initial time t o is reached. At the completion of this process a field of optimal control trajectories has been identified for all possible states of the system at all times in the time frame. To determine the appropriate control strategy, one need only identify an initial system state and initial time. The corresponding strategy then consists of a predetermined sequence of optimal actions requiring no additional information. Such a strategy is k n o w n as openloop control, so named because the strategy is selfdetermined once the initial state and time are specified (Intriligator, 1971). The dynamic p r o g r a m m i n g approach of decomposing a multistage optimization problem into a series of single-stage optimization problems yields tremendous gains in computational efficiency. To illustrate, consider a simple problem of finding the least-cost sequence of state transitions over [0, T], given that the transition from a state at time t to any other state at time t + 1 is possible. If initial and terminal states are fixed and k states are available at all other times, it is straightforward to show that there are k T-1 possible state trajectories for this problem. For a given trajectory, one must perform T - 1 additions to determine the accumulated transition costs. Thus, a search for the optimal trajectory via comparison of these costs for all possible trajectories would entail (T - 1)k T- 1 additions. On the other hand, the stage-specific optimizations of dynamic p r o g r a m m i n g require k2 additions at each stage between t = 1 and t = T - 1, plus k additions for the transition from t = 0 to t = 1. A total of only (T - 2)k2 + k additions therefore is required to determine the optimal trajectory. Even for very small problems, the computational savings are extraordinary. A problem with five possible states (k = 5) and six time periods (T = 6) requires only (4)(52) + 5 - 105 additions for a dynamic p r o g r a m m i n g solution, as opposed to (5)(5 s) = 15,625 additions for complete enumeration. It thus would be 149 times more costly to enumerate
Chapter 23 Modern Approaches to Decision Analysis
632
all possible trajectory costs than to use dynamic programming for the optimization. A comparison of the formulas (T - 1)kT-1 and (T - 2)k2 + k shows that the relative efficiency of dynamic programming increases exponentially with an increase in the number k of possible states and the length T of the time frame. Thus, expanding the number of states to k = 6 and the number of periods to T = 7 in this example increases the relative cost of enumeration from 149 to over 1500. Though computing demands sometimes are heavy for applications of dynamic programming, they pale in comparison to the demands for enumeration. Example
Managers of a seasonal fishery wish to minimize accumulated costs of stocking and maintenance, given a requirement to satisfy sport-fishing demand during a 4-month fishing season. Let x(t), u(t), and z(t) represent the size of the fish population, the level of stocking, and anticipated angler demand each month during the season. Assume for simplicity that population sizes, stocking events, and angler demand are measured in units of 10,000 fish. The size x(0) of the population, estimated via capture-recapture methods, is available prior to the season opening. There is no reproduction during the fishing season, and natural mortality is negligible over that time. On the other hand, population increases can occur through stocking, and angler demand for fish take is expected to fluctuate over the season. Population dynamics during the season are given by x(t + 1) = x(t) + u(t) - z(t),
where monthly stocking levels u(t) are under management control and monthly demand z(t) = {1, 3, 2, 4} is assumed known at the beginning of the season based on previous angler surveys. Stocking levels are limited by the hatchery capacity, assumed to be Umax - 5 , and population sizes are constrained by available fish habitat to be x(t) -< 5. Costs are incurred as a result of stocking at the beginning of each month, according to the cost function Cl[U(t)] = 0 for u[(t) = 0 and cl[u(t)] = 3 + u(t) otherwise. Costs associated with population and habitat maintenance accrue at the end of each month, according to c2[x(t + 1)] = [x(t) + u ( t ) - z(t)]/2. The objective functional accumulates stocking and maintenance costs over the fishing season. Thus, the optimization problem is 3
minimize
3
I[x(t), u(t)] = ~,{Cl[U(t)] + c2[x(t + 1)1} t=0
t=0
subject to x(t + 1) = x(t) + u(t) - z(t),
t = 0, 1,2,3;
O 1 - C~s~/. 1 - c
For both the additive and compensatory models, nonhunting mortality during the hunting season is assumed to be negligible (e.g., Cowardin and Johnson, 1979; Reineke et al., 1987). 25.4.1.2. Recruitment
Recruitment models are based on estimates of the annual fall age ratios of female mallards originating from the region of North America surveyed in spring,
674
Chapter 25 Case Study
1961-1993 (U.S. Department of the Interior, 1994). Let A t be the ratio of young females to adult females in the preharvest population, as estimated from the age ratio of the harvest corrected for relative harvest vulnerability (young:adult ratio of direct recovery rates of banded females), in year t (Martin et al., 1979). Age ratios of the harvest were calculated from parts-collection surveys (Martin and Carney, 1977) in those portions of the Central Flyway and Mississippi Flyway that derive ->80% of their harvest from the mallard population of interest (Munro and Kimball, 1982). The harvest vulnerability of young relative to adults was estimated for each of eight banding reference areas (Anderson and Henny, 1972) within the breeding range, then averaged for a single estimate of relative vulnerability for each year, with estimates of population size within reference areas as weights. Age ratios are linked to population and habitat conditions via models describing A t as a linear function of mallard population size xl(t) and the number of ponds x2(t). An interaction between xl(t) and x2(t) also was considered, and the linear relationship was allowed to vary between two unspecified "epochs" within the period 1961-1993. Weighted least-squares regression was used to identify the models, with the values A t inversely weighted by the variance of the annual harvest age ratio, which was considered to be proportional to variability in A t . All possible regression models induced by interactive combinations of xl(t), x2(t), and epoch were fitted, with epoch boundaries (i.e., the first year of the second epoch) adjusted to each year between 1965 and 1990 (14 models/epoch partition • 26 epoch partitions + 5 "no epoch" models = 369 models). Model selection was based on the lowest value of the Akaike Information Criterion (AIC) (see Section 4.4) and checked against model residuals for conformity with least-squares regression assumptions (Draper and Smith, 1981). The model with the smallest of the 369 AIC values contained xl(t) and x2(t), but no interaction. It distinguishes an epoch boundary at 1970, where the regression coefficient for xl(t) changed from a pre-1970 value of -0.0874 • 10 -6 (SE -- 0.0622 • 10 -6) to a post1970 value of -0.0547 • 10 -6 (SE = 0.0225 • 10-6). That portion of the model corresponding to the most recent epoch was selected as a weakly density-dependent model of recruitment: Al(t ) = (0.8249 - 0.0547 x 10-6)x1(t) (25.2) + (0.1130 x 10-6)x2(t). To express uncertainty about the degree of density dependence in recruitment [i.e., the magnitude of the coefficient for xl(t)], a strongly density-dependent
model of recruitment also was considered, based on the minimum parameter estimate for the coefficient of xl(t) for the post-1970 period located on the 95% confidence ellipsoid for all the parameters (Draper and Smith, 1981). The minimum estimate was selected as an alternative to Eq. (25.2) based on the most likely mechanisms for density-dependent recruitment (e.g., spacing behavior of pairs) (Dzubin, 1969). Thus, the strongly density-dependent model of recruitment was A2(t) = (1.1081 - 0.1128 • 10-6)xl(t)
(25.3)
+ (0.1460 • 10-6)x2(t). The number of young females in the fall population was modeled as a product of the predicted age ratio and the number of adult females in the fall. The number of adult females in the fall was given in turn by the product of summer survival and the number of females in the spring, which was determined based on the May estimates of population size and assuming a constant sex ratio of 1.2 males per female (Anderson, 1975a). To determine the number of young males and females, a sex ratio of 1.0 was assumed for young birds in the fall (Bellrose et al., 1961; Hestbeck et al., 1989). Thus:
y~(t) = y[(t) = gj[XSl(t), x2(t)]
(25.4)
= Aj(t)oL~xl(t)/Z2. The combination of two survival hypotheses (i = 1, 2) and two recruitment hypotheses (j = 1, 2) resulted in four alternative models of mallard population dynamics: (1) additive hunting mortality and weakly density-dependent recruitment; (2) additive hunting mortality and strongly density-dependent recruitment; (3) compensatory hunting mortality and weakly density-dependent recruitment; and (4) compensatory hunting mortality and strongly density-dependent recruitment (Johnson et al., 1997).
25.4.2. Environmental Variation The number of wetland basins containing surface water (ponds) in the Prairie Pothole Region during the breeding season is an important determinant of mallard production (Pospahala et al., 1974). Since 1961, the number of ponds in Prairie Canada during May has varied from 1.443 million (SE = 0.075 million) in 1981 to 6.390 million (SE = 0.308 million) in 1974 (U.S. Department of the Interior, 1994). Managers involved in harvest management cannot predict with certainty the number of ponds (and thus the mallard production) in the future. However, it is possible to make probabilistic statements about temporal changes in pond abun-
675
25.4. Modeling Population Dynamics dance, thereby allowing managers to assess the future consequences of current regulatory decisions. Thus, the estimated number of ponds in Prairie Canada and records of monthly (1 June 1974-31 May 1992) precipitation (millimeters) from five weather stations in southern Alberta, Saskatchewan, and Manitoba were used to construct an autoregressive model (see Section 10.8.4) of pond abundance: x2(t+l) = -3835087.53 + 0.45xa(t)
(25.5)
+ 13695.47r(t), where r(t) is total precipitation during the 12-month period from time t to t + 1. Pond numbers predicted by this model are nearly identical to those of the model provided by Pospahala et al. (1974). Annual (1 Jan-31 Dec) precipitation records were examined for the period 1942-1991 from the same five weather stations. The hypothesis that annual precipitation r(t) was distributed normally (Shapiro-Wilk W = 0.97, P = 0.36, range 304-574 mm, 2 = 418 mm, SD = 56 mm) was supported by the data, and results were virtually identical to those reported by Pospahala et al. (1974). Preliminary analyses with several data sets (some of > 100 years) failed to provide strong evidence of precipitation cycles (J. R. Sauer, U.S. Geological Survey, personal communication), supporting the conclusion of Pospahala et al. (1974) that annual precipitation in Prairie Canada can be described adequately as a normally distributed, independent random variable. Random draws from this distribution provided stochasticity in pond abundance according to Eq. (25.5).
25.4.3. Partial Management Control Managers control hunting regulations rather than harvest rates directly, and accounting for uncertainty in the functional relationship between the two is important. Early on, partial controllability was incorporated based on band recovery data from preseason banding as applied to three regulatory options. Later, a more complex procedure was used that would allow for variable bag limits and season lengths (see Section 25.6.1). In this section we describe the derivation of distributions of harvest rates for midcontinent mallards under each of three regulatory options. These options, characterized as liberal, moderate, and restrictive, corresponded to the regulations in effect in 1979-1984, 1985-1987, and 1988-1993, respectively. Each regulatory option contained flyway-specific season lengths and bag limits. The analysis of harvest relied on direct recovery rates of mallards banded before the hunting season in a representative portion of the midcontinent region
(banding reference areas 3-5) (Anderson and Henny, 2 1972). First the mean [fAM,p] and variance [Stota 1 (f,4~,p)] of recovery rates were estimated from the point estimates of direct recovery rate for adult male mallards for each of the three time periods (p). The analysis focused on adult males because they generally had the largest banded-sample sizes. The variances of the direct recovery rates are composed of both temporal and sampling components. For the purpose of choosing hunting regulations, interest focused primarily on the temporal component, which is a measure of the variability in recovery rates that could be expected when the same regulations are used in different years (i.e., partial controllability). This temporal v a r i a t i o n [S2emp(fAM,p)] was estimated using the approach suggested by Burnham et al. (1987): 2 St2mp(fAM'p) = S t ~
--
~t~l
s2(fAM,p,t) I np
(25.6)
where np is the number of years in period p and s2(fAM,p,t ) is the estimated sampling variance. Periodspecific harvest r a t e s [hAM,p] and their temporal vari2 a n c e s [Stemp(hAM,p)] then were estimated using a constant band-reporting rate of X = 0.32 (Nichols et al., 1991): St2mpGM,p) St2mp(hAM,p) --
ha
.
(25.7)
Based on this analysis, mean harvest rates for adult males were 0.090 [Stemp(hAM,p) = 0.016] for the restrictive option, 0.120 [Stemp(hAM,p) = 0.022] for the moderate option, and 0.156 [Stemp(hAM,p) = 0.025] for the liberal option. A closed season also was considered, in which the mean harvest rate was assumed to be 0 with no variation. The vulnerability to harvest for each of the other age-sex cohorts (adult females, young males, young females) was specified relative to that of adult males. The mean relative vulnerability Wa,p of each cohort a during each period p was calculated by averaging the ratio of annual recovery rates within the specified period: E n p fa,p,t t = lfAM,p, t Wa,p -Tip
Mean relative vulnerabilities did not differ among periods (asymptotic normal test of general contrast) (Sauer and Williams, 1989) for adult females, immature males, or immature females. Thus, constant rates of differential vulnerability were used, with 0.480 for adult females, 1.310 for young males, and 0.868 for young females.
676
Chapter 25 Case Study
25-
20-
X
15-
..~ ""...,.
/
10-
/iX....."",. / '
5-
--...'..
\
g
~ "\'~;~. O'
-
,
0.00
"
~
0.03
...,.. ... ,
'3
0.06
0.09
0.12
0.15
0.18
0.21
The harvest m a n a g e m e n t objective for midcontinent mallards is to maximize cumulative harvest value over the long term, given an aversion to harvest decisions that result in an expected population size below the goal of the North American Waterfowl Management Plan (NAWMP) (Fig. 25.9). The value of harvest opportunity decreases proportionally as the difference between the goal and expected population size increases. This balance of harvest and population objectives results in a more conservative harvest strategy than one maximizing long-term harvest, but a more liberal strategy than one seeking to attain the N A W M P goal regardless of losses in hunting opportunity. The current objective uses a population goal of 8.7 million mallards, based on the N A W M P goal of 8.1 million for the federal survey area and 0.6 million for the combined states of
(hAM,p)
Minnesota, Wisconsin, and Michigan (U.S. Department of the Interior, 1994). The utility function expressing the model-specific value of harvest u[Hi(htixt)] has the form u[H~(htlxt)]
=
if Ei[xl(t+l)] >- 8,700,000, if Ei[xl(t+l)]
m 40 population goal = 8.7
"r 20 0 0
1 --2
3
4
5
6
7
8
9
10
Expected population size next year (in millions) FIGURE 25.9 The relative value ~ of mallard harvest, expressed as a function of breeding population size expected in the subsequent year.
25.6. Regulatory Alternatives TABLE 25.1
677
Regulatory Alternatives Considered for the 1995 and 1996 Duck-Hunting Seasons Flyway
Regulation
Atlantic
Shooting hours Framework dates Oct 1-Jan 20 Season length (days) Restrictive 30 Moderate 40 Liberal 50 Bag limit (total/mallard / female mallard) Restrictive 3/ 3/ 1 Moderate 4/4/1 Liberal 5 / 5/ 1
Mississippi
Central a
Pacific b
One-half hour before sunrise to sunset for all flyways Saturday closest to October 1 and Sunday closest to January 20 30 40 50
39 51 60
59 79 93
3/2 / 1 4/3/1 5/ 4 / 1
3/3 / 1 4/4/1 5 / 5/ 1
4 / 3/ 1 5/4/1 6-7c/ 6-7c/ 1
The High Plains Mallard Management Unit was allowed 12, 16, and 23 extra days under the restrictive, moderate, and liberal alternatives, respectively. bThe Columbia Basin Mallard Management Unit was allowed seven extra days under all three alternatives. cThe limits were 6 in 1995 and 7 in 1996. a
a compromise over the range of population sizes below 8.7 million, in that neither the objective to maximize harvest nor the objective to maintain the mallard population at or above the plan goal w o u l d be realized fully (Fig. 25.9). Both the population size xl(t) and the capacity of available breeding habitat x2(t) to promote population growth during the interval t to t + 1 are considered in the determination of the optimal regulatory decision for x t. Thus, liberal hunting regulations could still be appropriate for a mallard population that is below the goal of 8.7 million, if current habitat conditions were expected to result in good production of young.
25.6. R E G U L A T O R Y
ALTERNATIVES W h e n A H M was first implemented in 1995, liberal, moderate, and restrictive regulations were defined based on regulations used during 1979-1984, 19851987, and 1988-1993, respectively (Table 25.1). These regulatory alternatives also were considered for the 1996 hunting season. However, in 1997 the regulatory alternatives were modified to include (1) the addition of a very restrictive alternative, (2) additional days and a higher duck bag limit in the moderate and liberal alternatives, and (3) an increase in the bag limit of hen mallards in the moderate and liberal alternatives. The basic structure of the regulatory alternatives has been u n c h a n g e d since 1997 (Table 25.2).
25.6.1. P r e d i c t i n g
Harvest Rates
Since 1997, harvest rates (and associated variances) for the A H M regulatory alternatives have been predicted using (1) linear models that predict total seasonal mallard harvest for varying season lengths and bag limits, accounting for numbers of successful duck hunters, and (2) adjustment of historical estimates (Section 25.4.3) to reflect differences in bag limit, season length, and trends in hunter numbers (Table 25.3). The adjustments are based on estimates of hunting effort and success from hunter surveys. The procedure utilizes linear models that predict total seasonal mallard harvest for varying regulations (daily bag limit and season length), while accounting for trends in numbers of successful duck hunters. Using historical data from both the U.S. Waterfowl Mail Questionnaire and Parts Collection Surveys, the resulting models allow one to predict total seasonal mallard harvests and associated harvest rates for varying combinations of season length and daily bag limits. The linkage between regulations and harvest rate involves two component models: a "harvest" model that predicts average daily mallard harvest per successful duck hunter for each day of the hunting season, and a " h u n t e r " model that predicts the n u m b e r of successful duck hunters. The "harvest" model uses as its d e p e n d e n t variable the square root of the average daily mallard harvest (per successful duck hunter), with independent variables that include the consecutive days of the hunting season (ignoring splits in the season), daily mallard bag limit, season length, and
Chapter 25 Case Study
678 TABLE 25.2
Regulatory Alternatives Considered for the 1999 Duck-Hunting Season Flyway
Regulation
Atlantic a
Mississippi b
Shooting hours
Central c
Pacific a
One-half hour before sunrise to sunset for all flyways
Framework dates
Oct 1-Jan 20
Season length (days) Very restrictive
Saturday closest to October 1 and Sunday closest to January 20
20
20
25
38
Restrictive
30
30
39
60
Moderate
45
45
60
86
Liberal
60
60
74
107
Bag limit (total/mallard/female mallard) Very restrictive 3/ 3/ 1 Restrictive
3/3/1
3/ 2/ 1
3/ 3 / 1
4/ 3/ 1
3/2/1
3/3/1
4/3/1
Moderate
6/4/2
6/4/1
6/5/1
7/5/2
Liberal
6/4/2
6/4/2
6/5/2
7/7/2
a The states of Maine, Massachusetts, Connecticut, Pennsylvania, New Jersey, Maryland, Delaware, West Virginia, Virginia, and North Carolina are permitted to exclude Sundays, which are closed to hunting, from their total allotment of season days. b In the states of Alabama, Mi3sissippi, and Tennessee, in the moderate and liberal alternatives, there is an option for a framework closing date of January 31 and a season length of 38 days and 51 days, respectively. c The High Plains Mallard Management Unit is allowed 8, 12, 23, and 23 extra days under the very restrictive, restrictive, moderate, and liberal alternatives, respectively. d The Columbia Basin Mallard Management Unit is allowed 7 extra days under the very restrictive, restrictive, and moderate alternatives.
the interaction of bag limit and season length. Terms for an opening-day effect, a week effect, and several other interaction terms also are included. Seasonal mallard harvest per successful duck hunter is obtained by back-transforming the predicted values from the model and summing the average daily harvest over the season length.
TABLE 25.3 Expected Harvest Rates (SE) of Adult Male Midcontinent and Eastern Mallards a Harvest rate (SE) Mallard population
Alternative
Midcontinent
Very restrictive
Eastern
1995, 1996
1997, 1998, 1999
N/A
0.053 (0.011)
Restrictive
0.067 (0.014)
0.067 (0.014)
Moderate
0.089 (0.020)
0.111 (0.027)
Liberal
0.118 (0.029)
0.131 (0.032)
N/A
0.121 (0.020)
Restrictive
0.133 (0.021)
0.135 (0.022)
Moderate
0.149 (0.023)
0.163 (0.025)
Liberal
0.179 (0.028)
0.177 (0.028)
Very restrictive
a Under different regulatory alternatives, based on mean hunter numbers during 1981-1995.
The "hunter" model utilizes information about the numbers of successful duck hunters (based on duck stamp sales information) from 1981 to 1995. Using daily bag limit and season length as independent variables, the number of successful duck hunters is predicted for each state. Both "harvest" and "hunter" models were developed for each of seven management areas: the Atlantic Flyway portion with compensatory days; the Atlantic Flyway portion without compensatory days; the Mississippi Flyway; the low plains portion of Central Flyway; the High Plains Mallard Management Unit in the Central Flyway; the Columbia Basin Mallard Management Unit in the Pacific Flyway; and the remainder of the Pacific Flyway, excluding Alaska. The numbers of successful hunters predicted at the state level were summed to obtain a total number (H) for each management area. Likewise, the "harvest" model results in a seasonal mallard harvest per successful duck hunter (A) for each management area. Total seasonal mallard harvest then is given by the product T=H•
To compare total seasonal mallard harvest under different regulatory alternatives, ratios of total harvest for different alternatives were formed for each management area and then combined into a weighted mean. Under the key assumption that the ratio of harvest rates realized under two different regulatory alternatives is
25.7. Identifying Optimal Regulations equal to the expected ratio of total harvest obtained under the same two alternatives, the harvest rate experienced under the historic "liberal" package (19791984) was adjusted by T to produce predicted harvest rates for the current regulatory alternatives. Harvest rates for each of the regulatory alternatives for 1999 were predicted assuming no change in the regulatory alternatives from 1997 and 1998 (Table 25.4). However, predicted harvest rates for 1997-1999 differ from those used previously as a result of revised analytical procedures, which rely on mean numbers of hunters during 1981-1995 rather than on short-term trends in annual hunter numbers. This change was made to prevent year specificity in harvest rates predicted for a given alternative and to better reflect uncertainty about hunter numbers in the future.
25.7. I D E N T I F Y I N G OPTIMAL REGULATIONS 25.7.1. A n A l g o r i t h m for A d a p t i v e Harvest Management
ai(dtlx t) -= ~ p(ht[dt){bl[Hi(ht[xt)3}, ht
(25.9)
i
= ~ Pi(t) ~ , p(htldt){u[Hi(htlxt)~},
i
ht
where pi(t) is the weight for model i [~,i Pi(t) = 1]. The n o t a t i o n a(dt]xt, Pt) in this expression indicates that the return accruing to decision d t depends on the model weights in Pt as well as system state x t. Similarly, system transitions can accommodate random effects and structural uncertainty. Thus, each of the transition models in Eq. (25.1) inherits stochastic behaviors from random environmental variation and partial controllability of harvests, on the basis of which a state transition probability structure can be derived. Let pi(x t +1 IX t, dt) represent the model-specific probability of transition from state x t to Xt+l, given regulatory decision d t. These transition probabilities can be aggregated across models into an average probability by
dt) = ~ pi(t)pi(xt+llXt, dt). i
(25.10)
The averages in Eqs. (25.9) and (25.10) are used in the Hamilton-Jacobi-Bellman algorithm (see Section 23.3) as if they represent the utilities and transition probabilities for a single model: V*(xt) = maxdt { -~(dtlXt'pt)+
~ ~ pi(t)pi(xt+ltXt'dt)g~'(Xt+l)} Xt+ 1
i "x
TABLE 25.4 Mean Harvest Vulnerability (SE) of Female and Young Mallards a Mean harvest vulnerability (SE) Young females
Young males
Midcontinent
0.748 (0.108)
1.188 (0.138)
1.361 (0.144)
Eastern
0.985 (0.145)
1.320 (0.264)
1.449 (0.211)
a
R(dtlxt, Pt) -~ ~_j pi(t)Ri(dtlxt)
(25.8)
where u[Hi(htlxt)] is the utility accruing to harvest Hi(htlx t) and p(htld t) is the probability of a specific harvest rate conditioned on the regulatory decision d t. The notation Hi(htIx t) indicates that harvest is a modelspecific function of system state x t and harvest rate ht, and the summation in Eq. (25.8) essentially averages harvest utilities over the possible harvest rates corresponding to regulatory decision d t. The model-specific
Adult females
utilities in Eq. (25.8) can be aggregated into an average utility by
F(Xt+I[Xt,
Implementation of a regulatory strategy for sport hunting yields annual benefits for waterfowl harvest, and the goal of management to provide as large a temporal sum of benefits as possible over an extended time frame. Assuming model i, the immediate harvest benefit at time t for state x t and regulatory decision d t is
Mallard population
679
Relative to adult males, based on band recovery data, 1979-1995.
(25.11)
~-a(dtlxt, Pt) + ~ -P(X_.t+llXt,dt)W*(Xt+l)~) dt I, xt+l
= max
(see Section 24.11). Algorithm (25.11) shows that an optimal regulatory strategy can be identified sequentially, in a manner that explicitly accounts for structural uncertainty, environmental variation, and partial controllability of harvests. After some finite number of iterations, continued application of Eq. (25.11) produces the same statespecific decisions, so that the decision structure becomes time independent. The result is a stationary open-loop feedback strategy (see Section 23.3) that identifies an optimal regulatory decision for each combination of population size and number of ponds on the breeding grounds. In 1995 the adaptive regulations process was implemented, and in that year the following steps were taken: 9 The population size and number of ponds were determined from the breeding grounds survey.
680
Chapter 25 Case Study
9 Initial model weights were chosen to be pi(O) = 0.25, thereby weighting each of the four models equally. 9 Average utilities and transition probabilities were identified as in Eqs. (25.9) and (25.10), based on these weights. 9 The average utilities and transition probabilities were used to identify an optimal policy with algorithm (25.11). 9 The appropriate regulatory decision was identified for the population and habitat conditions on the breeding grounds. The strategy identified in 1995 was specific to the set of equal model weights used in Eqs. (25.9)-(25.11). These weights have evolved over time, as information from the breeding grounds survey is used to compare model predictions against population status. Thus, in each succeeding year since 1995, the information from previous years has been incorporated in the process via the following actions: 9 Each spring the population size and number of ponds (xt+1) have been determined from the breeding grounds survey. 9 Population and habitat conditions have been used to identify the transition probabilities pi(Xt+llXt, dt). 9 The transition probabilities have been used to determine the model weights pi(t + 1) with Bayes' Theorem (see Section 24.5 and Appendix A), based on pi(t) from the previous year. 9 Average utilities and transition probabilities have been identified as in Eqs. (25.9) and (25.10), based on the weights in p(t + 1). 9 The average utilities and transition probabilities have been used to identify an optimal regulatory policy with algorithm (25.11). 9 The appropriate regulatory decision has been identified for the population and habitat conditions in xt+ 1. This sequence describes a "passive" adaptive approach to harvest management (see Section 24.11). The process involves incrementing the time index t by 1 each year, and then following the prescribed sequence of actions based on new information from the breeding grounds. This approach to harvest management has been in use since 1995. The sequence can be applied on a continuing basis, thereby allowing the model weights pi(t) to evolve over time. On assumption that the model set contains a model describing system dynamics appropriately, in theory the continued application of the sequence will lead to the convergence of model weights to unity for that model and to zero for the other models.
25.7.2. Optimal Regulatory Prescriptions The AHM process was implemented in 1995 based on equal weights for each model, and each year since then the strategy has been revised as the model weights have evolved. As shown in Tables 25.5-25.9, optimal harvest strategies for the 1995-1998 seasons have shown changes over time as model weights have changed (U.S. Fish and Wildlife Service Office of Migratory Bird Management, unpublished report). The 1999 AHM strategy is based on (1) regulatory alternatives that are unchanged from 1997 and 1998, (2) model weights for 1999, and (3) the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million birds (Table 25.10). This strategy provides optimal regulatory choices for midcontinent mallards assuming that all four flyways use the prescribed regulations. Blank cells in Tables 25.6-25.9 represent combinations of population size and environmental conditions that are insufficient to support an open season, given current regulatory alternatives. In the case of midcontinent mallards, the prescriptions for closed seasons largely are a result of an emphasis on population growth at the expense of hunting opportunity when mallard numbers are below the NAWMP goal. Of course, a decision to close the hunting season always depends on both biological and sociological considerations, recognizing that limited harvests at low population levels might well impact long-term population viability only slightly, if at all. Population dynamics were simulated with the harvest strategy in Table 25.10 with the four population models and current weights, to determine expected performance characteristics. Assuming that regulatory choices adhere to this strategy, the results indicate the annual harvest and breeding population size would average 1.3 million (SE = 0.5) and 8.3 million (SE = 0.9), respectively. Based on a breeding population size of 11.8 million mallards and 3.9 million ponds in Prairie Canada from the breeding grounds survey in 1999, the table indicates that the optimal regulatory choice for midcontinent mallards is the liberal alternative.
25.8. SOME ONGOING ISSUES IN WATERFOWL HARVEST MANAGEMENT 25.8.1. Setting Management Goals Natural resource management is a process of using biological information to predict the consequences of management and sociological information to value those consequences (Lee, 1993). When managers agree
25.8. TABLE 25.5
Some Ongoing Issues in Waterfowl Harvest M a n a g e m e n t
681
Temporal Changes in M o d e l Weights ("Likelihoods") for Alternative M o d e l s of Midcontinent Mallard Population D y n a m i c s a Model weights
Mortality hypothesis
Reproductive hypothesis
1995
1996
1997
1998
1999
Additive mortality
Strong density dependence
0.2500
0.65479
0.53015
0.61311
0.60883
Additive mortality
Weak density dependence
0.2500
0.324514
0.46872
0.38687
0.38416
Compensatory mortality
Strong density dependence
0.2500
0.0006
0.00112
0.0001
0.0001
Compensatory mortality
Weak density dependence
0.2500
0.0001
0.0001
0.0001
0.007
a Four models are included in the model set, each including a different combination of hypotheses about (1) the effects of hunting on annual mortality density, and (2) the magnitude of density dependence in reproduction.
on b o t h goals a n d c o n s e q u e n c e s , m a n a g e m e n t decis i o n s c a n b e b a s e d o n a n e s t a b l i s h e d r o u t i n e of g a t h e r i n g a n d e v a l u a t i n g i n f o r m a t i o n . W h e n t h e r e is d i s a g r e e m e n t a b o u t m a n a g e m e n t g o a l s , a p r o c e s s of n e g o t i a t i o n a m o n g s t a k e h o l d e r s is n e c e s s a r y to d e v e l o p a c c e p t a b l e policy. O n t h e o t h e r h a n d , w h e n m a n a g e m e n t g o a l s a r e b r o a d l y a c c e p t e d b u t t h e r e is d i s a g r e e m e n t o r u n c e r t a i n t y a b o u t t h e i m p a c t s of m a n agement actions, adaptive management can be a useful t o o l for a d d r e s s i n g a n d r e s o l v i n g t h e conflicts. I n effect, a d a p t i v e m a n a g e m e n t a l l o w s m a n a g e r s to a g r e e o n
TABLE 25.6 Optimal Regulatory Choices for Midcontinent Mallards during the 1995 H u n t i n g Season a Ponds b Mallards c
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
M L L L L L L L L L L L L L
M L L L L L L L L L L L L L
M L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
policy when they do not necessarily agree on the outc o m e s . O n t h e o t h e r h a n d , it is n o t b y itself a u s e f u l a p p r o a c h for a d d r e s s i n g d i s a g r e e m e n t o v e r m a n a g e m e n t goals a n d objectives. It s e e m s o b v i o u s t h a t a n y d e c i s i o n - m a k i n g p r o c e s s w i l l b e l i m i t e d in its e f f e c t i v e n e s s if t h e r e is a m b i g u i t y a b o u t t h e g o a l s o r o b j e c t i v e s of t h e p r o c e s s . Yet, m u c h of t h e h i s t o r y of w a t e r f o w l h a r v e s t m a n a g e m e n t in N o r t h A m e r i c a h a s b e e n m a r k e d b y a l a c k of explicit, u n a m b i g u o u s , a n d a g r e e d - u p o n o b j e c t i v e s ( N i c h o l s et al., 1995a). P e r h a p s b e c a u s e h a r v e s t e d w a t e r f o w l a r e not a commercial commodity, there always has been
TABLE 25.7 Optimal Regulatory Choices for Midcontinent Mallards during the 1996 H u n t i n g Season a Ponds b Mallards c
a This strategy is based on the regulatory alternatives for 1995, equal weights for four alternative models of population dynamics, and the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million. R, Restrictive; M, moderate; and L, liberal. b Estimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
R R M M L L L L L L L
2.0
R R M L L L L L L L L
2.5
R R M L L L L L L L L
3.0
3.5
4.0
4.5
5.0
5.5
6.0
R M L L L L L L L L L
R R M L L L L L L L L L
R R M L L L L L L L L L
R M L L L L L L L L L L
R R M L L L L L L L L L L
R M L L L L L L L L L L L
R M L L L L L L L L L L L
a This strategy is based on the regulatory alternatives and model weights for 1996 and the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million. R, Restrictive; M, moderate; and L, liberal. b Estimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
682
C h a p t e r 25
TABLE 25.8 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1997 H u n t i n g S e a s o n a
Case Study TABLE 25.10 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1999 H u n t i n g S e a s o n a Ponds b
Ponds b Mallards c
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
VR R R M M L L L L L
2.0
VR R R M M L L L L L
2.5
VR VR R M M L L L L L L
3.0
VR VR R M M L L L L L L
3.5
VR R R M L L L L L L L
4.0
VR R M M L L L L L L L
4.5
VR R M L L L L L L L L
5.0
VR R M M L L L L L L L L
5.5
VR R M L L L L L L L L L
6.0
VR R M L L L L L L L L L
a This strategy is based on regulatory alternatives and model weights for 1997 and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.7 million. VR, Very restrictive; R, restrictive; M, moderate; and L, liberal. bEstimated number of ponds in Prairie Canada in May, in millions. CEstimated number of midcontinent mallards during May, in millions.
TABLE 25.9 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1998 H u n t i n g S e a s o n a Ponds b Mallards c
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
VR R R M M L L L L L
2.0
VR VR R M M L L L L L L
2.5
VR VR R M M L L L L L L
3.0
VR R R M L L L L L L L
3.5
VR R M M L L L L L L L
4.0
4.5
5.0
5.5
6.0
VR R M L L L L L L L L
VR R M M L L L L L L L L
VR R M L L L L L L L L L
VR R M L L L L L L L L L
VR R M L L L L L L L L L L
a This strategy is based on regulatory alternatives and model weights for 1998 and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.7 million. VR, very restrictive; R, restrictive; M, moderate; and L, liberal. bEstimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
Mallards c
0 for all values of Yi, the latter requirement is met only on condition that ~ki ~ 0, i = 1, ..., m.
= [dyij/dt].
Finally, differentiation of the elements in the n-dimensional vector y by elements in the m-dimensional vector x can be expressed in matrix form as =
ax
roy/] Lax/i 3yl / c~X1
...
3y 1/ (~Xm
9
LOYn)OXm
..i
~
Oyn)OXmJ
A P P E N D I X
C Differential Equations
In this appendix we describe procedures for analysis of some differential equations that arise in population ecology. These equations express differential change in population size over a continuous time frame, with population trajectories that are given by their solution. The procedures discussed below apply to multiple populations or to a single population with multiple cohorts (or both). Without loss of generality, we refer to N(t) as a vector of populations, recognizing that N(t) also may represent a vector of population cohorts. More detailed treatments of differential equations can be found in a large number of references, such as Tenenbaum and Pollard (1985), Coddington and Landin (1989), and Rainville et al. (1996). Consider a set of populations (or population cohorts) N(t) that experience change through time according to the equation dN/dt
is nonlinear. Under some quite general conditions, the growth function for a single population can be written as a Taylor series expansion oo
F(N) =
so that the models can be expressed as (possibly infinite) polynomials. For example, the exponential model in Section 8.1 requires only a linear term for an exact representation of its growth function, the logistic model in Section 8.2 requires linear and quadratic terms, and the Allee effect F ( N ) = alN + a2N2 + a3N3
expressing depensatory population change can be modeled with linear, quadratic, and cubic effects. The growth functions for most population models are complicated mathematical expressions, polynomial approximations for which may require many terms. For example, growth functions for the Gompertz model of human mortality and the Ricker and the Beverton-Holt models from fisheries biology involve exponential terms, and therefore require infinite series of polynomial terms for an exact representation. If F(N) contains only terms that are functions of N, the equation is said to be homogeneous; otherwise, it is nonhomogeneous. Thus, d N / d t = r N is a homogeneous differential equation, whereas
= F(N),
in which F(N) expresses differential change in population status at any particular point in time. This formulation represents population change with a first-order differential equation, i.e., a differential equation that includes only first derivatives. It is assumed here that the growth function F(N) is well behaved, in that the derivatives of F _ with respect to the population units in _N exist. If F(N) contains no terms of degree higher than 1, then the equation is said to be linear; otherwise, it is nonlinear. For example, dN/dt
= rN
dN/dt
is a linear differential equation, whereas the logistic equation dN/dt
= rN(1 -
akN k, k=0
= rN + c
is nonhomogeneous. Homogeneity obviates population change in the absence of individuals in the population, e.g., spontaneous generation (or independent
N/K)
693
694
Appendix C Differential Equations
immigration). A homogeneous growth function F(N) can be expressed by a Taylor series: F(N) = a l N + a2 N 2 + a3 N 3 + ... = N(a I + a2N + a3N2 + ...
One need not use the artifice of choosing an appropriate function to derive this solution, because Eq. (C.1) is simple enough that it can be solved by straightforward integration. Thus, Eq. (C.1) can be rewritten as d N / N = rdt, and integration yields
= NG(N).
f dN
The function G(N) is the instantaneous rate of growth or per capita rate of growth, which varies with N for all polynomial growth functions except the constant growth rate of the simple exponential. Finally, a constant coefficient differential equation contains coefficients that are invariant through time. For example, the logistic equation above is a constant coefficient differential equation, but replacement of r with an oscillatory term such as r(t) 4: r sin(t/4) produces a differential equation
= In(N)
-frdt =rt +c,
with the result that N(t) = kert,
where k = ec. Substituting (0, N 0) for [t, N(t)] in this equation produces k = N 0, so that N(t) = N o eFt.
d N / d t = r(t)N[1 - N / K ] , Example
with a time dependent growth coefficient r(t).
Consider a logistic population as in Section 8.2, with population dynamics given by
C.1. FIRST-ORDER LINEAR H O M O G E N E O U S EQUATIONS
d N / d t = rN(1 - N / K )
C.1.1. Population Dynamics for One Species
and initial population size N(0) = N 0. A closed form for the population trajectory can be solved by rewriting Eq. (C.2) as
Consider first a single population with no cohort structure, for which population dynamics are characterized by a single linear homogeneous differential equation. Several methods are available for solving this equation, including the use of Taylor series expansion, numerical integration, and mathematical analysis.
dN =rdt N(1 - N / K )
and recognizing that
1 N(1 - N / K )
Example
Consider the exponential model described in Section 8.2, d N / d t = rN,
=
1
t
N
1 K-
N
Then
(C.1)
T N(I
-
dN N/K)=
with initial population size N(0) = N 0. A simple approach is to assume a solution of the form N(t) = ke ~t, so that
dN dN f --N + f K - N
=In
(N) N-K
+C
= rt,
d N / d t = Mke xt)
so that
= XN.
Comparing this expression with Eq. (C.1), we have X = r and N(t) = ke rt, with the constant k determined by
N N-K
-- ce rt"
Substituting the initial condition N(0) = N O into this expression yields c = N o / ( K - No), so that
N(O) = ke ~
= N O. Thus, a complete solution for the model is N(t)
(C.2)
= N o ert.
N K-N
o
K-
rt
No e "
C.1. First-Order Linear Homogeneous Equations After some algebra, the population trajectory can be rewritten as N(t) = 1 + Ce -r(t - to),
with C = K/N o - 1. Differentiation of this equation demonstrates that it satisfies Eq. (C.2).
C.1.2. Population Dynamics for Two Species The mathematical situation is somewhat more complicated with two populations. Let the population dynamics again be specified by linear homogeneous differential equations dN1/dt = a11N1 + a12N2,
(C.3)
dN2/dt = a 2 1 N 1 + a 2 2 N 2.
Two approaches are available for the solution of this system. The first approach utilizes the fact that these equations can be combined into a single second-order differential equation, and the second approach utilizes matrix theory. C.1.2.1. Second-Order Equations
A pair of first-order differential equations can be combined into a single second-order differential equation, which then can be solved by straightforward algebraic procedures. For example, N 2 can be eliminated from the pair of equations in Eq. (C.3) by a second differentiation of the transition equation for NI: d2Nl= dt 2
a dN1 dN2 11---~- + a12 dt a
dN1
--- 11--~- q- a12(a21N1 q- a22N2)
11---~ q- a12a21N1 + a12a22 - - ~
- a11N1
)/
a12.
Thus, elimination of N 2 in Eq. (C.3) results in the second-order equation d2N1/dt 2 -
otdN 1/dt
+ ~ N 1 = 0,
(C.4)
with ot = all -t- a22 and [3 = alia22 -- a12a21. It is easily shown that the elimination of N 1 leads to a differential equation in N 2 with the same coefficients. To solve the system of differential equations [Eq. (C.3)], assume a solution of the form Nl(t) = ke ~'t for Eq. (C.4). Substitution of the first and second derivatives of Nl(t) into Eq. (C.4) results in ~2 _ OLd. + [3 -- O,
(C.5)
695
a quadratic equation that is satisfied for the values K -- 0.5[or -+- (or 2 -- 4 ~ ) 1 / 2 ] . Thus there are two solutions to Eq. (C.4) of the form Nl(t) = ke xt. Furthermore, any linear combination N l ( t ) = k11e~,1 t + k12 e~'2t
also is a solution. Population dynamics for Nl(t) thus are determined by the parameters )~1 = (o~ + V ~ ) / 2 and )~2 = (Or -- V ~ ) / 2 , where y = ot 2 - 4~ is the discriminant of Eq. (C.5). The parameters K 1 and K2 a r e either both real or both complex, depending on whether ~/>- 0. As shown in Fig. C.1, several possibilities arise: 1. The parameters are both positive but not identical. This situation occurs if oL and y are positive and c~ > Vyy, where the latter condition is equivalent to f~ > 0. Under these conditions, the population trajectory for Nl(t) is a linear combination of two exponential components, both of which increase (but at different rates) through time. The exponential term in )k I dominates the trajectory as t increases. 2. One parameter is positive and the other is negative (or 0), i.e., )~1 ~ 0 and )~2 -< 0. This situation occurs if V ~ - > levi, which holds if and only if ~ -< 0. On condition that )~1 and )~2 differ in sign, the population trajectory for Nl(t) is a linear combination of two exponential components, one of which decreases as the other increases through time. 3. Both parameters are negative but not identical. This situation occurs if the discriminant y is positive and V~y < -e~. With y > 0, sufficient conditions for ~1 and )~2 to be negative are ot < 0 and [3 > 0. If )~1 and )k 2 a r e negative, the population trajectory for Nl(t) is a linear combination of two exponential components, each decreasing through time. 4. The parameters are complex conjugates" ~'1 = O~ + iX/-8 and )~2 = Ot -- iVS, where i = ( - 1 ) - 1 / 2 and 8 = ]y]. This situation occurs if the discriminant is negative, in which case the solution of Eq. (C.4) can be expressed as Nl(t) = k11e~,lt + k12e~'2t = kll e~t {cos(St) + i sin(St)} + k12 e~'t {cos(St) - i sin(St)} = e~t(k11 + k12) cos(St) + ie~t(k11 - k12) sin(St) = e at {ClCOS(St) + c 2 sin(St)}.
This combination of sinusoidal terms satisfies Eq. (C.4) for any choice of cI and c2. It is possible to describe Nl(t) in real terms only, by Nl(t) = e at {C 1 cos(St)
+ C2
sin(St)}.
696
Appendix C Differential Equations
NI(O
N,(t)
F I G U R E C.1
Possible system trajectories arising from solutions ~kl,2 -- 0.5[Or -+- (Or2 -- 4~) 1/2] of the equation and k 2 are real; the system trajectory is decreasing only if k 1 and k 2 both are negative. a r e complex; the trajectory is oscillatory with either increasing, decreasing, or stationary
~2 _ Ot~k -f- ~ -- 0. (a) K1 (b) ~1 and amplitude.
K2
The corresponding population trajectory exhibits oscillations of period 2~r/8, with the magnitude of the oscillations increasing, decreasing, or stable through time depending on whether ot is positive, negative, or zero, respectively. 5. The parameters }kI and k 2 are identical. This situation occurs if the discriminant is 0, which is equivalent to all -a22 = -4a12a21.
In this special case, a solution to Eq. (C.4) is given by Nl(t ) = kll ext + k12te xt, with population dynamics exhibiting exponential growth (or decay) scaled by the factor kll + k12t. In the preceding development, the two equations in Eqs. (C.3) were combined so as to eliminate N2(t). It is easy to show that a companion differential equation in N2(t), obtained by elimination of Nl(t) in Eqs. (C.3), has a solution of the form N2(t) = k21eMt + k22ex2t, with the same values kl and ~k2 in the exponential terms and therefore the same patterns in the trajectory of
N2(t). The values k21 and k22 a r e related to kll and k12 by k21 = kll (~1 -
a11)/a12
k22 = k12(~ 2 -
a11)/a12 ,
and
with specific values of kll and k12 determined by the system initial conditions. In summary, the differential equations [Eqs. (C.3)] give rise to a range of possible system behaviors, depending on the values of e~, I3, and ~/: 1. Weighted exponential growth with distinct exponential rates of growth: ~ > 0, [3 > 0, ~ > 0. 2. Weighted average of exponential growth and decay:
~ 0, ~/> 0. 4. Unstable (increasing) oscillation: a > 0, ~/< 0. 5. Damped (decreasing) oscillation: e~ < 0, ~/< 0. 6. Stable or neutral oscillation: a = 0, ~ < 0. 7. Exponential growth (or decay) with a single growth rate: ~/ = 0.
C.1. First-Order Linear Homogeneous Equations Because y is defined in terms of oLand [3, these system behaviors can be described in terms of the latter parameters alone. Thus, the plane defined by ((x, [3) can be partitioned into zones corresponding to the first six conditions described above, with condition 7 associated with the parabola oL2 = 4[3 (Fig. C.2).
697
C.1.2.2. M a t r i x A p p r o a c h
The system of equations shown in Eqs. (C.3) can be expressed in terms of matrices, by
aN1/at] dN2/dtJ
Example
Consider the population trajectories of N' = (N 1, N2) as defined by the transition equations
=
,21 [:12]
[all [a21
(C.6)
a22J
or d N / d t = A N. As above, assume a solution to this matrix equation of the form
[NI] iv1]
d N 1 / d t = 3N 1 - N2,
=
e at
N2
v2
d N 2/dt = 6N 1 - 4N 2.
Combining these transition equations results in the second-order differential equation
or N = ve ~'t. Then d N / d t = Mve xt) = A(ve~'t), resulting in the matrix equation A v = )~v or (A - )U)v = 0.
d2N1/ dt 2 4- d N 1/ dt - 6N 1 = 0.
Substituting first and second derivatives of ke ~'t into the latter equation yields )~2 + ) ~ _ 6 = 0 , with solutions is
~kI =
2 and
)k 2 - -
- 3 . The general solution
with specific values for k l l and k12 determined from the system initial conditions. Thus, the trajectory for the system is characterized by a combination of two components, one increasing exponentially and the other decreasing exponentially through time. Because = - 6 < 0, this behavior is consistent with condition 2 above.
"x asymptotic decrease
increasing oscillation /
X1, ~,20
asymptotic increase ;~l >0, ;L2 0 (because the trajectory of deviations exhibits exponential growth).
the higher degree terms are of negligible importance, and Eq. (C.15) reduces to F k ( N * + n) = nlFkl(N *) + n2Fk2(N*).
As in the single-species case, we can express d ( ~ / d t
Example
as
Consider the logistic model d N / d t = d(N* + n ) / d t d N / d t = rN(1 - N/K)
from Section 8.2 with constant growth rate r > 0 and constant carrying capacity K > 0. The model has two equilibria, N* = 0 and N* = K, and deviations in a neighborhood of N* are given as in Eq. (C.14), by dF , dn/dt = n -~(N ) = rn(1 - 2N*/K).
Population dynamics around N* = 0 are specified by dn / dt = rn (1 - 2N* / K)
= dn/dt,
so that the equation for population dynamics can be expressed in terms of the deviations n = N - N* as d n k / d t = n]Fk(N *) + n2Fk(N*),
k=l, 2. Thus, the nonlinear transition equations can be approximated by linear differential equations in a neighborhood of N*. Equation (C.16) is written in matrix notation as [dnl/dt]
-- TTl,
d n / d t = rn(1 - 2 N * / K )
--
~
OF2
L~-~ (N*)
tl 1
n2
(C.17)
= J(N*)n, and the properties of J(N*), known as the Jacobian matrix, determine the equilibrium stability of the system. Assuming a solution of the form n(t) = ve ~t, Eq. (C.17) reduces to
= rn(1 - 2 K / K ) --
OF 1
F 3F1 |-~11 (N*)
Ldna/dtJ = ] OF2
which exhibits simple exponential growth away from 0. Thus, N* = 0 is an unstable equilibrium, in that positive deviations from 0 increase over time. On the other hand, population dynamics around N* = K are specified by
(C.16)
[J(N*) - M]v = 0,
TYl,
for which the characteristic equation which exhibits simple exponential decay toward 0. Thus, N* = K is a stable equilibrium, in that deviations from K lead to asymptotic declines in the deviations [and thus to asymptotic convergence of N(t) to K].
I!(_N*) -
is a polynomial of degree 2. Thus, either of the pairs ()~1, Vl) and (~2, v2) of eigenvalues and eigenvectors corresponds to a solution
C.2.2. Stability Analysis for Two Species As before, the addition of another state variable complicates the analysis. Consider two populations with nonlinear growth functions F(N)' = [FI(N), F2(N)]. A Taylor expansion about an equilibrium value N* is F k ( N * + n__) = F k ( N *) + nlFlk(N *) + n2Fk2(__*) 2
-Jr-
n 2F k22'__" rN,~ kl(X~') -}- T
(C.15)
x_ l = o
n(t) = vi e•
of d n / d t = J(N*)n, and a general solution is given by
n(t)
= Cl(Vl exit) + c2(v2eX2t).
From the analysis of linear differential equations, N* is a stable equilibrium if the roots ~'1 and K2 of Ij(N, ) _ )~ii = ~ 2 _ )kO~ -Jr-
=0 + (nln2)Fk12(N *) + ...
for k = 1, 2, where F/k(_N)= 3 F k ( ~ / O N i and F~(N) = 3 2 F k ( ~ / O N i 3Nj. For "small" deviations n = N - N*,
are both negative. For positive discriminant ~/= ~2 _ 4[3, this condition is equivalent to oL = tr[J(N*)] = all + a22 < 0 and [3 = IJ_(N*)I = a11a22 - a12a21 > 0 (see
702
Appendix C Differential Equations
Section C.1.2). Thus, tr[J(N] < 0 and IJ_(N*)I > 0 ensure that deviations n I and n 2 both decrease asymptotically to 0 [so that N(t) converges to N*].
Thus, the population dynamics for this system exhibit neutral stability in a neighborhood of (48, 80), with trajectories given by
Example
N(t) = N* + n(t)
Consider a pair of populations with dynamics governed by the system of equations
_[N~] I_N~J
dN1
N1
N2
dt
3
5'
I
dN2 7N2(1 - N2/35 ) N1 = 4 dt 75 5
OF 1
i
1
12cost4t) 2 sin(4t) + 48
OF2 -~2 (N*)
_1
and
7(1 - 2N~/35) " 5 75
12cos(4t) 26sint4t) + 80.
For deviations in a neighborhood of N* = 0, the Jacobian is
J(N*) =
.
Example Consider the Lotka-Volterra competition equations
1
[ Nlj t] [F1NII N1 a12 2 J l]
,
Ygj
dN2/dt
for which tr[/(N*)] > 0 and IJ(N*)I > 0. From the analysis of linear differential equations, it follows that N* = 0 is an unstable equilibrium, in that positive deviations from 0 exhibit growth away from 0. On the other hand, the Jacobian for N* = (48, 80) is 1
'
where the coefficients C 1 and C2 are determined by the initial population sizes. For initial population sizes of, say, N~ = (60, 92), it is easy to show that the system oscillations are described by
;~11(N_*) ;G2 (N_*) J(N*) = I OF2(N,) LON 1 --
1
+ 5ClcOs(~t)+4C 1sin(4t)+5C2sin(4t)-4C2cos(~t)
It is easy to show that this system has two equilibria, N* = 0 and N* = (48, 80). The Jacobian for the system is OF
3C1cos(4t)+ 3C2 sin(4t)
r2N2(K2 -
N 2 -
a21N1)/K2
from Section 8.8, for which there are two equilibria, N* = 0 and m
K 1 - a12K2 IN, I= LN~_J
1
1 - a12a21
K2 - a21K1
9
1 - a12a21 The Jacobian for this system is for which tr[J(N*)] = 0 and IJ(N*) i > 0. These conditions indicate that the deviations n(t) exhibit stable oscillations, with the deviation trajectories described by n(t) =
I
()
3~,cos/~tt+3~2sin(4tt
()()
(
5C1cos 4t +4C 1sin 4t +5C 2sin ~t -4C 2cos ~t
tl
J(N*) r OF1
9 OF1 / ~ ~/1 = /oF 2
[r1(1 -
I
,
aF 2
. /
2N'~/K 1 - a12N'~/K1) -r2a21N~/K 2
-rla12N'~/K1
r2(1 -
]
2N'~/K 2 - a21N'~/K2) ~"
C.2. Nonlinear Homogeneous EquationsfStability Analysis For deviations in a neighborhood of N* = 0, the Jacobian takes the values
of Section 8.7. This system has two equilibria, N* = 0 and
[
J(N*)
-[
703
X ~ ] __ [ d 2 / b 2 ]
r111 - O/K 1 - a12(O/K1) ] - r2a21(0/K 2)
-rlal2(O/K1) r211 -
]
OIK2- a2~(OIK2)]
=Er,0 r0]"
N~J
kblldlJ"
where N 1 and N 2 characterize prey and predator populations, respectively. The Jacobian of the system is
J(N*) =
Because
OF
9
OF 1
9 -I
I 3F 2
9
3F 2
,/
tr[J(N*)] = r I + r 2 > 0
pbl-alN2 L b2N2
and
I!(N*)I =
rlr2 > O,
it follows that _N* = _0 is an unstable equilibrium, in that positive deviations from _0 exhibit growth away from 0. On the other hand, the Jacobian for
-alN1 -I b2N1 - d2 ]"
For deviations in a neighborhood of N* = 0, the Jacobian takes the values
r bl - dl(~
l(m*) - --
L
b2(O)
-dl(~
-I
b2(O)-d2]
K 1 - a12K 2
IN~}= LN~J
1 - a12a21 K2-
a21K1
1 -
a12a21
This corresponds to the simple differential equations dn 1 / d t = bin 1
and
](N*) =
dn2/dt = -d2n 2
(1 -- a12a21) -1 r - r l ( K 1 - a12K2)
t r2a21(K 2 -
rla12(K 1 - a12K2)]
a21K1)
- r 2 ( K 2 - a21K1) J '
for which tr[J(N*)] = - r l ( K 1 - a12K2) + r2(K2 - a21K1) 1 - a12a21 and
I/(N*)I
in the deviations//1 and n 2, so that//1 (and therefore N 1) increases in a neighborhood of 0, while n 2 (and therefore N 2) decreases. This accords with the biological sense of predator-prey interactions, whereby small numbers of predators allow for growth of a prey population, and small numbers of prey lead to predator declines. On the other hand, the Jacobian for N*' = (d2/b2, b 1 / d 1) is j(N,)
= rlr2(K1 - a12K2)(K2 - a21K1).
b2(bl/dl)
If competition is not severe (that is, if K 1 - a12K 2 > 0, a21K 1 > 0, and 1 - a12a21 ~ 0), it follows that tr[J(N*)] < 0 and IJ(N*)I > 0. The latter conditions ensure that deviations in a neighborhood of the equilibrium converge to 0, so that the population returns to N*.
Example The analysis of stability provides a mathematical justification for the oscillatory patterns observed with the Lotka-Volterra predator-prey equations
Ibl-dlN1][ =
b2N2
_d 2
N1] N2
(C.18)
-dl(d2/b2) ] b2(d2/b2) - d 2
I0
K2 -
dN1/dt] dNaldt ~
= [bl - d l ( b l / d l )
bib 2
for which tr[J(N*)] = 0 and IJ(N*)I bid 2. At this equilibrium, the system eigenvalues are the complex conjugates K1,2 = 0.5(-bid2 )1/2. The corresponding deviation trajectories are sinusoidal, and the populations exhibit stable oscillations about the equilibrium. =
Example It is straightforward to show that the stable oscillations of a Lotka-Volterra predator-prey system are not
704
Appendix C
Differential Equations
maintained in the presence of density-dependent birth. Let the p r e d a t o r - p r e y system in Eq. (C.18) be modified by
aN1/at dN2/dt]
= [bl(1-N1/K) b2N 2
-diN1 -d2 ] [X12]"
for k = 1, ..., m. For small deviations n = N - N*, the higher degree terms are of negligible importance, and Eq. (C.19) reduces to
m OFk Fk(N * + t l ) = s n i - ~ i (N*). i=1
As above, we can write d ( N ) / d t as
Equilibria for this system are N* = 0 and
d N / d t = d(N* + n ) / d t = dn/dt, =
LN~._]
bl dll
bid2 ' b-~-lK_]
so that the equation for population dynamics can be expressed in terms of the deviations n = N - N* as
with d 2 < b2K a necessary condition for N~_ to be positive. The Jacobian matrix is [b I - (2blN~/K) -
J(N*)
L
baN ~
diN'2 -dlN'~ 1 b2N'~ - d2_]"
For N* = 0, the Jacobian is identical to the matrix for the unmodified Lotka-Volterra system, so the dynamics of n 1 and n 2 in a neighborhood of 0 are the same as in the previous example. However, the Jacobian at N*' = [d2/b 2, b l / d I - bld2/(b2dlK)] is
OFk ni ~ i (N*),
dnk/dt =
(C.20)
i=1
k = 1, ..., m. Thus, the nonlinear transition equations can be approximated by linear differential equations in a neighborhood of N*. Equation (C.20) can be expressed in matrix notation as
n
J(N*) =
bid2 b2K bib______22 bid2 dl dlK
d n l d t = !(N*)n, where n' = (n I .... , n m) and
J(N*) = I_-~i (N*) .
dl___d2 l
I'
A s s u m i n g a solution of the form n_(t) = ve ~t, this system reduces to [/(N*) - XI]v = 0,
for which tr[J(N*)] = -bld2/(b2K ) < 0 and IJ(N*)I = bid2[1 - d2/(b2K)] > 0. As argued previously, these conditions ensure that _N* is a stable equilibrium, so that deviations in a neighborhood of N* are eliminated as N(t) returns to N*.
C.2.3. Stability Analysis for Multiple Populations
for which the characteristic equation IJ(N*) - ~/I = 0 is a polynomial of degree m. Thus, there are m combinations (h i, vi) of eigenvalues and eigenvectors for which Eq. (C.20) is satisfied. A n y of these combinations corresponds to a solution n(t)
Consider m populations with nonlinear growth F ( N ) ' = [FI(N), F 2 ( ~ , ..., Fm(N)l. A Taylor expansion
=
vi e~it
of d n / d t = J(N*)n, and the general solution is given by
about an equilibrium value N* is n__(t) = ~ . ci vi e~#
m OFk Fk(N * + t l ) = Fk(N *) + ~.~ n i - ~ i (N*) i=1
+ ~
m
n2 c92Fk
i=1 2 a-~i2(N*)
m c92Fk + ~ , nin;_ (N*) + ... i,j=1 i c~mi cgXj --
9
i=1
(C.19)
As above, the deviation trajectories in a neighborhood of N* are controlled by the values hi, i = 1, ..., m. The trajectories decline if all eigenvalues hi are negative, and they increase if at least one eigenvalue is positive. Oscillations in the trajectories follow from the occurrence of complex conjugate eigenvalues. As with the single-species and two-species models,
705
C.3. Graphical Methods it is possible to describe stability conditions for a multispecies system in terms of the characteristic equation. Consider again the deviation model
dn/dt = J(N*)n, the eigenvalues for which are given by the characteristic equation IJ_(N*)
-
=
0.
This equation can be expressed as the polynomial Km + al Kin-1 4- a 2 ) t m - 2
+
""
+ a m - - O,
with a i given in terms of the coefficients of J(N*). Define m matrices Hj, j = 1, ..., m, such that Hj contains the elements
a21_k, 1, 0,
I
0 ~ 2l -
k ~ m;
21 = k; 2lk+
For example, H 1 =
H3
with
aI =
-(all
+ a22) a n d
a 2 = alia22 -
a12a21 . T h e
conditions a I ~ 0 and a 2 ~ 0 previously were shown to result in exponential declines. Finally, after some algebra, the system of equations,
IdN1/dt I [-1.75-0.75-0.5] dN2/dt = -0.75 -1.75 -0.5 dN3/dt -0.5 - - 0 . 5 - - 2 . 0
IN1] N2 , XsJ
which was shown in a previous example to exhibit exponential declines, can be seen to meet the Routh-Hurwitz criteria for equilibrium stability.
C.3. GRAPHICAL M E T H O D S
il
[ al =
a3
a2
aI
1
0]
a3
a2
al
a5
a4
a3
I
.
It can be shown that the equilibrium value N* is stable [that is, the real parts of all the eigenvalues for Eq. (C.20) are negative] if the determinant of each of the m matrices defined by Eq. (C.21) is positive:
]Hj] > O,
(C.22)
j = 1,..., m. The matrices Hj are called Hurwitz matrices, and conditions (C.22) constitute the Routh-Hurwitz criteria for stability. The Routh-Hurwitz criteria for systems of dimension m = 1, 2, 3, and 4 are 1" 2: 3: 4:
~. )k2 q- a l h + a 2,
(C.21) m.
and
= = = =
an - K a12 ~- )k2 -- (all + a22)h + (alia22 - a12a21 ) a21 a22- K
al,
S2
m m m m
tially if a I ~ 0. For two populations, the characteristic equation is
al>0; a I >0, a2>0; a I >0, a3>0, ala2>a3; a 1 > O, a3 > O, a4 > O, ala2a 3 > a2 + a2a4.
These conditions are in accord with the equilibrium conditions previously described. For example, the linear differential equation for a single population is d N / d t + alN = 0, with a corresponding characteristic equation of h + a I -- 0. The solution for this equation is
N(t) = No e-alt, so that the population trajectory decreases exponen-
One sometimes can obtain useful information about population dynamics without actually obtaining solutions for the corresponding differential equations. Often it is sufficient to recognize the direction of movement for a population of a given size at each point in time. The graphical representation of this information is called a direction field, consisting of direction vectors at each point in the (t, N) plane. A direction vector at (t, N) is simply the vector (1, dN/dt), with d N / d t = F(N) evaluated at (t, N). It represents the direction of change of the population in a neighborhood of (t, N). Curves of constant directional vectors in the (t, N) plane are given by F(N) = C, with different vectors specified by different values of C. This is illustrated in Fig. C.3 for the model d N / d t = N 2 - t. Note that the curves for which the directional vectors are unidirectional are given by N 2 - t = C (Fig. C.3a). Population trajectories coincide with points of tangency to the directional vectors (Fig. C.3b). If the differential equation is autonomous, i.e., if F(N) does not contain an explicit reference to t, then the direction vectors vary only with N over the direction field. The direction field for an autonomous growth function is illustrated in Fig. C.4a for the logistic equation d N / d t = N(1 - N). Because the direction field for an autonomous function varies with population size but not with time, one can essentially collapse the directional information in the direction field into a one-dimensional phase representation, with F(N) plotted against N. This is illustrated in Fig. C.4b for the logistic equation dF/dt = N(1 - N). Thus, the direction of change for N between 0 and 1 is positive (irrespective of the time at which the N achieves that
706
Appendix C Differential Equations
////
/////////..
\
\\\
\\\\\\\\\\\\\\\\\ \ \ \
\\\\\\\\\\
\ \
\ \
\\\\ ' // // // // / / / / , /
\\\
\ \\
N(t) a
\ K
\ \\\
\
\\\\ \\\\ \\ \\\ \\\
//Z
\\\~
//~/////,,
//X///~/////////// / /
/
/
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/
/
/
/
/
/
/
/
~
/
dN/dt
b
rKI4
F I G U R E C.3 Direction field for d N / d t = N 2 - t. (a) Directional vectors at each point [t, N(t)] are given by (t, N 2 - t). Directional vectors are constant along parabolic curves for which N 2 - t = C. (b) Population trajectories coincide with points of tangency to the directional vectors.
value), whereas the direction for N > 1 is negative. At the values 0 and 1, of course, the change is 0.
C.3.1. Stability Assessment with Null Clines The notion of a phase representation for autonomous differential equations can be extended naturally to two equations. In this case, a phase plane is described, with directional vectors at each point that are given by the growth functions F I(N) and F2(N) of the system of equations. Null clines are defined by the equation F I(N) = 0, which specifies curves in the phase plane for which the rate of change of N 1 is 0, and F2(N) = 0, which specifies curves for which the rate of change of N 2 is 0. System steady states are given by the intersections of the respective null clines. Vectors along the null cline F I(N) = 0 are of the form [0, F2(N)] and therefore are represented as
K/2
K
F I G U R E C.4 The direction field for logistic equation d N / d t = N(1 - N). (a) Because the logistic function is autonomous, its direction field varies with population size but not with time. (b) Onedimensional phase representation of the directional information in the direction field, with d N / d t plotted against N.
vertical arrows in a phase plane. Similarly, vectors along the null cline F2(N) = 0 are of the form [FI(N), 0] and are represented as horizontal arrows in a phase plane. Because the growth functions FI(N) and Fa(N) are assumed to be continuous in N, the direction vectors change smoothly along the null clines and therefore can change direction only at a steady state. Consider, for example, the system depicted in Fig. C.5 with null clines FI(N) = 0 and F2(N) = 0 and a unique steady state N*. Because the point P1 on the null cline FI(N) = 0 satisfies F2(N) > 0, all points on the null cline to the right of _N* must satisfy F2(N) > 0. Furthermore, the direction vector must reverse direction at N*, so that F2(N) < 0 for all points on the null cline to the left of N*. Similarly, FI(N) > 0 for the
C.3. Graphical Methods
707
F I G U R E C.5 Phase plane for a system with null clines FI(_N) = 0 and F2(_N) -- 0 and a unique steady state _N*. Both populations increase in region I, both decrease in region II, and the populations move in opposite directions in regions III and IV. These directional tendencies correspond to oscillatory system behavior.
III
G" eq
II
/ ,v
1
>4 1
/2
i
..q Nl(t)
point P2 on null cline F2(_N) = 0, and therefore FI(N) > 0 for all points on the null cline to the left of N*. Furthermore, the direction vector must reverse direction at N*, so that FI(_N) < 0 for all points on the null cline to the right of N*. An analogous logic can be applied to systems with multiple steady states, and in this w a y the pattern of direction can be deduced in a fairly straightforward way, with little calculation. The directions of the arrows along the null clines also are indicative of the direction of m o v e m e n t throughout a direction field. Thus, both populations increase in region I of Fig. C.5, both decrease in region II, and the populations move in opposite directions in regions III and IV. These directional tendencies correspond to oscillatory system behavior.
Example The use of direction fields can be illustrated with the Lotka-Volterra competition equations
d X l / d t ] __ [FINI(K1- X 1 --a12N2)/K1] dN2/dt Lr2N2(K2 - N 2 - a21N1)/K2] from Section 8.8. The null clines dN 1/dt = 0 for population N 1 are given by N 1 = 0 and N 1 = K 1 - a12N2, whereas the null clines dN2/dt = 0 for population N 2 are given by N2 = 0 and N 2 = K2 - a21N 1. The null clines N 1 = 0 and N 2 = 0 coincide with the axes of the (N 1, N 2) plane, limiting the operative values of N 1 and N2 to the set of nonnegative population values. The other two null clines are arranged in the (N 1, N 2) plane in one of four configurations, depending on the magnitudes of the carrying capacities and competition coefficients: Case 1. K 1 < a12K 2 and K2 > a21K 1. As s h o w n in Fig. C.6a, the null clines do not intersect, and the null cline for N 2 is to the right of the null cline for N 1. Direction vectors on dN1/dt = 0 point in the direction of growth for N 2, whereas direction vectors on
dN2/dt = 0 point in the direction of decline of N 1. The corresponding direction field suggests that population 1 will become extinct and population 2 will attain its carrying capacity. This accords with results highlighted in Section 8.8. Case 2. K 1 > a12K2 and K2 < a21K1. Again, the null clines do not intersect, but n o w the null cline for N 1 is to the right of the null cline for N 2. As s h o w n in Fig. C.6b, the direction vectors on dN2/dt = 0 point in the direction of growth for N 1, whereas the direction vectors on dN 1/dt = 0 point in the direction of decline of N 2. The direction field n o w has N 1 increasing and N 2 decreasing, so that population 2 becomes extinct as population 1 attains its carrying capacity. Again, this accords with results highlighted in Section 8.8. Case 3. K 1 > a12K2 and K2 > a21K1. In this case, the null clines intersect at an equilibrium point N* at which the direction vectors on the null clines switch direction. As s h o w n in Fig. C.6c, the direction vectors on dN 1/dt = 0 indicate growth in N 2 for points to the right of N* and indicate declines in N 2 for points to the left of N*. On the other hand, the direction vectors on dN 2/dt = 0 indicate declines in N 1 for points to the right of N* and indicate growth in N 1 for points to the left of N*. This partitions the (N 1, N 2) plane into four regions, one in which N 1 and N 2 both are increasing, one in which N 1 and N 2 both are decreasing, one in which N 1 is decreasing and N 2 is increasing, and one in which N 2 is decreasing and N 1 is increasing. The corresponding direction field suggests that the populations will converge on N* irrespective of initial population sizes (as long as both are positive). Case 4. K 1 < a12K2 and K2 < a21K1. The null clines again intersect at positive population sizes. As s h o w n in Fig. C.6d, the direction vectors on dN 1/dt = 0 indicate declines in N 2 for points to the right of _N* and growth in N 2 for points to the left of _N*. On the other hand, the direction vectors on dN2/dt = 0 indicate growth in N 1 for points to the right of N* and declines
708
Appendix C
Differential Equations
Kt/a12
K1/a12
~._
b
.
KI
K2/a21
K2/a21
K1
K2/a21
K1
K2,
C
K lla 1 2 ~ , , , ~
K1/a12~~~'~i.~ K2
.I-
0. D.1.2.1.1. Discriminant y > 0
If the discriminant ~/is positive, then ~'1 and )~2 both are real, and therefore both components of Eq. (D.8) are as well. The behavior of each component depends on the m a g n i t u d e of the exponential term )kt (Fig. D.1). Thus: 9 For ~ > 1, ~.t grows exponentially. 9 For 0 < ~ < 1, )~t declines exponentially. 9 For - 1 < )~ < 0, ~t oscillates each time period between positive and negative values, with amplitudes that decline over time. 9 For K < - 1 , )~t oscillates each time period between positive and negative values, with amplitudes that increase over time. The trajectory N l ( t ) is influenced by both components in Eq. (D.9) and inherits transient characteristics from both. However, one component eventually dominates the trajectory over time. Values of the parameter pairs (ha,)~2) can be grouped according to the asymptotic
x(O b
x(t)
FIGURE D.1 Trajectoryof x(t) = )~t as influenced by the sign and magnitude of )~. (a) ~ is greater than 1. (b) is positive but less than 1. (c) ~ is negative with magnitude greater than 1. (d))~ is negative with magnitude less than 1.
712
Appendix D Difference Equations
behaviors of the corresponding trajectory (Fig. D.2). Because ~kI is always larger than )~2,feasible parameter combinations lie below the line )kI = ~k2,as shown in Fig. D.2. Four regions are defined. Region I. h I > 1, )kI > IK2I. In this region, K1 is greater in magnitude than ~'2, and K 1 exceeds unity (Fig. D.2). Thus the component h~ dominates )~t over time, and the trajectory exhibits asymptotically exponential increases. Region II. 0 < ~'1 1, )H > [K2I, and the system trajectory exhibits asymptotically exponential increases. In region II, 0 < K1 < 1, K1 > ]K2], and the trajectory exhibits asymptotic declines attendant to the small magnitude of )~1.In region III, - 1 < K2 < 0, ]K2 ] > K1, and the trajectory exhibits declining oscillations of period 2 over time. In region IV, ~'2 < - - 1 , ]~'2 ] > ~q, and the trajectory exhibits increasing oscillations of period 2 over time.
h l , 2 -- 0.5 [0~ -+ (OL2 -- 4 ~ ) 1/2] o f t h e
and combining these equations yields
'
\
IV
t(a11 + a22)h t+l 4- 0.25t(a11 4- a22)2h t = O.
From h = (all 4- a22)/2, we have
/ \
-2
th t+2 --
/
=0, demonstrating that tK t is a solution of Eq. (D.6), along with hr. The population trajectory thus is given by N l ( t ) = k11)k t + k12(t)~t),
with population dynamics that exhibit exponential change or oscillatory behavior as scaled by the factor kll 4- k12t.
D.1. First-Order Linear Homogeneous Equations
713
It is easy to show that the companion difference equation in N2(t), obtained by elimination of Nl(t), has a solution with a form analogous to that of Nl(t). For example, if y ~ 0, the trajectory of N2(t) is given by
nent that increases exponentially through time and a component that decreases exponentially through time.
N2(t) = k21)` ~ 4- k22)`t2,
The system of equations shown in Eq. (D.5) can be expressed in terms of matrices, by
with the same values )`1 and )`2 a s in the solution for Nl(t). Thus, the trajectory for N2(t) exhibits the same patterns as Nl(t), with the values k21 and k22 related to kll and k12 by k21 = k11()` 1 - a11)/a12
D.1.2.2. Matrix Approach
N2(t+l)j =
[all a121[ 11 a21
a22J
Nl(t)
N 2 ( t ) ] = [ vl V2
On condition that y = 0, the solution again includes t)` t along with )`t: N2(t ) = k21)` t 4- k22(t)`t),
with )` = (all 4- a22)/2 and with k21 , k22 given by the system initial conditions.
(D.10)
or N(t+l) = A N. As above, assume a solution to Eq. (D.10) of the form
and k22 = k12()` 2 - a11)/a12.
N2
])`t
or N(t) = v)` t. Then N(t+l) = )`(v)`t) = A(v)`t), resulting in the matrix equation A v = )`v or
(A - )`/)v = 0.
(D.11)
This equation has a nontrivial solution for values of )` satisfying the characteristic equation
IA_- _II =
-- (all 4- a22))` 4- (alia22 - a12a21 )
(D.12) =0,
Example Consider the population trajectories of _N' = (N 1, N 2) for two populations with interactions defined by the transition equations N l ( t + l ) = 2N1/3 + N2/3, N2(t+l ) = 2N1/3 + N 2. Combining the transition equations results in the second-order difference equation 9N1(t+2) - 15N1(t+1) + 4Nl(t) = 0,
with the corresponding vectors v produced from Eq. (D.11). The parameters )` and v satisfying Eqs. (D.11) and (D.12) are the eigenvalues and eigenvectors of A (see Appendix B). The values of )` solving Eq. (D.12) are )`1,2 = (0L "4- ~ / ~ ) / 2 ,
where oL = tr(A) = all 4- a22 and y = oL2 - 4 ~ , with = IA] = alia22 - a12a21. N o t e that t h e s e are the s a m e values produced from Eq. (D.7) above. With some algebra, it can be shown that, for a12 ~ 0,
and substitution of k)` t into the latter equation yields 9)`2 - 15), + 4 = 0, with solutions )`1 = 4 / 3 and )`2 -- 1/3. The trajectory Nl(t) for population 1 is therefore N l ( t ) = k11(4/3) t 4- k12(1/3) t,
with specific values for kll and k12 determined from the system initial conditions. An analogous derivation for population 2 yields a trajectory N2(t) with the same exponential components and with constants again determined by initial conditions. Thus, the population trajectories for this particular system include a compo-
vi
=[
a12 ]
(D.13)
) ` i - au
is an eigenvector corresponding to )`i" Either of the pairs ()`i, v i) corresponds to a solution _N(t) = vik~ of Eq. (D.10), so that a general solution is given by N(t)=
Cl(Vl)`~)4- c2(vaKt~).
From this expression, it is clear that the trajectories for both populations are controlled by the values of k 1 and k 2.
714
Appendix D Difference Equations
Example
selves complex conjugates of the form Vl, 2 a_ + _bi. Expressing )~ and kt2 as k~ = rt[cos(~pt) + i sin(~pt)] and kt2 = rt[cos(~pt) - i sin(~pt)] from DeMoivre's Theorem, a general solution to Eq. (D.10) may be written as ~-
To illustrate the matrix approach, consider two competing populations with population transitions defined by
N(t + 1)
Nl(t+l ) = N 1 - 0.25N2, N2(t+l ) = - N 1 + N 2,
=
ClVl~.~ if-
C2Va)ktp
= rt{cl(a+ bi)[cos(~pt) + i sin(~pt)]
which can be expressed in matrix notation as ~[_N2(t N l ( t ++ l )1)]
= [ _ 1 -01"25] [X12]"
With an assumed solution of the form N system reduces to [l-k-0.25] -1 1-h
=
_ _ [cos(q~t) -sin(q~t) i +c2(a-bi) ]}
vX t, this
=rtcl{acos(~pt)+a[isin(~pt)]
Iv1] =0 ' v2 -
+ b[i cos(~pt)]
- b sin(~pt)} +rtc2{acos(~pt)-a[isin(q)t)]
which is satisfied for values of k such that - b [ i cos(~p)t]- b sin(~pt)}, I A - kI[ = --
-
1 - k
-0.25 1-k
- 1
= k2-2k
with ~p = tan-l(o~/X/'8). Choosing cI = c2 = 0.5 yields the real solution
+ 3/4
= (2k-3)(2k=0.
whereas cI
From Eq. (D.13), the eigenvectors corresponding to ~k1 = 3/2 and )k2 = 1/2 are v~ = [1, -2] and v~ = [1, 2] respectively. Thus, the general solution for the system of equations is
N2(t)
= Cl
[ 1] -2
u(t) = a cos(~pt) - b sin(q)t),
1)/4
(1"5)t q-
c2
[12]
5 0 = C1 q- C2
produces
w(t) = a sin(q~t) + b cos(q~t). Because any linear combination of these expressions is a solution, N(t + 1) = rt[ClU(t) + C2w(t)]
(0"5)t'
with cI and c2 determined from the system initial conditions. For example, if _0 N' = (50, 80), then
-- --C 2 = 0 . 5 i
(D.14)
is a general real solution to Eq. (D.10).
Example Consider the system of equations
Nl(t + 1 ) = 2N 1 + 3N2/2,
and 80 = -2c I + 2c2,
N2(t + 1 ) = - 2 N 1 / 3 + N 2.
so that (c1, C2) = ( 5 , 45). The particular trajectories of Nl(t) and N2(t) therefore are
As above, an assumed solution of the form N = vh t allows this system to be expressed in matrix form as
Nl(t ) = 5(1.5) t + 45(0.5) t and N2(t) = -10(1.5) t + 90(0.5) t, with population 1 exhibiting exponential growth and population 2 quickly driven to extinction. Of particular interest are systems for which )k I and complex conjugates. With complex eigenvalues, the corresponding eigenvectors in Eq. (D.11) are them~'2 a r e
m
[2-k -2/3
3/2 ] [Vl] = 0 ' 1- k v2 -
which is satisfied for values of k such that IA-
_II =
2- k -2/3
3/2 1 - k
= k 2 - 3k + 3 =0.
(D.15)
D.1. First-Order Linear Homogeneous Equations The latter is satisfied by k = V ~ ( V ~ / 2 ___ i / 2 ) = V 3 [cos('rr/6) + i sin(-rr/6)], and substitution of these values into Eq. (D.15) produces Vl, V 2 -- a -4- bi
E3
E 3]i
Defining
E B1
=
ci]k~,
i=1
- 1 cos (6t) -
E3]
with a specific solution determined by the system initial conditions. Alternatively, a matrix approach can be used to determine the population trajectories. For n populations, the equation N(t + 1) = A N is of dimension n, which results in a characteristic equation with n roots. Thus, there are n combinations (ki, vi) of eigenvalues and eigenvectors that satisfy Eq. (D.11), any of which corresponds to a solution
sin(6t)
and w ( t ) = a sin(q~t) + b cos(q~t)
[:1 sin(6t, [ 31 cos6, from Eq. (D.14) a general population trajectory is given by N(t) = 3t/a[ClU(t) + C 2 w(t)]
with the coefficients C 1 and C 2 determined by system initial conditions. For example, if _N'0 = (40, 50), then
[ Cl ] _C 1 if- V~C2
[401 50'
C1
=40,
C2 =
30 V~, and
3j2(E40] 50
= Vik ~
of N(t + 1) = A N. The general solution is given by
[30] )
cos(6t) + X/3 _70
(D.16)
Ci Vi~. I .
]
(-C1 + C2X/3)cos (6t) - (C1V3 + C2) sin (6t) '
X(0)--
N(t)
N(t) -- i~
C1 cos (6t)+ C2 sin(6 t)
= 3t/2 [
--
and two approaches can be taken to determine the population trajectories. The first involves transformation of the equations into a single nth-order equation, in the same manner as for two equations. This produces an nth-degree polynomial equation with n roots k i, i = 1, ..., n, each of which corresponds to a solution cik ~. The general solution is Ni(t) = ~
u ( t ) = a cos(q~t) - b sin(q~t)
so that
715
As above, the population trajectories are controlled by the values ki, i = 1..... n. For example, the populations decline if 0 ~ ~ki < 1 for all eigenvalues, and increase if all eigenvalues are positive and at least one eigenvalue exceeds unity. Oscillations occur if there is at least one pair of complex conjugate eigenvalues a n d / o r at least one negative eigenvalue. It is clear that the inclusion of additional populations into a system, increasing its dimensionality and thus increasing the number of eigenvalues, can lead to greater complexity in system behaviors. Because the exponential terms in Eq. (D.16) all have a value of unity when t = 0, the population initial state is simply
sin(6t) .
tl
_N ( 0 ) = ~ The oscillatory nature of this trajectory is most easily understood in terms of oscillations about some equilibrium system state (see below). The trajectory then exhibits increasing oscillations about the equilibrium population levels, with an oscillation period of 12.
CiVi,
i=1 indicating that the constants c i are directly related to population initial conditions. This relationship can be expressed in matrix notation as c1
D.1.3. Population Dynamics for Multiple Species
c2
__N(0) = [Vl
The population dynamics of n species can be characterized by n first-order difference equations N ( t + 1) = A N ( t ) ,
v2
"'"
Vn] Cn
= Vc,
716
Appendix D Difference Equations
so that c = V - i N ( 0 ) . On condition that A is symmetric, it can be shown that the eigenvectors v i are both real and orthogonal, in that v' i vj = 0 for i ~ j (see Appendix B). For example, a system of three populations with symmetric transition equations has
Fv l .
.
i
=
C1
(-1) t
if- C 2
--
(--0.5)
t 4-
C3
(-0.25) t,
N3(t)J
V~_V2
1
~
[Cl] Ivy/a|i25o1
.
9
=
N2(t) /
with Cl, c2, and c3 determined by the system initial conditions. For N(0)' = (250, 50, 100), the constants are given by
v~vl
=
(1, 1, 1), v~ = (1, - 1 , 0), and v~ = (1, 1, - 2 ) , so that a general solution is
C2
:
C3
9 V3V 3
Lv;/6J
~1331
:[lOOl,
~
K2
and the corresponding population trajectories are Nl(t ) = 133(-1) t + 100(-0.5) t + 17(-0.25) t,
=)~,
N2(t ) = 133(-1) t -
from which it follows that K-iV'N0
L 00j
100(-0.5) t + 17(-0.25) t,
N3(t) = 133(-1) t - 34(-0.25) t. =c.
Thus, an eigenvector decomposition of A sometimes provides a convenient way to compute the constants in c corresponding to a set of population initial conditions. Example
It is instructive to note the similarities between this solution and that of an analogous example in Section C.1 for conditional time. Thus, the coefficients of the two solutions are identical, so that the only effect of a discretized time frame is that the exponential function e ~it in the continuous-time solution is replaced by the power function K~ in the discrete-time solution.
To illustrate, consider the system N2(t+l) N3(t + 1 )
=
[_1.75075 o5]rNlt] 0.75 0.5
1.75 0.5
0.5 2.0
|N2(t) .
LNB(t)
This system is analogous to an example in Section C.1, which there was described in terms of continuous time. For the present case, a solution is assumed to be of the form N(t) = v)~t, so that the system equations reduce to
[175 075 05][Vl] -0.75 -0.5
-1.75 - )~ - 0 . 5 -0.5 -2- K
v2 V3
= 0,
(D.17)
which is satisfied for values of )~ such that - ~ -0.75 -0.5 ] -0.75 -1.75- h -0.5 =0. -0.5 -0.5 -2-
D.2. NONLINEAR HOMOGENEOUS EQUATIONS~ STABILITY ANALYSIS An important extension concerns homogeneous difference equations that include nonlinear terms. In this case, the function F(N) in N ( t + 1) = F(N) includes terms such as N12,N i Nj, and other mathematical expressions that are nonlinear in the population values N i. Examples include the logistic and Lotka-Volterra models in Chapter 8. We restrict attention here to an analysis of population dynamics for populations that are "near" an equilibrium, for the purpose of assessing equilibrium stability.
-1.75
The latter is a polynomial equation with the three roots )~ = - 1 , -0.5, and -0.25. Substituting these values back into Eq. (D.17) produces the eigenvectors v~ =
D.2.1. Stability Analysis for One Species Consider the dynamics of a population with a nonlinear growth function F(N) for which derivatives exist over some operative range of population size. Assume that the population is in equilibrium at a value N*, so
717
D.2. Nonlinear Homogeneous Equations--Stability Analysis that F(N*) = N*. Then population dynamics can be expressed in terms of a Taylor series expansion of F about N*:
about the equilibrium are given by Eq. (D.19), with ( d F / d N ) ( N * ) determined by d ( l n F ) = 1 dF dN
N ( t + 1) = F(N* + n t) dF n2t daF = N* + nt-d-~(N*) + - - ~ - ~ ( N * )
1
=N
+ ...,
with n t = N ( t ) - N * describing "small" deviations about N*. The higher degree terms in Eq. (D.18) are of negligible importance, leading to the simplified expression
d--n
= 1-
dF nt--d-~(N*),
dF
nt--~(N
,
-
lnot.
Thus the equilibrium condition N* = In or/[3 is stable for all values of c~ such that I 1 - log ~1 < 1, in that small deviations from N* decrease through time to 0. Example
so that the equation for population dynamics can be written in terms of deviations n t = N ( t ) - N*, as --
-1
ot
Expressing N ( t + 1 ) = N* + nt+l, we have
lit+l
- ~ e-~N*
= a In oL
dF , = N* + n t -d--~(N ).
= N* +
f~"
Then
N ( t + 1 ) = F(N* + II t)
N* + lit+l
F dN
(D.18)
Consider the logistic model, Eq. (D.4), which can be reparameterized as N ( t + 1) = r N ( 1 - N / K ) ,
).
(D.19)
In this way a nonlinear transition equation can be approximated by a linear difference equation in a neighborhood of N*. It follows that N* is a stable equilibrium if F' (N*) < 1 (because the trajectory of deviations exhibits exponential decay a n d / o r damped oscillations) and N* is an unstable equilibrium if F' (N*) > 1 (because the trajectory of deviations exhibits exponential growth a n d / o r increasing oscillations).
with constant growth parameter I < r < 2 and constant carrying capacity K > 0. The model has two equilibria, N* = 0 and N* = K(1 - l / r ) , and deviations in a neighborhood of N* are given as in Eq. (D.19), by dF
nt+ 1 = n t - ~ ( N
,
)
= rnt(1 - 2N*/K).
Population dynamics around N* = 0 are given by F/t+ 1 - - t.litr
Example
A model of broad applicability for fish population dynamics is the Ricker model N(t+l) = otN(t)e -~N(t) with the parameter ot representing a m a x i m u m population growth rate and [3 inhibiting growth with increasing population size. Population steady state N* for the Ricker model is given by N(t+l) = N ( t ) = N*, so that N * = otN*e-f3N* or
1 =
~e-~N*.
After some algebra, N* = In ot/ [3 is seen to be a nontrivial equilibrium. The dynamics of small deviations
which exhibits simple exponential growth away from 0. Thus, N* = 0 is an unstable equilibrium, in that positive deviations from 0 increase in magnitude. On the other hand, population dynamics around N* = K(1 - l / r ) are given by nt+ 1 = rnt(1 -- 2 N * / K ) = rnt[1 - 2(1 - l / r ) ] = nt(2 - r),
which exhibits simple exponential decay toward 0. Thus, N* = K(1 - l / r ) is a stable equilibrium, in that deviations from K(1 - l / r ) lead to asymptotic declines in the deviations (and thus to asymptotic convergence of N ( t ) to K(1 - 1 / r ) . It is useful to consider the influence of the parameter
718
Appendix D
Difference Equations
r in the stability conditions N* = 0 and N* = K(1 1/r) for the logistic model. From nt+ 1 = rn t it follows that N* = 0 is an unstable equilibrium only for r > 1, because all other values of r produce declining (or negative) deviations and hence population extinction. On the other hand, N* = K(1 - 1/r) is a stable equilibrium only for values of r such that 1 < r < 3; for all other values, nt+ 1 = nt(2 - r) produces deviations that fail to converge to zero [and populations that do not to return to K(1 - l / r ) ] . In particular, as values of r increase from 3 to 4, the population exhibits stable limit cycles with increasing periodicity. It can be shown that values of r beyond 4 produce unstable behaviors that can lead to extinction (May, 1976).
D.2.2. Stability Analysis for Two Species The addition of another state variable complicates the analysis of system stability. Consider two populations with nonlinear growth functions F(N)' = [FI(N), F2(N)]. A Taylor expansion about an equilibrium va~ue N* is 1 k , k , F k ( m * nu Fit) = Fk(N *) + n tFl(N_ ) + n 2tF2(N )
(nl) 2 +
2
Fkl(N*)
-+-
(nt2) 2 2 Fk2(N*)
Ht+l
3F 1
L 3F
3F2
2
aVii("_*) ;G2("_*) = J(N*)nt,
and the properties of J(N*), k n o w n as the Jacobian matrix, determine the equilibrium stability of the system. For example, the analysis of linear difference equations above indicates that N* is a stable equilibrium if the r o o t s ~.1 and ~k2 o f I a - ~._/I -- ~.2 _ o/.K -+= 0 are both of m a g n i t u d e less than 1. If so, then deviations (n I, n 2) from N* will decay through time, and N* is a stable equilibrium. Determination of the stability properties of a nonlinear system does not require one to solve the determinantal equation above for K1 and )~2. Because ot = ~k1 nt- ~k2 = tr[](N*)] (see A p p e n d i x B), a necessary condition for stability is - 2 < oL < 2 or Ic~/21 < 1. Additional conditions are that 0.5(or + , ~ 1 / 2 ) ~ 1 if ot > 0, and 0.5(OL _ ~ / 1 / 2 ) > - 1 if ot < 0, where y = OL2 -- 4[3 and [3 = IJ(N*)I. The latter two inequalities can be combined into a single inequality ]o~/2[ + X/yy/2 < 1, which, after some algebra, simplifies to 1 + [3 > Ic~I. Because ~ = ~.lK2 = I ! ( X * ) l < 1, w e therefore have
for k = 1, 2, where Fk(N) = oFk(N)/ONi and Fk(N) = c92Fk(N)/ONi cONj. For "small" devi--ations __Ht = N(t) N*, the higher degree terms are of negligible importance, and Eq. (D.20) reduces to +
[ni+I]
3F
(D.20)
+ (n~n 2t)F12(N k , ) + ...
Nk(t + 1 ) = Fk(N *
neighborhood of N*. Equation (D.21) is written in matrix notation as
levi < 1 + ~ < 2 .
Example Consider a system of two populations with (scaled) dynamics given by
F/t)
Nl(t + 1) = Nl(t)exp{-0.2511 - N2(t)]}
= N~ + n lFk(N *) + r l 2t F 2k ( X , ). and As in the single-species case, we can express Nk(t + 1 ) = N ' ~ + n kt + 1 , SO that N~
+
F/k+l
--
g'[ + n~Fk(N *) + n t2f a (kN
,
),
so that the equation for population dynamics can be expressed in terms of the deviations F/t = X ( t ) - X * ,
The equilibrium condition _N (t + 1 ) = N* yields exp{0.2511 - N~]} = 1.0 and N~ = 0.5N~(3 - N'~/N1), with the resulting nontrivial equilibrium point N*' = (N~, N~) = (1.0, 1.0). The behavior of small deviations about N* is governed by
n,,1]:[ 1 0.25] rn'l
as 2 k , ntk+l = n t1F lk( N , ) + ntF2(N ),
N2(t + 1) = 0.5N2(t)[3 - N2(t)/Nl(t)].
(D.211
k = 1, 2. Thus, the nonlinear transition equations can be approximated by linear difference equations in a
I
n2+1
--0.5
0.5
Ln 2]
for n(t) = N(t) - N*. Because oL = ( a l l + a22) = 1.0 + 0.5 = 1.5 and ~ = (1.0)(0.5) - (-0.5)(0.25) = 0.625, we
D.2. Nonlinear Homogeneous Equations--Stability Analysis have Jo~J< 1 + ~ < 2 and the equilibrium (1.0, 1.0) is stable. This result is confirmed by an eigenanalysis of the transition matrix, which reveals that the system eigenvalues are complex conjugates of magnitude less than unity.
and so on. Each list of subscripted coefficients is shorter than the list that precedes it alphabetically, until there are only three quantities that relate to their predecessors by the rule qn
D.2.3. Stability Analysis for Multiple Species
qn-1 qn-2--
As above, a description of the population dynamics for n species requires n transition equations, one for each species. In theory the stability of small perturbations about an equilibrium point can be determined by linearization of the transition equations as above. Nevertheless, a stability analysis for n species still involves finding the zeros of a polynomial equation of degree n, a difficult task for large values of n. However, it is possible to specify necessary and sufficient conditions such that all zeros are of magnitude less than unity. Thus, consider the polynomial P(k)
=
~n
4- a l k n - 1
4- a 2 k n - 2
4- . . .
719
=
p2_
2 Pn-3,
= PnPn-1
-- Pn-3Pn-2,
PnPn-2-
Pn-3Pn-l"
Then necessary and sufficient conditions for all zeros of P(M to be of magnitude less than unity are as follows: 1. P(1) = 1 + a I 4- a 2 4- ... 4- a n _ 1 4- a n ~ O. 2. ( - 1 ) nP(--1) = ( - 1 ) n [ ( - 1 ) n 4- a 1 ( - 1 ) n - 1 4a2(--1) n-2 4- -'- 4- an - 1(-1) + an > O. 3.
lan] < 1,
[b l > [bl[, ]On]
>
]C2[,
]dn]
>
]ds],
Iqn]
~
Iqn-l[
an
4- a n _ l K 4-
of degree n. Let bi, ci, di, etc. be defined by bn = 1 - a 2,
c n = b2 - b2,
d,, = c2 - c2,
bn-1 - al - a,,an-1,
%-1 = bnb,-I - bib2,
dn-1
=
bn_ k = a k -
Cn_ k = b n b n _ k -
dn-k
= CnCn_ k -
anan_k,
bl = an-1 - an-1 - anal,
blbk+l,
c2 = bnb2 - blbn_l,
d3
=
CnCn-1
CnC 3
--
--
C2C3,
C2Ck+2,
C2Cn-1,
(Jury 1971). As an example, consider the zeros of the polynomial P(M = )k4 4- ~3 4- K2 4- ... 4- K 4- 1. Conditions (1) and (2) above are satisfied, because P(1) > 0 and (-1)P(1) = 1 (1 - 1 + 1 - 1 + 1) > 0. However, condition (3) fails, because a 4 = 1. Thus, there is at least one zero of P(M that is not smaller than unity, and the corresponding system equilibrium is not stable.
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A P P E N D I X
E Some Probability Distributions and Their Properties
parameters: the sample size n and probabilities Pl, ..., Pk-1 (conditional on Pl, ..., Pk-1, the parameter Pk is given by s pj = 1). Means and variances for the random variable x i of a multinomial distribution are ~i - - npi and 0-/2 = npi( 1 _ Pi), respectively, and the covariance between x i and xj is cov(xi, xj) = - n p i P j . If n is assumed known, the m a x i m u m likelihood estimate of Pi is ]9i -- x i / n . In some applications (for example, when the parameter n is identified as the size of the population rather than the size of the sample), n is u n k n o w n and must be estimated. If ~i is an estimator of n, the conditional m a x i m u m likelihood estimator (conditional on ~i) of Pi is given by Pi = Xi/19l" When there are only two categories of individuals in the population, the multinomial distribution reduces to the binomial distribution, with probability density function
In this appendix we describe some statistical distributions that often arise in modeling and estimation of animal populations. Because of the emphasis in this book on count data for estimating parameters, we describe several distributions that are appropriate for counts. We also focus on distributions that arise in the application of m a x i m u m likelihood estimation and likelihood testing procedures. Probability density functions for these distributions are described in terms of their moments, shaping parameters, and other relevant statistical properties. Where appropriate, parameter estimators also are given. A more detailed treatment of statistical distributions can be found in references such as Evans et al. (2000) and Johnson and Kotz (1969, 1970a,b). Distributions of linear and quadratic forms are covered in detail by Searle (1971) and Graybill (1976).
n~ f(xlp'n)
E.1. DISCRETE D I S T R I B U T I O N S
Consider a trial for which k distinct outcomes are possible, and denote the probability associated with each outcome as Pi, with s Pi = 1. Suppose there are n trials, and let x i denote the number of trials for which outcome i is observed. If the trials are independent, then the resulting probability density function is n Xlr
"'"
Xk
i=1
P)
yl m X
"
The binomial sometimes is denoted by B(n, p) or B(xln, p) to emphasize the roles of n and p. The parameter n determines the number of values that x can take, and p influences the probability mass associated with each of these values. The mean and variance for the binomial distribution are ~ = np and 0 -2 = np(1 - p). Figure E.la shows the binomial distribution for different values of n, and Fig. E.lb shows the distribution for different values of p.
E.1.1. Multinomial Distribution
f ( x l p , n) =
x!(n - x)! px(1
xi Pi ,
E.1.2. Poisson Distribution
with ~i Xi -- Yl. Note that if x 1, ..., Xk_ 1 are given, then the value of Xk is determined by ~i Xi = Y/. Note also that the distribution is parameterized by k independent
The Poisson is a discrete distribution that corresponds to the counting of occurrences of some event 721
722
Appendix E Some Probability Distributions and Their Properties
F I G U R E E.1 Binomial probability density function. (a) Effect of the parameter n (number of trials), for p = 0.7. (b) Effect of the parameter p (probability of success), for n = 10. The binomial mean varies with changes in both n and p, according to E ( x ) = tip.
(e.g., birth, death, or migration) over some continuous time frame T. Because a Poisson r a n d o m variable is restricted to nonnegative integer values, the Poisson is a candidate for the distribution of any counting process. For temporal processes it arises u n d e r the following conditions: 1. For an arbitrary time t in the time frame T, the probability of exactly one occurrence in a "small" interval [t, t + h] is approximately ~h:
3. The n u m b e r s of occurrences in n o n o v e r l a p p i n g time intervals are independent: if there are Xl occurrences in [tl, tl + hi], x2 occurrences in [t2, t 2 + h2], and t 2 ~ t I 4- h i , then Prob(Xl, x 2 ) = Prob(Xl) • Prob(x2). If these three conditions are satisfied, the n u m b e r x of occurrences in a period of length t has a Poisson distribution with probability density function f(x[h,) = e-"~.X/x!,
Prob(one occurrence in It, t + hi) = ~h + o(h), where o(h) is some value with limiting m a g n i t u d e of degree less than h:
o(h)
lim T h-~0
= 0.
2. The probability of more than one occurrence in [t, t +h] is negligible w h e n c o m p a r e d to the probability of a single event: Prob(two or more occurrences in It, t + hi) = o(h).
where K = ~t (h, is referred to as the m e a n rate of occurrence). A Poisson r a n d o m variable can take any nonnegative integer value and the distribution parameter K can a s s u m e any positive value. The p a r a m e t e r ~, influences the spread of the Poisson distribution, such that distributions with smaller values of ~, are more peaked (Fig. E.2). The Poisson sometimes is denoted by P(M or P(x[M to e m p h a s i z e the role of ~. The m e a n and variance of a Poisson r a n d o m variable x are identical, with E(x) = var(x) = ~,. For r a n d o m samples of size k, the m a x i m u m likelihood
E.1. Discrete Distributions
723
FIGURE E.2 Poissonprobability density function. Effects of changes in the parameter )~.
estimator of ~ is ~. count for sample j.
= ~,j xj/k,
where
xj
represents the
E.1.3. Geometric and Negative Binomial Distributions The geometric distribution represents the n u m b e r of failures before the first success in a sequence of independent Bernoulli trials (see Section 10.1). R a n d o m variables with geometric distribution take nonnegative integer values according to the probability density function
f(xlp)
= p(1 -
p)X,
where the parameter p can assume any value in the interval 0 < p < 1. As illustrated in Fig. E.3, the probability density function declines geometrically for all values of x _> 1, with the rate of decline specified by the parameter p. The mean and variance for the geometric distribution are given by E(x) = (1 - p)/p and var(x) = (1 - p)/p2. The m a x i m u m likelihood estimator for p is/~ = k/(k + ~,j xj), based on a sample of k r a n d o m variables. The negative binomial distribution is closely related to the geometric, in that the sum of independent geometric r a n d o m variables is distributed as a negative
binomial. The probability density function of the negative binomial distribution is
f(x[r'P) = (r + x - 1 )
pF(I -
where r is any positive integer, 0 ~ p ~ 1, and x can take any nonnegative integer value. The r a n d o m variable x in this distribution can be thought of as the n u m b e r of additional trials (beyond the m i n i m u m possible number, r) required to record r successes in a sequence of independent Bernoulli trials. As s h o w n in Fig. E.4, the parameter r influences the location of the m o d e of the distribution and the parameter p plays a similar role as in the geometric. The mean and variance of the negative binomial are given by E(x) = r(1 - p)/p and var(x) = r(1 - p)/p2. The m a x i m u m likelihood estimator for p is ]~ = kr/(kr + ~,j xj), based on a sample of k r a n d o m variables. As mentioned above, the sum of r identically distributed geometric r a n d o m variables has a negative binomial distribution: if the r a n d o m variables xj, j = 1, ..., r are distributed as geometric and y = ~,j xj, then y is distributed as a negative binomial. It follows that the geometric distribution is a special case of the negative binomial, in that the geometric distribution is simply a negative binomial distribution with r = 1.
FIGURE E.3 Geometricprobability density function. Effects of changes in parameter p.
724
Appendix E Some Probability Distributions and Their Properties
F I G U R E E.4 Negative binomial probability density function. (a) Effect of parameter r on the distribution (p = 0.3). (b) Effect of the parameter p (r = 2).
E.1.4. Hypergeometric Distribution Like the multinomial distribution, the hypergeometric distribution generally is applicable to sampling situations in which k distinct outcomes are possible. However, the hypergeometric differs from the multinomial in the size of the population a n d / o r the manner of sampling. A hypergeometric distribution is appropriate under the following conditions: 1. A finite population consists of k different categories with sizes M' = (M1, ..., Mk). It is assumed that every individual is in one and only one of the categories, i.e., ~ i M i -- N . Thus the population is assumed to be finite. 2. Sequential sampling of the population is without replacement; i.e., once an individual has been selected, the individual no longer is available for subsequent selection. The size of the sampled population thus is effectively reduced by one. It follows that the probability of selection of any individual is influenced by the selection of others, so that individual selections are not statistically independent events.
3. All combinations of n individuals are equally likely to arise in a sample of size n from the population. For example, any individual in the population is equally likely to be chosen in a sample of size 1; any combination of two individuals is equally likely to be chosen in a sample of size 2; and so on. Under these conditions, a hypergeometric distribution is defined as follows. For a random sample of size n, let x i denote the frequency of occurrence of individuals from category i, i = 1, ..., k. The vector x' = (x I .... , x k) of frequencies is described by the probability density function m
H:lt -
f(xlM) =
X i
where ~'i Xi -- n. Note that the distribution is parameterized by k + 1 independent parameters: the population size (N), sample size (n), and category sizes (M1, ...,
E.2. Continuous Distributions
Mk_ 1) for k - 1 of the categories (conditional on M 1, 9.., Mk-1, the parameter M k is given by ~,j Mj = N). If there are only two categories of individuals in the population, then the hypergeometric distribution with multiple categories reduces to the standard hypergeometric distribution, with probability density function
f(xln, N, M) =
t xl
725
and var(x) = n ( M ) t N
NM)(N
-11
9
Figure E.5a shows the standard hypergeometric distribution for different sample sizes, and Fig. E.5b shows the influence of category size M on the distribution.
E.2. C O N T I N U O U S In this expression, the parameter N is the population size, M is the size of one of the two population cohorts, and n is the sample size. The parameter n determines the n u m b e r of values that the hypergeometric r a n d o m variable x can take, and both N and M influence the relative probability mass associated with these values. The mean and variance for the standard hypergeometric distribution are M
E(x) = n m N
DISTRIBUTIONS E.2.1. Normal Distribution The normal distribution is appropriate for continuous m e a s u r e m e n t s with m e a s u r e m e n t frequencies that decline rapidly as the m e a s u r e m e n t s deviate from some central value. The normal also is a limiting distribution in the central limit theorem and arises in the theory of m a x i m u m likelihood estimation. It therefore is used extensively in statistical modeling and estima-
FIGURE E.5 Hypergeometric probability density function. (a) Influence of sample size (n) for fixed category size (M = 25) and population size (N = 100). (b) Influence of category size (M) for fixed sample size (n = 25) and population size (N =100).
726
Appendix E Some Probability Distributions and Their Properties
tion. The probability density function of the univariate normal distribution is
f(xll~, 0-) -
1 V'2"rr0-
exp
[1( )21 X--Ia,
-
,
size the role of the mean and variance in specifying the distribution. M a x i m u m likelihood estimates of I~ a n d 0 -2 a r e ~ = ~ i x i / F l a n d 4 2 -- ~ i ( X i -- t~)2/F/ for a r a n d o m sample of size n. Because 42 is biased, the adjusted estimator s 2 = ~ i ( x i - ~)2/(F/ -- 1) typically is used in its place. An intuitive extension of the univariate normal distribution involves m e a s u r e m e n t on individuals of two or more attributes instead of one. If the corresponding r a n d o m variables are normally distributed, then the vector of variables is said to have a multivariate normal distribution. For example, the bivariate normal distribution is defined by two r a n d o m variables, Xl and x 2, with probability density function
(E.1)
0-
which is parameterized by the population mean # and the standard deviation 0- (or equivalently, the variance 0-2). The distribution is bell-shaped, symmetric about IJ,, and more or less peaked d e p e n d i n g on 0-. The mean IJ, is a location parameter, in that it specifies the location (but not the shape) of the distribution. The variance is a shape parameter, in that it specifies the shape (but not the location) of the distribution. Small values of 0correspond to distributions that are highly peaked, with probability mass concentrated about the population mean. Figures E.6a and E.6b show the influence of D and 0- on the univariate normal. Typically the normal is denoted by N(I~, 0-2) o r N(x I ~ , O'2), to empha-
f ( x l ' Xa]lJt,1 ' ill,2, 0-2, 0-2, 0-12)
= (2,rr) -1 I ~ 1 - 1 / 2 exp{--(2i~l) -1 [0-2(X 1 -- IJbl)2 + 0-2(X2-
]Jb2)2 + 20-12(Xl -- ~ 1 ) ( X 2 -
f(x) a
p.=-I
~
0.4
0.3
/ 0.2
/
0.1
!
/i
i ,'
I "
/
l.l=O
/ //""\\
, A' \
/ \/ ,' ~ ,; /\./\ ,,' ,
/
/,,~,
iJ.=l
/
"-'..\ ,, ', ',,,
\
\
\
'\.
',
\,, \x
k
0.0
f(x)
0.4] b 0.3 o2=1
0.2
0.1
.:'Y/ .//// ././ /
/-"
\'%, \ x,/
a2=3
\ \~,"\ a2=2 X ",,~
/
0.0 -2
0
2
4
FIGURE E.6 Normal probability density function. (a) Influence of mean I~on the distribution. (b) Effect of the standard deviation or.
I-1'2)]}
E.2. Continuous Distributions
in Fig E.7 corresponds to a probability mass of I - oL, i.e.,
or
f(xlp_, Y-)
727
= ( 2 ~ ) -1
lY.I-~/2
exp [-(x
- ~_)' ~ _ - l ( x - ~_)/2],
P r o b [ ( x - D)'~-l(x
where Ix' = (I/,1, 1.1,2) is a bivariate vector of means for x I and x 2 and 0-1, 0-2, and 012 are the variances and covariance for x I and x 2, respectively. The expression ]~___1 = 2 2 0-10.2 0"22 represents the determinant of the dispersion matrix
_
2
O.12
0-2
l
consisting of the variances and covariance of x I and x 2. As shown in Fig. E.7, the bivariate normal corresponds to a bell-shaped surface that is centered at Ix, with the spread and orientation of the distribution determined by the variances and the covariance, respectively. The ellipse (X-
IJ,)'~-I(x-
__~) -
X2_.(2)
- la,) ~ X2_~(2)] = 1 - oL,
where X12_~(2) is the 1 - oL quantile of a chi-square distribution with 2 degrees of freedom (see below). M a x i m u m likelihood estimates (MLEs) of the means in ix and the variances in ~_ are given as above, and the MLE for covariance is 412 = ~ k ( X l k -- ~l)(X2k -- ~ 2 ) / n. Because 612 is biased, the bias-adjusted estimator $12 = ~k(Xlk~1)(X2kt3"2)/(n -- 1 ) t y p i c a l l y is used in place of ff12" Similar expressions hold for multidimensional systems for which the n u m b e r of variables is greater than 2. If samples are characterized by k variables, x' = (Xl, ..., Xk), the probability density function is given by f(xIth, E) = (2,rr)-k/2[~[ -1/2 e x p [ - ( x - I.I,)'E-I(x-
where IX' = (I Jr,l, ..., I.l,k), ~___is a k-dimensional dispersion matrix of variances and covariances, and [E] is the de-
F I G U R E E.7 Bivariate normal distribution. The 1 - oL probability ellipse for the bivariate normal random variables is parameterized by the 1 - oL quantile of a chi-square with 2 degrees of freedom, according to ( x - IX)' ~ _ - l ( x - IX) =
x~_o(2).
ix)/2],
728
Appendix E Some Probability Distributions and Their Properties
terminant of the dispersion matrix ~. By analogy with the two-dimensional case, a k-dimensional ellipsoid (X-
I,t,)'~-l(x-
___~) =
a
Axlk)
X2_o,(k) k=3
corresponds to a probability mass of 1 - oL: 0.2
P r o b [ ( x - }.l,)'~-l(x__- ____~) 1 random variables. It can be shown that this estimator is biased; i.e., E(h) = [k/(k - 1)]k. A bias-adjusted estimator for h is given by h = (k - 1) / E j xj. The exponential distribution arises naturally in Poisson stochastic processes, which track the occurrences of some event subject to the Poisson conditions described in Section E.1.2. Thus, the number x of such occurrences in an interval of length t is distributed as P(K) with X = vt, whereas the time between occurrences
729
is distributed as Exp(v), with 1/v the mean time between occurrences (see Section 10.3 for a discussion of Poisson processes). The gamma distribution is closely related to the exponential, in that the sum of independent exponential random variables is gamma distributed. The probability density function of the gamma distribution is f(xlh, r)
where r > 0 and F(r) is the gamma function, defined as above. The gamma distribution is unimodal, with mode influenced by the parameter r. The parameter k plays a similar role as in the exponential. Figure E.10a shows probability density functions for gamma distributions with different values of r, and Fig. E.10b shows probability density functions for different values of k. The gamma distribution sometimes is denoted by F(k, r) or by F(xlk, r), to emphasize the role of k and r in influencing the shape of the distribution. The mean
flxlX, r) X=I 0.8
0.6 r=l O.4 \
0.2
\
r = 5
f--\
/V\
r=lO
r-15
--............. .",,f ~ - \ ............... ..-ffxlX)
0.3
k ~-C~(;kx)r-le-xX,
=
/
/
/"
\
\\
\
"~:,0 \ \ .............::.-. 0 restrict the feasible solutions in nonlinear programming to the nonnegative orthant of the n-dimensional Euclidean space E n. Feasible solutions are restricted further to a subset of the orthant by each constraint gi(x) ~ b i. The opportunity set X consists of values x in the intersection of these subsets.
to zero (for x~ = 0). Then ..., n, so that ar (x,)x
= ~,
ax_--
j
(aF/Oxj)(x*)xj = OF
~(x*)xj
for j - 1,
= o.
-
It follows that a maximizing value x* must satisfy OF re(x*) - 0, Ox-
OF
E. G(x,)x
(H.13)
=o
(H.14)
]
H.3.1. Nonnegative Constraints Only
and
A special case of the nonlinear p r o g r a m m i n g problem restricts the constraints to nonnegativity conditions only: maximize
0
x* - 0.
(H. 15)
F(x)
H.3.2. General Inequality Constraints subject to
In this case, the optimization problem is x_>0.
maximize
For a local m a x i m u m x*, F(x* + hAx) _O. These are the well known Kuhn-Tucker conditions of nonlinear programming. The direction of the inequalities in the Kuhn-Tucker conditions indicates that (x*, _K*)is a saddle point of the Lagrangian, in that L(x, ~) is maximized with respect to the decision variables x and minimized with respect to the Lagrangian multipliers _h. As with classical programming, the optimal Lagrangian multipliers can be interpreted in terms of a marginal change in the objective function with respect to the constraint coefficients:
The idea with gradient projection is to search in the direction of the gradient, but with suitable reorientation of the search direction hs needed to account for the constraints. It starts with an initial value x 0 in the opportunity set X and moves at each step in the direction of the gradient of F, provided that direction remains in X. A formula describing each step in the algorithm is Xk+l -- Xk q- ~kVF(Xk )'
with step size
m
OF (x*)= ~* 0b -"
(H.17)
Note that the Kuhn-Tucker conditions reduce to conditions (H.13)-(H.15) in the absence of inequality constraints.
H.3.3. Solution
Approaches
[VF(xk)l [VF(xk)l' [VF(xk)] [H__p(Xk)] [VF(xk)]" provided Y_k+1 remains in the opportunity set X._ If at some step in the iteration the gradient direction is infeasible [i.e., if x k is on a boundary of X and VF(x k) points away from X], the direction of movement is altered to follow the projection of the gradient vector on the tangent to the boundary of X. The corresponding step size is chosen to increase the value of the objective function while remaining in the opportunity set. The sequence {Xk}can be shown to converge to x* provided the objective function is concave and the opportunity set is convex. H.3.3.1.2. Reduced Gradient
The overall logic with iterative methods for constrained optimization is to move at each step in a "best" (unconstrained) direction unless a constraint forces a change in that direction. Approaches to constrained optimization can be grouped into four broad classes, roughly corresponding to a focus on (1) the decision variables constituting the objective function, (2) a subset of those variables as specified by the constraints, (3) the Lagrangian multipliers, or (4) the combination of both decision variables and Lagrangian multipliers.
This approach is a variant of gradient projection, in that the gradient VF(xk) again is adjusted at each iteration to ensure that feasible approximations of x* are generated. The added feature here is that one takes advantage of the constraints to reduce the dimensionality over which the search is conducted. To illustrate, consider a constrained optimization problem with n decision variables and m constraints"
H.3.3.1. P r i m a l M e t h o d s
subject to
Primal methods involve the search for an optimal value x* via procedures for unconstrained optimization, as adapted to ensure the search remains within the opportunity set X. For classical programming with
maximize
F(x)
A x = b, m
where the constraints are linear and m < n. The decision variables can be grouped into vectors x I and X__2
754
Appendix H
The Mathematics of Optimization
of dimension m and n - m, respectively, with x' = (x~, x~). Then A x can be partitioned as ay
= a l x 1 q- a 2 x 2 ,
where the matrix a I is of dimension m • m. To simplify notation we assume that the constraints are linearly independent, so that A is full rank and therefore nonsingular (see Appendix B). Thus, the constraints can be rewritten as a l x I q- a2x__2 -- b or Y1 - - a l I [ b
-
-
a2x__2].
This allows us to rewrite the optimization problem absent the constraints, in terms of the n - m decision variables in x 2 only: maximize
direction of vector Dk is chosen to ensure that a search from x k in that direction at least initially yields candidates for Xk+l in the opportunity set. Each step in the algorithm thus consists of a constrained line search in a feasible direction, with the selection along the line of an optimal value Xk+1 via an optimizing choice of 8k. A useful criterion for selection of the direction vector D k is that it be aligned with the gradient dF(xk)/dx = VF(xk)' as closely as possible and still remain feasible. We illustrate with a problem that has linear constraints. Assuming D k is normalized by requiring its elements to sum to unity, the optimizing choice of D k can be obtained as the solution of the linear programming problem: minimize
VF(xk)D
subject to
F(a11[b -- A2x2] , X2).
AD- 0',
The gradient of L(x, ~) is
m
h~O,
VL(x, )0 = [VLx, VL~]
where the minimization is now with respect to _h. But the partial gradient VLx(x, ~.) of the Lagrangian L(x, ~) can be written as
= [VF(x) - )~J, h (x)'], u
where
VL x = VF(x) - h_J_h(X),
Vh 1
where J_h(X) is the Jacobian matrix
l= Vhl(X)
LVhmJ
=
is the Jacobian matrix for h(x). Necessary conditions for optimization are therefore
LVh2(/
VF(x) + XV/= 0'
Thus, the dual programming problem is minimize
F(x)- kh(x)-
[ V F ( x ) - kJh(X)]X
and
h(x) = O.
subject to
m
lh(x) -
VF(x) _> 0',
D
This constitutes a system of m + n equations in x and ~. As with the other classes of constrained optimization methods, there are several methods based on the Lagrangian. Among others, these include first-order methods, conjugate directions, Newton's method, and modified Newton's method, which are extensions of methods described above to accommodate searches in (n + m)-dimensional space. An example is the differential gradient method, which uses the gradient and Hessian of the Lagrangian in a manner analogous to the method of steepest ascent for unconstrained optimization (see Section H.1.4). m
K&0. In essence, the dual method of solving constrained optimization problems is to focus on the latter formulation, with a goal of finding a solution ~*. It sometimes is possible to describe the dual problem in terms of the Lagrangian multipliers only, based on a prior conditional optimization with respect to x. Then the solution of the dual problem corresponds to that of the original "primal" problem, with the solution of the latter obtained directly from the solution of the former. m
H.4. Linear Programming It should be clear from the above discussion that there are strong interconnections between and among the methods in the different classes of iterative approaches, both in terms of implementation and performance. In fact, many of the procedures for constrained optimization were adapted from procedures for unconstrained problems. Much as the rates of convergence for unconstrained problems are determined by the Hessian of the objective function, so are the rates of convergence for constrained optimization determined in most cases by the structure of the Hessian matrix of the Lagrangian (Luenberger, 1989).
757
H.4.1. Kuhn-Tucker Conditions for Linear Programming As a special case of nonlinear programming, the linear programming problem is amenable to the use of Lagrangian multipliers for derivation of the Kuhn-Tucker conditions [Eq. (H.16)]. The Lagrangian function for linear programming is L(x,
h) =
cx +
h(b -
Ax),
and the corresponding Kuhn-Tucker conditions at (x*, h*) are OL/Ox = c -
hA
_O,
k_>0'. all a21 a
a12 a22
... ...
aln a2n
.__
H.4.2. Dual Linear Programming m
and b = (bl,
..., bin)'.
am1
am2
...
amn
. .
A statement of the problem is
maximize
F(x) = c x
subject to Ax
An important consequence of linearity in the objective function and inequality constraints is that a second, or dual, linear programming problem can be defined by switching the role of the Lagrangian multipliers and the decision variables. The dual problem involves the minimization of an objective function in the Lagrangian multipliers:
_0. subject to As before, the nonnegativity constraints x _ 0 restrict feasible solutions to the nonnegative orthant of E n. Additional restrictions are imposed by the linear constraints a i l x I + ai2x 2 + ... + a , x n
