Alan Hastings, University of California, Davis, USA
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Alan Hastings, University of California, Davis, USA
Anthony R. Ives, University of Wisconsin~Madison, USA Peter Chesson, University of California, Davis, USA J. M. Cushing, University of Arizona, USA Stephen P. Ellner, Cornell University, USA Mark Lewis, University of Alberta, Canada Sergio Rinaldi, Politechnic of Milan, Italy
Cushing, Costantino, Dennis, Desharnais, Henson, 2003
J.
Shandelle M. Henson
Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
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This book is printed on acid-free paper. | Copyright 2003, Elsevier Science (USA). 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. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. On the cover: Symmetry Drawing E70 by M.C. Escher. 9 All rights reserved.
Cordon Art - Baarn - Holland.
ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press 84 Theobald’s Road, London WC1X 8RR, UK http://www.academicpress.com Library of Congress Control Number: 2002104259 International Standard Book Number: 0-12-198876-7 Printed in the United States of America 02 03 04 05 06 07 MB 9 8 7 6 5 4 3 2 1
Contrary to what Isaac Newton may have believed, the deterministic equations of classical mechanics do not imply a regular, ordered universe. Although most m o d e r n physicists and gamblers would concede that dynamical systems with large numbers of degrees of freedom, such as the atmosphere or a roulette wheel, can exhibit random behavior for all practical purposes, the real surprise is that deterministic systems with only one or two degrees of freedom can be just as chaotic. m RODERICK V. IENSEN [96] When observation and theory collide, scientists turn to carefully designed experiments for resolution. Their motivation is especially high in the case of biological systems, which are typically far too complex to be grasped by observation and theory alone. The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens. m EDWARD O. WILSON [192a]
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1 1.1 WhatIs Chaos? 1.2
4
Bifurcations a n d C h a o s
1.3 The Hunt for Chaos 1.4
6
17 21
Mathematical Models and Data
27 2.1
The Deterministic LPA Model
2.2
The Flour Beetle
2.3
Dynamics of the LPA Model
36
2.4 A Stochastic LPA Model 2.5
Parameter Estimation
2.6
Model Validation
58
2.6.1
Model Fit
59
2.6.2
Prediction Fit
2.7
2.8
29
Predicted Dynamics
41
46 53
60 62
2.7.1
Deterministic Dynamics
2.7.2
Stochastic Dynamics
Concluding Remarks
3
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81 3.1
A Bifurcation Experiment
81
3.1.1
The Experimental Design
3.1.2
Parameterization
85
3.1.3
Model Evaluation
88
3.2
The Experimental Results
3.3
Concluding Remarks
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91
84
viii
Contents
4
101 4.1 A Route-to-Chaos
102
4.2 Demographic Variability
107
4.3 Analysis of the Experiment 4.3.1
Parameter Estimation
4.3.2 Model Evaluation 4.3.3
112 113
120
Population Dynamics
123
4.3.4 Transients and Effects of Stochasticity 4.4
Concluding Remarks
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147
5 5.1
Sensitivity to Initial Conditions
5.2 Temporal Patterns 5.3 Lattice Effects 5.4
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Concluding Remarks
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6 183 195 A The Desharnais Experiment
195
B The Bifurcation Experiment
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C The Chaos Experiment 223
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138
This past year, chaos in ecology marked its 25th anniversary. From the start, complex ecological dynamics have been an up-down affair. In the beginning were Robert May’s landmark papers [122, 124, 125] in and These articles called attention to the fact that simple, deterministic models can evidence large-amplitude, aperiodic fluctuations which, when viewed over the long term, are indistinguishable from the output of a stochastic process. One might imagine that this observation, with its potential implications for fluctuating natural populations, would have provoked immediate and intense interest. In the event, it did not. One reason was the belief that complex ecological dynamics arise only in difference equations and that they require rates of increase far in excess of those observed in nature [83]. Another was the fact that no one knew how to look for complex determinism in real world data or even that one look for i t - - i.e., the fact that chaotic processes have characteristic field marks was yet to be appreciated. Accordingly, and for the next 10 years, the nonlinear revolution, which had derived much of its initial impetus from ecological models [81], proceeded apace in the physical sciences, but not in ecology. Eventually, the ripening fruits of nonlinearity, what Mark Kot and I [161a] called "the coals that Newcastle forgot," were reintroduced to the ecological consciousness. Among the points we stressed were the following: The early emphasis on single-species difference equations notwithstanding, complex dynamics are a general feature of nonlinear dynamical systems. As such, they are readily observed in a wide range of ecological models including the multispecies differential equations (Lotka-Volterra models) that had long been a staple of ecological theory. In such cases, dynamical complexity reflects the totality of interactions among species rather than a capacity for excessive reproduction by any one of them. Inferences regarding the dynamics of real-world populations based on the parameterization of ad models [83] are only reliable to the extent that such models adequately describe the forces to which said populations are subject [13 la]. Low-dimensional chaos has a characteristic signature that can be detected in univariate time series via phase-portrait reconstruction [174]
X
Foreword
provided that the time series in question are of sufficient length and quality. In the specific case of childhood epidemics, mechanistic models (SEIR equations) generate simulated time series which bear remarkable resemblance to historical notifications for chickenpox, measles, and mumps [162a]. In retrospect, the response to this brief was predictable. Except in the case of childhood diseases, the data sets were meager, and mechanistic models, with which the data might be compared, were nonexistent. There followed a period of statistical wrangling from which emerged the consensus that chaos in ecologywas a murky business at best [144]. Fundamentally, the issue was what it has always been when chance and determinism confront each other in ecology: ecological time scales are long, which makes for a paucity of data, and the systems themselves, subject to major disturbance, which makes for an abundance of noise. In such circumstances, attempting to ferret out evidence for determinism is an ambitious, some might say, an undertaking. One approach to dealing with such difficulties is to scale back one’s aspirations and bring nature into the laboratory. Then one can do what, in other disciplines (e.g., [86a]), has become almost routine: formulate a mathematical model reflecting one’s opinion as to the essential interactions, determine the model’s behavior under different conditions, and perform experiments whereby the model’s predictions can be tested. In fact, just this approach was adopted by George Oster and his students who studied sheep blowfly dynamics in the late 1970s [138a]. Unfortunately, this work is now largely forgotten, in part because much of it remains buried in unpublished doctoral dissertations. Enter Costantino, Cushing, Dennis, and Desharnais (later joined by Henson and King), affectionately known to their friends as "The Beetles." In short order, these investigators produced unequivocal evidence for complex dynamics in laboratory populations of the flour beetle, Key to their success has been the abiliW to manipulate their system experimentally and to replicate the manipulations. In addition, they have developed a workable methodology that allows for the simultaneous incorporation of random and deterministic forces in ecological models. It is the latter accomplishment which is perhaps the most significant. In the first place, it underscores the importance of modeling the mechanisms and the noise. And it goes beyond the context in which it was developed. This brings us to the present volume, the principal subject of which is the Beetles’ "route-to-chaos" experiment. Clearly, and in detail, the
Foreword
xi
authors lay out the experiment itself, its historical and intellectual context, and the techniques whereby the data were analyzed. As such, it will likely serve as a textbook example for years to come. By showing what can be done in the laboratory, this work additionally lays the groundwork for the challenging task that remains: venturing out of the lab and into the real world which, after all, is the subject which interests most ecologists. But that task is for the future. For the present, it is a privilege to commend the pages that follow both to the individual reader and to the scientific community at large. W. M. Schaffer Tucson, Arizona
oo
Xl!
It is widely appreciated that the dynamics of biological populations are nonlinear and that nonlinearity can be the source of complexity. The mathematical notion of "chaos" has captured the imagination of scientists during the past several decades. Ecologists, in particular, have mused, argued, and debated over the role that chaos might or might not play in biological populations and ecosystems. Although chaos is only one example from what is broadly referred to as complexity theory, it embodies a fundamental idea from that t h e o r y - - namely, that dynamic complexity can be the outcome of simple deterministic rules. In this way, chaos theory offers hope that at least some of the observed complexity in the dynamics of ecological systems might be understood on the basis of simple laws. For a variety of reasons, however, it has been difficult to test this idea. Obstacles include the insufficient length of available time-series data and the inherent difficulty in manipulating and experimenting with ecological systems ~ a difficulty that precludes controlled replicated studies from which one can make firm conclusions. Perhaps the most fundamental obstacle, however, has been the lack of biologically based models that are closely tied to data and that provide quantitatively accurate predictions. For over a decade the authors have collaborated on a series of interdisciplinary projects in population dynamics and ecology with a focus on complex nonlinear dynamics. This book reports on some of these projects. Although we try to place our studies in historical context, we do not attempt a general survey of all that has been done and written on the subject of chaos in ecology. Instead, we confine our attention to those of our projects that have chaos as an organizing theme. Broadly speaking there have been two approaches to the study of chaos in ecology: the investigation of historical data sets either by statistical methods of time series analysis or by methods based on the "reconstruction of attractors" from time series data by using the famous Takens theorem. In this book we take a different a p p r o a c h ~ o n e that harkens back to the seminal work of Lord (Robert) May of Oxford and his coauthors whose influential papers in the 1970s helped popularize the notion of chaos and stimulated the renaissance of nonlinear science that took place during the subsequent decades. This approach centers on transitions in dynamic behavior (bifurcations) that occur when demographic parameters of a population change. Such bifurcations can cascade into an increasing complexity of dynamic patterns that can result in chaotic dyn a m i c s - a cascade called a "route-to-chaos." Our approach focuses on the bifurcations and a route-to-chaos predicted by a mathematical model
Preface and on data obtained from experiments designed to test the occurrence of these bifurcations in a real biological population. To accomplish these goals we will necessarily become involved with a variety of topics, including deterministic and stochastic modeling methodology, dynamic attractors ofassorted types, stability and instability, transient dynamics, parameter estimation, model validation, and stochasticity. We will find that the mix of nonlinearity and stochasticity produces a level of complexity, a full understanding ofwhich cannot be gained by the study of deterministic attractors alone. Even in controlled experiments such as ours, population data are stochastic mixes of patterns influenced not only by attractors, but also by unstable entities, transient dynamics, and other unexpected factors. Nevertheless, the bottom line in our studies is the assertion that a simple (low-dimensional) deterministic model can provide accurate descriptions and predictions of the complex dynamics exhibited by a biological population. Although the book can be read as a report on the details and conclusions of our investigations into nonlinear and chaotic dynamics, it can also be read as a study of modeling methodology in population dynamics. Central themes include deterministic and stochastic models, the connection of models to data, the evaluation ofmodels using data, and the use ofmodels to design and implement experiments that test model predictions. The importance ofthese general themes extends beyond the particular studies detailed in this book. In our view a rigorous study of population dynamics, in which one hopes to associate observations and data with mathematical predictions, requires a strong connection between models and data. This is true even for simple dynamics, but it is particularly true for complicated and exotic dynamics such as chaos. The strongest case is made when a mechanistic model can be identified and shown capable of not only accurate descriptions (fitting) of data, but also accurate predictions of data. This approach is, of course, very much in the tradition of the "hard" sciences (an adjective unfortunately not often associated with the ecological sciences). We hope our studies provide a cornerstone example of a mathematical model in population dynamics whose p r e d i c t i o n s ~ often subtle, unexpected, and nonintuitive ~ are borne out by controlled experiments. By their very nature the studies reported in the book are interdisciplinary. This places some demands on those readers who, like the authors, were trained in disciplinary settings. We hope these demands are not so burdensome as to be a deterrent. Indeed, we hope the reader finds rewarding, as did the authors, those efforts necessary in dealing with new concepts from unfamiliar disciplines. We have benefited greatly from collaborations with many other researchers and students. William Schaffer’s probing critiques stimulated
Xlll
xiv
Preface deeper insights and improvements in our work. Aaron King made invaluable contributions to the analysis of patterns in our data. The list of people who, over many years, influenced our work and helped to clarify our thinking during many discussions and debates, as well as casual conversations and communications, is a long one. It includes Hal Caswell, Joel Cohen, John Delos, Jeffrey Edmunds, Steve Ellner, John Franke, Tom Hallam, Alan Hastings, Dave Jillson, Brian McGill, Laurence Mueller, Joe M. Perry, Jim Selgrade, William Stoeger, Gene Tracy, Michael Trosset, Peter Turchin, Joe Watkins, Aziz Yakubu, and undoubtedly others whose names we have (inadvertently and apologetically) overlooked. A number of graduate and undergraduate students also made significant contributions, including Scott Calvert, Lyn Curtis, Tivon Jacobson, Paul Mayfield, Naoko Nomura, Derek Sperry, David Wood (University ofArizona); Ruth Bernard, John Fitchman, Pao Her, Christene Kendrick, Michael Ledoux, Michele Ledoux, Sheree LeVarge, Nichele Mullaney (University of Rhode Island); Jonnie Burton, Warren Cheung, Juan Coleman, Karen Joseph, Anny Ku, Tai Luu, Roy Morita, Enrique Nufiez, Chau Phu, Karina Preciado, Gabriel Rodas, Luis Soto, Robert Tan, Rebecca Tatum, Yervand Torosyan, Timothy Weisbrod, Thomas Wong, Timothy Yeh (California State University, Los Angeles); Eric Davis, Viva Miller, James Reilly, Suzanne Robertson, Matthew Schu (College of William and Mary). Our work would not have been possible without the generous support of the National Science Foundation. In particular, we are extremely grateful to Michael Steuerwalt at NSF for his efforts on our behalf. We also express our appreciation to Alan Hastings for his support of our work and for the invitation to write this book. J. M. Cushing
R. E Costantino
and Brian Dennis
and Robert A. Desharnais
Shandelle M. Henson
1 If it was.., straightforward, then simple laws operating in simple circumstances would always lead to simple patterns, while complexlaws operating in complex circumstances would always lead to complex patterns .... This no longer looks correct, but it’s taken time to find out because we seem to be predisposed toward such a principle. m IAN STEWART [171]
A central goal in population biology and ecology is to understand temporal fluctuations in population abundance. Such fluctuations, however, often appear to be erratic and random, with levels of variation ranging from small percentages to several orders of magnitude. 1 In the 1970s Lord (Robert) May of Oxford put forth a bold new hypothesis concerning the possible explanation of the complex dynamic patterns so often observed in biological populations [122, 124, 125]. The prevailing point of view had been that complex patterns have complex causes and simple causes have simple consequences. May’s hypothesis implies, on the other hand, that complex patterns can result from simple rules. To a few mathematicians and scientists this thesis had been known at least since the pioneering work of Henri Poincar6 in the late 19th century [2, 3, 12]. However, to most ecologists the assertion was novel; it raised the intriguing possibility that (at least some of) the complexity of nature might arise from simple laws. The complexity about which May wrote is a result of nonlinearity. Although the classical models of theoretical ecology from the first half of the 20th century are nonlinear, the theories derived from them were centered on equilibrium dynamics. The famous logistic differential equation and the Lotka-Volterra equations for competition and predation are the 1 For example, one literature survey has found that a n n u a l adult recruitment could vary by factors of over 30 in terrestrial vertebrates, 300 in plants, 500 in m a r i n e invertebrates, and 2000 in birds [78].
2
I ]INTRODUCTION
prototypical examples. Fundamentally, the mind set at the heart of these theories encompassed the notion of a "balance of nature" in which ecological systems are inherently at equilibrium and the erratic fluctuations and complexity observed in data are due to "random disturbances" or "noise." From this point of view, ecosystems are "stochastic perturbations" of underlying equilibrium configurations (in which processes are in some kind of optimal efficiency). The point of view suggested by May, however, was not based on "noisy equilibrium" states. As he put it, the fact that a simple, deterministic equation can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship... [124]. May’s hypothesis opened the door to new ways of thinking about population dynamics and ecological systems ~ ways that bring nonlinearity to the forefront and make it a major role player. Unexplained noise will always be present in ecological data. However, May’s insight provided a new point of view: fluctuation patterns observed in the abundances of some population systems might be explained, to a large extent, by relatively low-dimensional nonlinear effects as predicted by simple mathematical models. Despite the fact that mathematical and theoretical ecology developed and expanded profusely during the decades following May’s seminal work, his hypothesis has proved both controversial and elusive to test [85, 146, 147]. Mathematicians have invented a plethora of ecological models and proved complicated theorems about them. Theoretical ecologists have applied methods from dynamical systems theory to ecological problems and drawn implications from the results. Nonetheless, as a whole, the community of empirical ecologists remains unconvinced. They are unconvinced that one can effectively reason about ecological systems using mathematical models, that there are reliable ecological "laws" available for such an enterprise, and that mathematical ecology is anything but peripheral to real populations in real ecosystems. This point of view of deep skepticism is not surprising, given the lack of evidence. Where are the examples in which mathematical models provide convincing explanations of real biological systems and accurate predictions for actual populations or ecosystems? Despite the optimism generated by famous experiments carried out by such notable figures as G. F. Gause [70], T. Park [139, 141], and P. H. Leslie [114] decades before
1 ] INTRODUCTION
May’s work (and still cited in most ecology texts), theorists admit there are few if any such examples. Does this mean there is some inherent property of the ecological world that precludes the application of the methods that have proved so successful in other scientific disciplines? Must ecology remain primarily a descriptive endeavor, in which mathematical reasoning cannot hope to provide quantitatively accurate predictions, and theoretical issues remain purely semantic? Can ecology ever take its place among the "hard" sciences? We will not attempt to answer such sweeping philosophical questions in this book. However, we will address some issues lying at the core of these problems, one of the most fundamental ofwhich is a serious gap or "disconnection" between theory (mathematical models) and data. We will do this in the context of a study in complex nonlinear dynamics that we have conducted over the past d e c a d e ~ a study motivated by and designed to address May’s hypothesis. Prerequisite to this project is the establishment of a strong connection between population data obtained from a particular biological organism and a mathematical model that describes the population’s dynamics. By showing that simple, lowdimensional nonlinear models can "work"--that is to say, can provide quantitatively accurate descriptions and of the dynamics of a real biological population--we will then be in a position to explore nonlinear phenomena in a rigorous way. The results of these explorations will document a variety of nonlinear effects (including chaos) whose occurrence, although well known in theoretical models, was mere speculation in real populations. However, beyond these specific phenomena, the project will provide an unequivocal example of how nonlinearity is absolutely central to the understanding of the dynamics of a real biological population. In some cases, in fact, we will see how surprisingly subtle nonlinear effects are required to obtain a more complete understanding of observed patterns ~ effects whose observation in real population data would be thought highly unlikely. We hope that these studies will supply new insights into how nonlinearity, particularly when coupled with stochasticity, can provide a new level of explanatory and predictive power in population dynamics. The studies will focus on a particular biological system which, in the tradition of experimental science, we are able to control, manipulate, replicate, and accurately measure. However, as in other scientific disciplines, that tradition provides the insight into fundamental concepts, the understanding of principles, and the verification of hypotheses that can then serve as guidelines for the investigation of other systems and other situations and circumstances. In this way, we hope our studies provide a small step toward raising the explanatory and predictive power of ecological science.
3
4
1 [ INTRODUCTION
1.1 I WHAT IS CHAOS? It seems appropriate to call a real physical system chaotic if a but one with the system’s inherent randomness suppressed, still appears to behave randomly. [italics added] EDWARD
LORENZ
[120]
Chaos is a name for any order that produces confusion in our minds. m
G E 0 R G E SAN TAYANA
and
Mathematicians have identified and studied many kinds of complex patterns that can arise in simple dynamical systems. A special type of complex d y n a m i c ~ a type called "chaos" ~ has, however, become in many ways an icon for complexity. What exactly is chaos? One can find many definitions, formal and informal, throughout the scientific and mathematical literature. Li and Yorke originally coined the term in their now-famous study of certain kinds of mathematical equations of the type studied by May [116]. Edward Lorenz, who had encountered chaos a decade earlier in a different context [119], offers several informal definitions of chaotic dynamics in a more recent book [120], including processes "that appear to proceed according to chance even though their behavior is in fact determined by precise laws" or "behavior that deterministic, or is nearly so if it occurs in a tangible system that possesses a slight deterministic." In his popua m o u n t of randomness, but does not lar book on the subject [71], Gleick provides several definitions of chaos formulated by various investigators, one of which (attributed to Philip Holmes) is: "complicated, aperiodic, attracting orbits of certain (usually low-dimensional) dynamical systems." Ian Stewart describes chaos as "apparent randomness with a purely deterministic cause"; as "[u]nruly behavior governed entirely by rules," it "inhabits the twilight zone between regularity and randomness" [171]. Although mathematicians have formulated several technical definitions of chaos, none have yet become universally accepted [153]. However, it is generally agreed that a rigorous mathematical definition of chaos must include a dynamic property called "sensitivity to initial conditions." By sensitivity to initial conditions it is meant that a disturbance, no matter how small, in the state of the system becomes highly amplified over time. Or, put another way, trajectories initiating from two nearby states diverge and become significantly dissimilar as time passes. Sensitivity to initial conditions was in fact the p h e n o m e n o n that led Lorenz to an encounter with chaos in his computer study of a mathematical model describing atmospheric weather patterns. A basic property of differential
1.1 [ What Is Chaos? or difference equation models is that they are deterministic, i.e., from any state of the system there is one and only one possible trajectory of future states. The future is completely d e t e r m i n e d by the present. Lorenz was puzzled by some numerical calculations obtained from his c o m p u t e r that seem to produce different trajectories from the same initial state. It t u r n e d out that in a s u b s e q u e n t calculation, however, he h a d r o u n d e d the initial state to fewer decimals t h a n he h a d used in the original calculation, thinking such a small change would m a k e no noticeable difference in the outcome. Instead, the two trajectories eventually evolved to b e c o m e so dissimilar that the differences between t h e m b e c a m e uncorrelated and seemingly random. This puzzling o u t c o m e resulted from the sensitivity to initial conditions in his model, caused by the presence of a chaotic attractor (the now famous "Lorenz attractor"). Because it implies a serious restriction on the practical ability to predict future states, sensitivity to initial conditions is the property of chaotic systems that has m o s t caught the attention of scientists and philosophers. This crucial property did not escape the penetrating intellect of Henri Poincar6, the founding father of m o d e r n dynamical systems theory, who stumbled across what we now call chaos in the 1890s [12]: A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But, even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation If that enabled us to predict the succeeding situation that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. ([145], p. 404) The property of sensitivity to initial conditions is not sufficient, however, for a system to be called chaotic. For example, an u n b o u n d e d , exponentially growing trajectory from a linear model is sensitive to its initial condition, since a small difference grows without bound. However, such exponentially growing trajectories, diverging from a source, possess regularity that is not associated with the behavior of chaotic trajectories, which are b o u n d e d and random-like. A chaotic trajectory is a b o u n d e d trajectory "that forever continues to experience the unstable behavior that an orbit exhibits near a source, but that is not itself fixed or periodic"
5
6
1 [ INTRODUCTION
and "at any point of such an orbit, there are points arbitrarily near that will move away from the point..." ([7], p. 106). A formal definition of chaos must deal with, and somehow capture, what is meant by bounded and irregular dynamics. It is therefore not surprising that formal definitions of chaos involve sophisticated mathematical concepts. For our purposes it is not necessary to delve into mathematical technicalities, and we take the following as a working definition: a a and Unfortunately it is difficult to prove rigorously that a dynamical system possesses chaotic trajectories. In practice, researchers generally rely on evidence obtained from computer simulations. Studies often focus on establishing the key property of sensitivity to initial conditions. This is often done by estimating a diagnostic quantity called the (dominant) "Lyapunov exponent," a n u m b e r whose positivity implies sensitivity to initial conditions. (For a discussion of this n u m b e r see [7].) We emphasize that chaos is a deterministic phenomenon. In fact, chaos is of interest precisely because it is deterministic. No one is surprised, of course, if complicated fluctuations arise in systems containing randomly changing elements or perturbations ~ that is to say, in "stochastic" systems. However, the discovery that deterministic systems can produce fluctuations that are seemingly indistinguishable from random fluctuations was surprising to most scientists and mathematicians. This fact blurs in many ways the distinction between two paradigms of modern science according to which, from one point of view, the fundamental nature of our world is deterministic while, from the other point of view, it is probabilistic. The discovery challenges determinism, in both a practical and a theoretical way, by attacking the ability to predict, which is a fundamental aspect of science [99]. One the other hand, it also causes one to rethink what is meant by "random" or "stochastic" events [104].
1.2 If the Lord Almighty had consulted me before embarking on creation I should have recommended something simpler. m attributed to ALPHONSO X May came to chaos in a way different from that of Lorenz. May was interested in how the long-term dynamics of a simple mathematical model changes as one ofits parameters (coefficients) changes. He considered equations (called "difference equations" or "maps") that recursively
1.2 [ Bifurcations and Chaos
7
define a population’s a b u n d a n c e at discrete c e n s u s t i m e s t = 0, 1, 2, 3 . . . . The "discrete logistic" e q u a t i o n (1.1) is a f a m o u s example. 2 Starting from an initial a b u n d a n c e (initial condition) Xo s u c h a f o r m u l a defines a u n i q u e "solution" or "orbit" (or "trajectory"), 3 i.e., a u n i q u e s e q u e n c e of future p o p u l a t i o n a b u n d a n c e s Xo, Xl, x2, . . . . In the simplest instance, this s e q u e n c e a p p r o a c h e s a limit point. However, there are other possibilities. The s e q u e n c e might, for example, settle into a n oscillation b e t w e e n two points, t h a t is to say, at even times the s e q u e n c e a p p r o a c h e s one n u m b e r , w h e r e a s at o d d times it app r o a c h e s a different n u m b e r . A p o i n t a p p r o a c h e d by s o m e s u b s e q u e n c e of the orbit is a "limit point" a n d the collection of all limit points is the "limit set" of the orbit. One of the f u n d a m e n t a l goals in d y n a m i c a l s y s t e m s t h e o r y is to describe the limit sets of orbits. A basic fact is t h a t a limit set is a "collection of orbits. ’’4 A limit set c a n consist, for example, of a single p o i n t x*. In this case the solution starting at this p o i n t m u s t r e m a i n at this point for all time: Xo = x*, Xl = x*, x2 = x*, x3 - x * , . . . , a n d we say the p o i n t x* is an "equilibrium" point. As a n o t h e r example, consider the situation m e n t i o n e d above in w h i c h an orbit a p p r o a c h e s one limit p o i n t x~ at even times a n d a different limit p o i n t x~ at o d d times. In this case, the limit set of the orbit consists of these two limit points. The orbit starting at one of the limit points, say x~, alternately oscillates b e t w e e n the two limit * . - - a n d is called a periodic points--xo "2-cycle." Thus, in this case, the limit set of the orbit is (the range of) a 2-cycle. Limit sets are n o t always as simple, it turns out, as t h e y are in these two examples. T h e y c a n in fact be extraordinarily c o m p l i c a t e d a n d even consist of infinitely m a n y points. z The name "discrete logistic" has its roots in the similarity of the formula for population abundance at time t + 1 to the formula for the rate of change of population abundance at time t found in the famous logistic differential equation However, the time series of population abundances that results from the solution of this differential equation does not satisfy the discrete logistic equation. Instead, the sequence satisfies the so-called Beverton-Holt equation - - a x t ( 1 d- ( a - 1)Xt)-1, a = exp(b)--a difference equation all ofwhose solutions, it turns out, approach an equilibrium. This equation might therefore be more deserving of the name "discrete logtistic." a Technically,a "forward" orbit or trajectory. 4 More precisely, a collection of ranges of orbits.
8
1 [ INTRODUCTION
Another fundamental goal of dynamical systems theory is to describe "attractors." An attractor is an orbital limit set that is also approached by the orbits starting from every point in a neighborhood of the set. In other words, an attractor contains the limit sets of all nearby orbits. Not all limit sets are attractors. For example, an equilibrium (which is a limit set, namely, its own limit set) might not be a limit point of any orbit starting nearby, or it might be a limit point of only some, but not all, orbits starting nearby. As May discovered, attractors can be extraordinarily complicated, even for dynamical systems defined by simple maps such as the discrete logistic map. For each value of the parameter b the discrete logistic (1.1) has an attractor. May was interested in how the attractors change as b changes. By increasing the parameter b > 0 he discovered a cascade of c h a n g e s ~ o r "bifurcations" ~ i n which the attractors become more and more complicated, progressing from equilibria and periodic cycles to complicated erratic and random-like oscillations, i.e., to "chaos." To describe this situation, May constructed a graph, called a "bifurcation diagram," like that appearing in Fig. 1.1, which we have constructed using the difference equation b > O, c > O.
(1.2)
This equation, called the Ricker equation [152], is similar to the discrete logistic, but is more useful for population models. The bifurcation diagram in Fig. 1.1 shows a plot of attractors against the value of the parameter b. This plot, which has virtually become a logo for chaos theory, illustrates a "period-doubling cascade to chaos" or a "period-doubling route-tochaos." As b increases from very small values, the attracting state of the system changes from the "extinction" equilibrium x = 0 to a positive equilibrium as b exceeds the critical (or bifurcation) value b - 1. The attractor undergoes another change, this time to a 2-cycle, as b increases through another bifurcation value at approximately b = 7.4. With further increases in b come further changes in the attractor m changes to a 4-cycle, then to an 8-cycle, and so on through cycles with periods equal to powers of 2. This cascade ofperiod-doubling cycles eventually ends, and as bincreases further there arises a complex array of complicated attractors. In this latter range of b values exist chaotic attractors interspersed with periodic cycles~cycles that occur in distinctive regions of the bifurcation diagram called "period-locking windows." Although mathematicians have proved many facts about the dynamic properties of orbits for these larger values of b, we will not concern ourselves with the intricate details. We simply point out that this particular sequence of bifurcations turns out to
1.2
9
600
X
600
200
1.1 0.01. a 13
a 5 10
a a
22 a
a a
a a
25.
10
1 I INTRODUCTION Ricker equation. One-dimensional systems describe the dynamics of a population characterized by a single state variable, such as total population numbers or density. Instead, the populations in our studies will be characterized by several state variables (specifically, three state variables representing three life-cycle stages). Thus, we require a system of three difference equations (one for each state variable) to describe the population dynamics. Since time is discrete in these equations, such systems are called discrete dynamical systems. Higher dimensional dynamical systems such as these can also undergo attractor bifurcations as parameters are changed. They can, for example, display period-doubling routes-tochaos similar to those displayed by the one-dimensional Ricker equations. However, higher dimensional models can also display other types of bifurcations and other kinds of bifurcation sequences. The route-tochaos that plays a crucial role in our study is not of the period-doubling type. To describe it, we must briefly discuss other types of bifurcations that can occur in higher dimensional discrete dynamical systems. The equilibrium and cycle bifurcations shown in Fig. 1.1 occur when the equilibria and cycles cease being attractors. We can determine these bifurcations and the points where they occur by using fundamental methods in stability theory. For a one-dimensional dynamical system defined by a single difference equation
(1.3) the simplest attractors are stable equilibria. Equilibria are constant solutions and therefore correspond to the roots of the algebraic equation x*--
(1.4)
We say x* is a "fixed point" of f(x), since the dynamic rule (1.3) fixes x* by mapping it to itself. A fundamental theorem states that an equilibrium x* is stable if the "linearization" of the equation (1.3) at the equilibrium is stable. That is to say, x = 0 is stable as an equilibrium of the linear difference equation in which ~ is the derivative evaluated at x = x*. All solutions of this equation tend to 0 as t increases without bound if (and only if) the n u m b e r )~ satisfies [~[ < 1.
(1.5)
This fundamental theorem also tells us that an equilibrium is unstable if the linearization is unstable. Thus, an equilibrium x* of (1.3) is stable if the inequality [~[ < 1 holds and is unstable if [~[ > 1.
1.2 [ Bifurcations and Chaos
11
As an example, consider the Ricker equation (1.2), for which exp(-cx) and
dx
= b(1 -
Applying the stability criterion (1.5) we find that the equilibrium x* = 0 is stable if ~ - b < 1 and unstable if b > 1. Note this equilibrium loses stability at b - i where ~ - 1. This is c o m m e n s u r a t e with the bifurcation diagram in Fig. 1.1. The Ricker e q u a t i o n also has the equilibrium x* - c -1 In b, which is o b t a i n e d by solving Eq. (1.4). This equilibrium is positive if b > 1. The stability of this equilibrium is d e t e r m i n e d by the n u m b e r Z - 1 - In b and the criterion (1.5), which tell us the equilibrium is stable if I < b < e 2 7.389. Note this equilibrium gains stability w h e n the equilibrium x* = 0 loses it, n a m e l y at b - 1where ~ - 1. Its s u b s e q u e n t loss ofstability occurs at b = ea where X - - 1 . This is also c o m m e n s u r a t e with the bifurcation diagram in Fig. 1.1. A loss of equilibrium stability occurs w h e n a change in a p a r a m e t e r causes an inequality change from ]~] < 1 to I~l > 1. Thus, bifurcation points are f o u n d by d e t e r m i n i n g those p a r a m e t e r values at which Ix[ - 1. Since for o n e - d i m e n s i o n a l m a p s X is a real number, this m e a n s equilibrium bifurcations can arise from only two cases: either )~ = 1 or )~ = - 1. As we have seen, b o t h possibilities occur in the Ricker equation. The result caused by the loss of equilibrium stability is quite different in each case. W h e n )~ - I the bifurcation involves a change from one stable equilibrium to another. Biologically this corresponds to passing from a prediction of extinction (when the equilibrium x* = 0 is stable) to survival (when the positive equilibrium is stable). A bifurcation of this type, in which two equilibrium "branches" intersect and cause an exchange of stability b e t w e e n them, is called a "transcritical" bifurcation. This is one of several basic types of bifurcations, all of which involve only equilibria, that can occur in the case ~ - 1. 5 In the s e c o n d case, w h e n ~ = - 1 , the loss of equilibrium stability results in the creation of a stable 2 - c y c l e - - a 2-cycle or "period-doubling" bifurcation ~ which, it turns out, is typical for this case. For more a b o u t these basic bifurcation types see [76] or [192]. In principle the study of periodic cycles and their stability is no more difficult t h a n that of equilibria. The crucial observation is that since a cycle, by definition, repeats after finitely many, say n, steps, it follows that its initial state is a fixed point of the recursion formula applied n times. 5 Other types include "saddle node" and "pitchfork" bifurcations.
12
1 I INTRODUCTION
For example, 2-cycles correspond to roots of the equation 6 x*= Thus, a 2-cycle is an equilibrium of the "first composite" equation
and we can apply the same stability analysis we apply to equilibria, but now with X equal to the derivative evaluated at x*. We conclude that 2-cycle bifurcations involve either other 2-cycles, when ~ = 1, or 4-cycles, when ~ = - 1 . For 4-cycles we can repeat the analysis by considering equilibria of the third composite equation, and so on. In this way, we obtain an explanation for the period-doubling cascade of bifurcations observed in the Ricker equation (and discrete logistic equation). Needless to say, the calculations involved in the analysis of higher composite equations generally become impossible to carry out algebraically and must be done numerically with the aid of a computer. A parallel analysis is possible for equilibria and cycles of higher dimensional systems. For example, equilibria = (x*, y*) of a two-dimensional dynamical system defined by the pair of difference equations
correspond to the roots of the two simultaneous algebraic equations
An equilibrium is stable if its linearization is stable, i.e., if the equilibrium = (0, 0) of the associated pair of equations
6 A n y r o o t defines a 2-cycle orbit x0 = x*, xl = f(x*), Xz = x*, x3 = f(x*), x4 = x*, . . . . A r o o t of (1.4) is also a r o o t of this e q u a t i o n . T h a t is to say, an e q u i l i b r i u m is also a 2cycle, a l t h o u g h it is n o t a n oscillatory 2-cycle. To yield a "genuine" 2 - c y c l e m o n e t h a t oscillates m a root m u s t n o t be a r o o t of (1.4), i.e., m u s t satisfy
1.2 [ Bifurcations and Chaos
1:
is stable where the coefficients are obtained from the matrix
c
d
ag
ag
in which the four derivatives are evaluated at the equilibrium point. This matrix is called the "Jacobian" matrix of the system at the equilibrium. The linearized system, and hence the equilibrium of the original system, is stable if eigenvalues ~ of the Jacobian matrix satisfy the stability criterion I~1 < 1; it is unstable if at least one eigenvalue satisfies I~1 > 1. Critical bifurcation values for parameters in the equations occur when at least one eigenvalue satisfies I~1 = 1. This bifurcation criterion includes the two cases )~ - +1 and the corresponding types of attractor bifurcations encountered in the one-dimensional case. However, in this two-dimensional case, and in higher dimensional cases, eigenvalues can be complex and, as a result, there exist other possibilities. The criterion I)~l - 1 means that the eigenvalue lies on the unit circle in the complex plane and therefore has the form X - exp(i0) where 0 is its polar angle (Fig. 1.2). Equilibrium stability is lost if a parameter appearing in the equations is changed in such a way that at least one eigenvalue moves from inside to outside the unit circle. If this crossing occurs atX - 1 or - 1 (i.e., ifthe polar angle is0 - 0 or n), then we have the two bifurcation cases already discussed. If, however, the crossing occurs at different points on the unit circle, i.e., at a point exp(i0) where 0 ~ 0 or n, then a new type of bifurcation, important to our studies, occurs. For complicated attractors, analysis is best carried out in what is called "state space" or "phase space." Patterns generated by complicated attractors may be difficult to observe when the state variables are plotted against time, whereas in state space they may be considerably easier to discern. For two-dimensional systems state space is the familiar Cartesian plane in which, for each census time, the pair (xt, Yt) can be plotted as a point in the usual way. An orbit arising from the dynamical system consists of a collection of points in the plane visited according to a certain itinerary. For three-dimensional systems state space is a three-dimensional Cartesian system in which trajectories of three state variables can be plotted using a three-dimensional set of coordinate axes. In state space, an equilibrium is a single p o i n t ~ a point that "doesn’t move" as time flows. A 2-cycle consists of two distinct p o i n t s that are visited temporally in an alternating fashion. A 3-cycle consists of three distinct points (visited in a specific order), and so on for cycles of longer periods. The collection of points in state space corresponding to a cycle (necessarily finite in number) is an invariant set.
14
1 [ INTRODUCTION
r
a
eiO
FIGURE 1.2 [ Complex numbers have the form a where a and b are real numbers and i satisfies 2 = -1 (and is sometimes written i = V~l). One can also write a complex and represent it geometrically as in this figure. The real number number in the form X r is the "magnitude" of X (i.e., r - [kl = v/a2 + b2). The real number 0 is the "polar angle." Using this geometric representation, we see that the stability criterion for a linear system is that all eigenvalues lie inside the unit circle defined by r = 1. At a bifurcation point, the linearization has an eigenvalue on the unit circle and, therefore, of the form ~ . =
=
S u p p o s e a n e q u i l i b r i u m loses stability b e c a u s e an eigenvalue L (actually, a c o m p l e x c o n j u g a t e pair of eigenvalues) crosses the u n i t circle at a p o i n t o t h e r t h a n + 1 or - 1 . In this event, w h a t typically o c c u r s is t h e c r e a t i o n in state s p a c e of a n i n v a r i a n t set w h i c h has the f o r m of a closed, o n e - d i m e n s i o n a l loop [76, 192]. Near the b i f u r c a t i o n p o i n t this " i n v a r i a n t loop" is n e a r l y elliptic in shape, b u t f u r t h e r a w a y it c a n b e c o m e c o n s i d erably distorted. If the loop is a n attractor, t h e n the final state of n e a r b y orbits d e p e n d s o n t h e d y n a m i c s t h a t take place o n the i n v a r i a n t l o o p itself. O n e possibility is the existence o n the l o o p of a n orbit w h o s e limit set is the entire i n v a r i a n t loop. Such a "quasiperiodic" orbit m o v e s a r o u n d the loop, n e v e r quite r e p e a t i n g a n d in the p r o c e s s c o m i n g arbitrarily close to every p o i n t o n the loop. In this case, the entire i n v a r i a n t l o o p is a n attractor. The c o m p l i c a t e d oscillatory d y n a m i c resulting f r o m a n chaotic, however. It d o e s n o t p o s s e s s " i n v a r i a n t loop" b i f u r c a t i o n 7 is sensitivity to initial c o n d i t i o n s . 7 Sometime called a "discrete Hopf" bifurcation or a"Neimark-Sacker" bifurcation [55,133, 154].
15
1.2 [ B i f u r c a t i o n s a n d C h a o s
We can use the two-dimensional system
=
(1 -
(1 -
to illustrate invariant loop bifurcations and quasiperiodic oscillations. We can view this system of difference equations as a generalization of the Ricker model (1.2). In the Ricker model all individuals are treated as identical and the population is described by a single state variable, the total population abundance. The model (1.6), on the other hand, distinguishes two types of individuals belonging to two distinct life-cycle stages, an immature juvenile stage ] and a reproductive adult stage A. In addition this model, unlike the Ricker model, allows for the overlapping of generations, since a fraction 1 - #a of the adults survive from one census time to the next. The unit of time is that of the maturation period so that no juvenile remains a juvenile longer than one unit of time. The exponential terms model the effects of population density on vital rates. The term b exp(-ClA) is the per unit time production of juveniles per adult (incorporating fecundity and survival) in the presence of A adults. The term (1 - lzj) exp(-c2]) equals the fraction ofjuveniles that survive and mature to adulthood in a unit of time (in the presence o f ] juveniles). Figure 1.3 shows a bifurcation diagram generated by the system of equations (1.6). Attractors are plotted against the parameter b, while the other parameters remain fixed at certain assigned values. As in the Ricker model, when b increases from 0, a transcritical bifurcation occurs with an accompanying exchange of stability between the extinction equilibrium (], A) = (0, 0) and a survival equilibrium with positive components for both juveniles ] and adults A. This positive equilibrium destabilizes with further increase in b and an invariant loop bifurcation occurs (at approximately b = 7.6). The resulting oscillations, although nearly periodic, never exactly repeat. In state space they trace out the loop shown in Fig. 1.3 (i.e., the orbit comes arbitrarily close to every point on the loop). Another possibility for the dynamics on an invariant loop is the existence of a periodic cycle that attracts all orbits on the loop. This situation is called "period locking." If, in addition all nearby orbits approach the loop (and therefore they approach the cycle on the loop), then this cycle is an attractor. In state space, the attractor consists of finitely m a n y points lying on the loop, which is now "invisible." Typically, parameter intervals on which such period locking occurs are interspersed with intervals of quasi-periodic dynamics, forming period-locking windows in the bifurcation diagram. See Fig. 1.3. The stable loops that result from an invariant loop bifurcation can, upon further changes in the model parameter, lose their stability and
16
1
g g 40 ~- 20
,
,
,
,
,
,
,
,
,
,
20 10
40
5 b=9 4
g3
1
J
1.3
a 0.01,
0, #a
1.1,
7.6.
a a a a
a
a
a
1.3 I The Hunt for Chaos Thus, in progressing from one to just two dimensions we can encounter a considerable increase in dynamic complexity. From a bifurcation theory point ofview, chaos is often embedded within parameter regions that include a complicated variety of other dynamic possibilities (such as periodic and quasiperiodic cycles). Moreover, as attractors destabilize across a bifurcation diagram they often do not disappear, but may survive as unstable invariant sets or leave behind their influence in the form of transient dynamics (i.e., the temporal route that orbits take to the attractor). In this way, cycles, quasiperiodic orbits and even chaotic sets can leave their mark on the dynamics of a system even when they are unstable or only present for nearby parameter values. In such a regime it is difficult, and may make little sense, to relate a population’s dynamic to a specific type of attractor. Parameter estimates come with confidence intervals that are likely to incorporate a range of different types of dynamic characteristics. Under these circumstances, an attempt to identify and label a particular time series of data as chaotic becomes problematic, even in a deterministic setting. Such an attempt is made even more difficult in the presence of stochasticity. We will need to deal with stochasticity in some detail in the following chapters. For now, we only point out that random disturbances, applied to an orbit during its journey toward an attractor, induce continual transient behavior and even allow for visits to regions of state space far from the attractor. As a result, a population’s dynamics may not be dominated by a deterministic attractor. Instead the dynamics might involve a mix of characteristics-- deriving from transients and even unstable invariant sets, in addition to attractors. This is particularly likely when the deterministic component of the dynamics is complicated, involving multiple, quasiperiodic, or chaotic attractors. From this point of view, even if deterministic chaos plays a role in a population’s dynamics, it is unlikely to be the sole player. Perhaps the most significant punch line resulting from nonlinearity is the potential for a complicated array of complex dynamics, in which perturbations in state variables and parameters can lead to unusual and perhaps unexpected dynamic consequences.
1.3 It is ... worth noting that most previous connections between the theory of nonlinear dynamics and natural populations have been aimed at simply establishing that chaos is evident in ecological time series. Only within the past decade have researchers used their understanding of nonlinear
17
18
1 ] INTRODUCTION
dynamics to interpret key features of observed population fluctuations. These reports have all been retrospective, however. In no case were predictions about exact boundaries in parameter space that demarcate different system behaviours tested by manipulating a population’s parameters, with the object of seeing whether the dynamics actually did shift between stable equilibria and aperiodic cycles. PETER KAREIVA [98] Robert May’s influential papers raised the possibility that some of the observed complexity in nature might be understandable by means of simple deterministic rules. His suggestion entailed the use of lowdimensional models (specifically difference equations) "where one can seek to use field or laboratory data to estimate the values of the parameters"[122], in this same paper he reports the results of this approach when utilizing data from a n u m b e r of insect populations (alSo see [83]). However, only one laboratory species out of 24 species used in the study lay in the chaotic region of the model’s bifurcation diagram. May concluded, at least for those species in his study, that natural populations "tend to have stable equilibrium behaviour" and that laboratory populations "tend to show oscillatory or chaotic behaviour." With regard to ecology in general, however, he cautioned "that these remarks are only tentative and must be treated with caution for several reasons," including problems associated with the selection and use of data, the specialized biological circumstances that the utilized models assume (for example, nonoverlapping generations8), and the fact that most natural populations do not live in isolation from other populations. Despite these caveats, this study seems initially to have d a m p e n e d the investigation of chaos in ecological data. 9 William Schaffer and his colleagues led a resurgence of interest in identifying chaos in ecological data with their studies in the 1980s of a n u m b e r of time series data sets [71]. The flowering of nonlinear dynamics that occurred during the last decades of the 20th century, stimulated to a large extent by the influential papers of May and Li and Yorke, included a number of significant theoretical advances. Schaffer recognized that one of these advances opened a door to the possibility of detecting chaos in ecological time series data. Ecosystems are by nature high-dimensional systems; time series measurements of selected variables from an ecosystem, however, generally consist of only a few variables (often only one). 8 Some early mathematical studies of models that relax this assumption are found in [77]. 9 The noveltyand difficultyof using nonlinear theory to study time series of data may also have had a lot to do with the initial lack of enthusiasm [144].
1.3 [ The Hunt for Chaos The problem of state space dimension versus the dimension of the measured variables is addressed by a famous theorem of E Takens [174]. The Takens theorem extends a deep theorem from topology (the Whitney embedding theorem [191]) so as to allow a "reconstruction" of a surrogate for the higher dimensional dynamics from a single time series of data. This is done by "lagging the data against itself," i.e., by, for example, plotting each data point together with a predecessor in two dimensions, or each data point together with two predecessors in three dimensions, and so on (stopping when the data trajectories no longer self-intersect). The Takens theorem guarantees that the resulting trajectories and attractors accurately portray those in the higher dimensional state space, so that if the reconstructed trajectories appear to be chaotic, then one has some evidence that chaos is present in the dynamics of the higher dimensional ecosystem. One can look for evidence of chaotic dynamics in the reconstructed trajectories by plotting "return maps." Return maps are obtained by observing recurrent trajectory intersections with a lower dimensional ("planar") slice placed across their path, a powerful analytical method invented by Poincar6. In this way, one often arrives at a study of maps which, if mathematically characterized by difference equations, can be of the same type studied by May. There is, however, a fundamental difference in the interpretation of equations obtained in this manner: the coefficients need not have the biological interpretations that one would associate with them if they were assumed to describe the dynamics of the original state variables. Instead, the coefficients have to be viewed as incorporating fundamental parameters, such as per capita birth and mortality rates, in some complicated and unknown way. The approach taken in the papers by May [124] and Hassell [83], in which estimates of parameters were obtained from empirical data, is likely to yield reliable results only if one has strong reason to believe that the dynamics (of the measured state variables) are adequately described by the difference equations. Another way to analyze reconstructed trajectories is to study how well mechanistic models for the population dynamics reproduce their properties. To some extent, one can study low-dimensional nonlinear models analytically, but for the most part one must rely on numerical simulations to compute and plot trajectories in state space (this is especially true for chaotic dynamics). Of course, in such an exercise one must somehow obtain estimates for the parameters appearing in the mechanistic model. Encouraged by successes of methods based on the Takens theorem in other disciplines (for example, the famous Belousov-Zhabotinskii chemical reaction), Schaffer and his co-workers applied these methods to a
19
20
1 [ INTRODUCTION
number of available data sets for childhood diseases and to the famous data set for the Canadian lynx and snowshoe hare interaction. The results, appearing in a series of influential papers [157-162, 179], helped establish chaos in ecology as a plausible hypothesis for explaining real population phenomena. However, in carrying out their analyses, Schaffer had to contend with two significant difficulties: the shortness of the available data time series and the "contamination" of the data by noise (measurement error, random environmental perturbations, etc.). In the end, they concluded that in only one of the cases investigated, measles in human populations, "was there sufficient data to justify our initial enthusiasm" [144]. Nonetheless, Schaffer and his co-workers had reawakened interest in chaos in ecology. Although progress was limited and empirical ecologists continued to view claims of chaos with a great deal of skepticism, there was an ongoing effort during the 1990s by many researchers (including S. P. Ellner, B. T. Grenfell, and P. Turchin) to find evidence of chaos in available ecological data sets [56, 58-62, 65, 66, 85, 86, 118, 126, 137, 173, 182-187]. The approach taken by these investigators is different from that taken by Schaffer in that it is based on the attempt to determine if a time series of data indicates sensitivity to initial conditions, the hallmark of chaotic dynamics. This is done by utilizing statistical m e t h o d s - - b a s e d on fitting the data with flexible phenomenological models ~ for the ultimate purpose of estimating a single diagnostic quantity, the dominant Lyapunov exponent (whose positivity indicates this signature property of chaos). This method, however, is also susceptible to the difficulties encountered by Schaffer namely, the shortness and noisiness of available time series. Because of these difficulties (and others), it is perhaps not surprising that the results of this hunt for chaos in ecology were equivocal [144, 194]. The consensus of opinion seems to be that no data sets were found to be convincingly chaotic (according to the tests used), although some were judged tantalizingly the edge of chaos" [57, 187]. By the end of the century, various opinions came to be formulated concerning the role of chaos in ecology, opinions ranging from "chaos is rare in nature" [132] to jury is still out" [144]. From the first opinion arises the question "Why is chaos rare in nature?", especially in light of the fact that ecological models abound with chaotic dynamics. Holding the latter opinion, one concedes that the formidable difficulties involved in detecting chaos in data have not been adequately overcome by phenomenological methods oftime series analysis, but that researchers have had to rely on such methods "during the early stages of an investigation, before a limited set of competing hypotheses has been delineated" [188] and before adequate mechanistic models and data are available. As ]. N. Perry puts it,
1.4 I Mathematical Models and Data
"the consensus [among population ecologists] is that there is no substitute for a thorough understanding of the biology of the species, allied to mechanistic modelling of dynamics using analytic models, with judicious caution against overparameterisation" ([144], p. 177). This brings us full circle back to the spirit of May’s seminal papers and the approach of seeking to use data to estimate parameters in low-dimensional models. It is clear that there are formidable difficulties to be overcome in order to provide a convincing argument for the presence of chaos in a biological population. We have mentioned the shortness of the data time series typically available in ecology. We have also mentioned the troublesome issues that arise because of the ubiquitous presence of noise in ecological data and the similarity of deterministic chaos and stochasticity. (How does one distinguish between the two? In what sense can a noisy system possess the deterministic property of chaos? What is m e a n t by "noise"?~~ Other difficulties include the identification of the appropriate state variables (state space), the lack of replicated data sets, and missing data (for example, for a state variable or a relevant species). Another major impediment is the unavailability of mechanistically based models that are closely tied to data, that is to say, models that one can parameterize, can statistically validate, and can show provide quantitatively accurate descriptions a n d ~ i m p o r t a n f l y ~ predictions of a population’s dynamics. In short, missing are most of the fundamental ingredients necessary for a rigorous treatment of the question in the tradition of the "hard sciences." It comes then as no surprise that the relevance to ecology of nonlinear science and chaos theory remains controversial, even as it flourishes in many other scientific disciplines [81, 82, 105].
1.4 For the things of this world cannot be made known without a knowledge of mathematics. Argument is conclusive ... but ... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment. ROGER BACON 10 In thinking about noise and stochasticity, one is forced to think about randomness and chance. It is an amusing philosophical aside to note that, in the quote reproduced in Section 1.1, Poincar6 defines chance in deterministic terms, by invoking what we now term sensitivity to initial conditions. From his point of view randomness (and hence noise) is deterministic chaos!
21
22
1 [INTRODUCTION In this book we take an approach to May’s hypothesis that differs from those based on the examination ofindividual historical time series of data for evidence of chaos. The complexity of natural systems, along with the inherent difficulties in confidently linking data from such systems with theory and models, points to the need for controlled laboratory experiments ~ experiments designed and analyzed with the specific intent of testing the predictions of nonlinear population theory. May’s hypothesis embodies a theoretical possibility or hypothesis. It is widely recognized in science that laboratory experiments are one of the best ways to test theory (although this is perhaps less recognized in ecology than it is in other scientific disciplines [132]). In the laboratory, one can carefully control environments and eliminate, or at least reduce, the effects of confounding elements, identify important and unimportant mechanisms, make accurate census counts and other measurements, replicate results, and manipulate parameters. Although laboratory microcosms are no substitute for field experiments, they are useful for testing or disproving basic ecological concepts, mechanisms, and hypotheses. Another feature of our approach is that it does not focus on individual data sets and ask whether they possess one kind of dynamic characteristic or another, such as equilibrium dynamics, periodicity, or chaos. Instead we return to the bifurcation diagram, as a mathematical metaphor for May’s hypothesis, with an intent to investigate the range of dynamic possibilities that a real biological population can display and whether these dynamics can include the dynamic bifurcations predicted by a simple mathematical model. A bifurcation diagram is a summary of a mathematical theory’s predictions as to how a population will respond to disturbances of a specific kind. Disturbances can generally be more easily invoked and studied in laboratory experiments. While such manipulated experiments might seem unduly remote from circumstances in the natural world, one need only reflect on the nonstatic natural world, in which populations and their physical and biological environments are continually subjected to perturbations, disturbances, and manipulations. Of special note, of course, are the deliberate and inadvertent disturbances caused by humans. What are the consequences of such disturbances for the dynamics of populations and their ecosystems? How do populations respond if survival and recruitment rates are changed? Such changes can be the result of any number of causes: changes in availability of food and habitat resources, changes in mortality rates, competition pressures, exposure to predation and diseases, genetic modifications, harvesting by humans, and so on. Nonlinear theory says that responses may be unexpected, nonintuitive,
1.4 ] Mathematical Models and Data
and c o m p l e x m and, as ecologists in general recognize, the natural world is nonlinear [144]. The laboratory setup used in our studies is in fact less artificial than it might appear, in that it is rather similar to the natural environment of the biological organism we use. Beetles species of the genus or "flour beeries," as the insects in this genus are often called, have lived in containers of stored grain products produced by humans for literally thousands ofyears. Furthermore, the demographic manipulations we impress on the populations in our studies are not unlike those that would result from natural or human-induced c a u s e s - - for example, a pest eradication program, a disease, or a genetic mutation. An important role played by laboratory experiments is that of providing a detailed understanding of the dynamics of a particular species. As in other scientific disciplines, controlled experiments, in which one probes, manipulates, p e r t u r b s - - e v e n d i s t o r t s - - a system and observes how it responds, can lead to an understanding of how that system works and to insights into the mechanisms and causes that drive its dynamics. However, understanding the flour-beetle system is not the primary goal of the experimental projects described in this book. We use our laboratory system as a tool to assist in the development of modeling methodologies and mathematical and statistical techniques upon whose foundations we can study, document, and provide insights into the role of nonlinearity in population d y n a m i c s - - particularly into the complexities that can result from nonlinearity. There is a demanding prerequisite to testing a theoretical hypothesis about dynamic transitions and bifurcations, namely, the identification of adequate mathematical models. We require models that provide more than adequate statistical fits to data. We need biologically based models derived from mechanisms known to be important for the dynamics of the particular organism under consideration and in whose predictions we have a high degree of confidence. Unfortunately, models of this type are rare in ecology. The track record of relating mathematical models to ecological data is not good, particularly with respect to the formulation of testable hypotheses and predictions. In this regard, ecology differs from most other mature scientific disciplines in which verifiable quantitative predictions of models play a central role. This may be the single most important reason why a great many ecologists do not use models as serious tools in their work and are skeptical or disbelieve the insights of modeling exercises [1]. Besides the lack of prediction, shortcomings common to modeling endeavors include weaknesses in (or lack of full disclosure of) model structure, failure to incorporate biologically relevant mechanisms, poor model parameterization or calibration (including the use of
23
24
1 ]INTRODUCTION
overparameterized models), lack of model validation against independent data, the absence of a robustness or sensitivity analysis, and the failure to include stochasticity. The first step in our investigation, then, will be to confront these difficulties and to build a convincing model of the biological system we use. Only with that preliminary goal accomplished will we be in a position to investigate, both analytically and experimentally, complex nonlinear p h e n o m e n a predicted by the model (including a route-to-chaos). A mathematical model is not sufficiently connected to population data unless it accounts for variability in the data. By containing a description of how data deviate from its predictions, a model can provide the means for parameter estimation, model evaluation, and generation of realistic predictions. One source of deviation is measurement error, which affects our estimation of the values of state variables. This kind of "noise" can be present even in a fully deterministic system. A different cause of variability in data is "process error." Process errors change the values of the state variables themselves because no deterministic model can account exactly for the dynamics of a biological population ~ m a n y extrinsic and intrinsic processes and forces are inevitably left unaccounted for by any model. We describe these deviations probabilisticaUy, and the resulting system is stochastic. Since measurement errors are negligible in our experimental studies, it is primarily process errors that account for "noise" in our data. Thus, in this book what we m e a n by "stochasticity" is deviation from deterministic model predictions due to process errors. Ecologists distinguish different sources of stochasticity. Two fundamental types that have been delineated and widely discussed as important to biological populations are environmental stochasticity and demographic stochasticity [11, 13, 14, 63, 64, 123, 156, 164]. These two sources of noise act in different ways to produce random variations in population numbers from census to census. Environmental noise involves the chance variation in population numbers arising from extrinsic sources that affect all (or at least many) m e m b e r s of the population. Demographic stochasticity, on the other hand, is the variability in population numbers caused by independent random contributions of births, deaths, and migrations of individual population members. In all populations both types of stochasticity are undoubtedly present. Whatever their source, random variations in population n u m b e r s cause deviations from predictions of a deterministic model. To describe these variations requires the inclusion ofa probabilistic term in the model. The mathematical descriptions of environmental stochasticity and of demographic stochasticity are different, however, as are the resulting
1.4 [ Mathematical Models and Data
stochastic versions of the model. We will have occasion, it turns out, to model both kinds of noise in the following chapters. In addition to providing a quantitative connection between model and data, a validated stochastic model can provide stochastic predictions for the time evolution of population numbers and be used for simulation studies of a population’s dynamics. As we will see, even in our controlled laboratory situation a full understanding of a population’s dynamics requires a mixture of stochastic and deterministic elements. The addition of stochasticity to nonlinearity brings a new level of complexity to the dynamics. It provides random perturbations that continually stir the system, bringing into play far more than just the attractors of the deterministic "skeleton." Although the underlying deterministic attractors can exert their influence in the form of discernible temporal patterns, the effect of stochasticity is to prevent a system from remaining on an attractor. Stochasticity allows the system to visit locations in state space that are not on or near the attractor, including the locations on or near invariant sets. This introduces transient dynamics that can produce observable patterns in data. For example, a random perturbation that places a population sufficiently near an unstable equilibrium can cause the population to linger near the equilibrium before it attempts a return to an attractor. Moreover, there can be regions in state space where orbits are actually attracted to an unstable equilibrium (the so-called "stable manifold" of the equilibrium). In this case, an unstable equilibrium is called a "saddle." A population randomly placed near the stable manifold is tugged toward the saddle equilibrium before it is repelled, resulting in a saddle "flyby." A similar p h e n o m e n o n can occur with an unstable periodic cycle (a "saddle cycle"). In this way, the transient behavior due to stochasticity can produce distinctive temporal patterns in the data that are unrelated to attractors. Thus, in time series data under the influence of nonlinear dynamics and stochasticity, one should expect to see a complicated dance of attractors, transients, and unstable entities. It is more fruitful, in attempting to explain patterns observed in data, to study the relative influences of these various components, rather than try to explain the data in terms of a specific type of deterministic attractor. Stochastic models provide the means by which to do this. By their formulation and application we can obtain useful predictions that combine deterministic and stochastic aspects. With regard to complicated dynamics such as chaos, the blend of stochasticity with nonlinearity can create a particularly complex array of patterns that is challenging to sort out. Chaotic attractors generally exist in the presence of unstable invariant sets and sometimes in the presence ofother attractors. Furthermore, within confidence intervals ofparameter
25
26
1 I INTRODUCTION
estimates there can be a variety of other types of attractors and unstable sets m even unstable chaotic sets m and hence a whole suite of different kinds of transient dynamics. Throughout the following chapters we will see how a mix of stochastic and deterministic ingredients is needed to provide a complete explanation of data obtained from our experimental populations. Given that these issues arise in studies of "simple" biological systems, in controlled laboratory settings, it is no wonder that in a natural setting there are formidable difficulties in "finding chaos in nature." These difficulties include, but go beyond, the similarity of chaos to stochasticity, which May warned would make chaos difficult to observe in ecological data. Diagnostics calculated from time series data that are based on characteristics of attractors can be highly contaminated with the influences of other dynamic entities. Quantities averaged over deterministic attractors, such as Lyapunov exponents, are prime examples [51]. ~ It has become clear that questions such as this biological system chaotic?" or this data set chaotic?" are too narrow. Instead one must find ways to sort out the extent to which various deterministic forces can contribute to the dynamics of particular populations or ecosystems w h e n e m b e d d e d in specific types of stochasticity [45, 57, 194]. The expectation is that m a n y properties of nonlinear systems will influence the dynamics of an ecological system, including attractors, transients, unstable invariant sets, stable manifolds, bifurcations, and multiple attractors, as well as new patterns that emerge from a stochastic mix of these ingredients. From this broader point ofview, one might discover that the role of chaos in ecology is more substantial than it would otherwise appear and, as a result, one might formulate different answers to questions such as "Is chaos found in natural populations?", it c o m m o n or rare?", and populations evolve away from chaos?". However, it is a significant challenge to devise and apply methods that identify those ingredients that play significant roles in data patterns. Controlled laboratory and field experiments are ideally suited for such an endeavor. The studies in the following c h a p t e r s - - b e s i d e s addressing specific points such as May’s hypothesis, dynamics bifurcations, and routes-toc h a o s - serve to illustrate and document the issues just discussed. The first step in these endeavors is the construction of an adequate model, to which we turn our attention in the next chapter.
1~ Researchershave recently made modifications to time series methods in an attempt to surmount these difficulties (although the short lengths of available data sets is a serious drawback) [144].
2 Philosophy (nature) is written in that great book which ever lies before our eyes. I mean the universe, but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language.., without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth. 0
Mathematics... was repugnant to m e . . . [but] I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seemto have an extra sense. m CHARLES DARWIN [38] Our a p p r o a c h to the investigation of nonlinear p h e n o m e n a involves connecting m a t h e m a t i c a l models with data in a rigorous way. The first step in this a p p r o a c h is the derivation of a biologically based model. The key to this derivation is an identification of those m e c h a n i s m s that are most i m p o r t a n t and influential in determining the dynamics of the particular population u n d e r investigation, u n d e r the environmental circumstances ofinterest. The goal is to build as "simple" a model as possible that "works." That is to say, we seek a low-dimensional model that adequately describes and the population’s dynamics. To begin the model derivation we m u s t first select a small n u m b e r of descriptive state variables whose temporal changes will be used to study the dynamics of the population. This selection requires careful consideration since "[t]he state space is often the most i m p o r t a n t postulate of a scientific theory, for it defines the subject m a t t e r u n d e r investigation" ([10], p. 18). The choice is dictated by a n u m b e r o f b o t h theoretical and practical considerations. The availability and accurate measurability of data for the state variables are i m p o r t a n t considerations.
27
28
2 / MODELS Another consideration is a knowledge of the relationships among the state variables and of the mechanisms that drive their dynamics. After choosing the state variables, we deterministically model how they change in time, and do so with as few parameters as possible. This step requires a choice for the unit of time to be used in the model. The resulting m o d e l ~ t h e "deterministic skeleton" ~ will not, of course, describe or predict exactly the dynamics of the population. No low-dimensional deterministic model is able to do that; biological populations are too complicated, even under controlled experimental circumstances. D a t a - - even highly accurate data ~ will almost certainly deviate from the predictions of the deterministic model. Therefore, the next step in our model building process is to account for these deviations by constructing a stochastic version of the m o d e l - - or, in other words, our next step is to also "model the noise." The resulting stochastic model provides for a rigorous connection with data, i.e., procedures for parameter estimation, hypothesis testing, and model evaluation. Should the stochastic model pass sufficient statistical testing and be deemed "accurate," then one has a solid foundation on which to use the model, and its deterministic skeleton, to simulate the population’s dynamics. This foundation is further solidified if one can corroborate model predictions by experiments or observations. Suppose the derived model fails to pass its confrontation with data. If it is not to be abandoned entirely, it must be modified in some w a y ~ either in its deterministic skeleton or in its stochastic component, or both. This may involve minor adjustments of the model (e.g., of its nonlinear structure or its stochastic properties) or it may involve more serious adjustments and perhaps, as a result, an increase in its complexity (e.g., an increase in the n u m b e r of state variables and parameters). Thus, one needs to know and understand not only what is put into a model, but also what is left out. The new and, one hopes, improved model is then confronted with data and the process iterated until one arrives at a sarisfactory result. The models developed in this chapter are the consequence of several such iterations, although we will not discuss the interim models that were found lacking. A model that provides accurate simulations can be put to a variety of uses. The model might, for example, provide explanations for observed dynamic patterns in d a t a - - n e w explanations that were previously unavailable. Conversely, the model might reveal patterns in existing data that had previously gone unnoticed. In both cases, our understanding of the population’s dynamics is deepened. Another use ofan accurate model is the formulation of predictions. The ability to predict is, of course, an important asset of a mathematical model, one that lies at the heart of
2.1 [ The Deterministic LPA Model
the scientific method. The predictive capability of a model can be put to practical application with regard to the specific biological population the model describes, applications such as pest control and resource management. However, a model accurately wedded to a specific biological population which is available for controlled experimentation also provides a powerful system that can be used to explore a variety of general issues in population dynamics and ecology that extend beyond the specific population modeled. (The highly regarded chemostat system is an example [165].) Using such a system one can study, for example, various nonlinear phenomena and determine circumstances under which they might be expected to occur in biological populations. With new insights and perhaps newly discovered nonlinear effects, one can put forth, or rule out, possible explanations for complicated dynamic patterns observed in population and ecosystem data. In this chapter our goal is to apply the modeling methodology just described to a specific biological organism, namely, species of beetles from the genus Although these "flour beeries" are, from a human point of view, a significant pest with regard to grain products, our ultimate purpose in this modeling exercise is not to further the understanding of this insect’s biology and population dynamics. (Indeed the model we develop is a rather general one appropriate for a large class of organisms possessing a three-staged life cycle.) Instead we will use the model, in conjunction with laboratory experiments, to investigate a variety of nonlinear phenomena in population dynamics. These investigations, which are the subjects of the following chapters, will include model predicted bifurcations, a route-to-chaos, and other distinctive nonlinear effects, all of which we can document in populations of flour beeries by means of carefully designed experiments.
2.1 Nay, it may even be said that the true value of mathematics m that pride of human reasonm consists in this: that she guides reason to the knowledge of nature. IMMANUEL
KANT
All mathematical models necessarily involve simplifying assumptions. The goal in building an accurate and usable model is to include those mechanisms that are known to be the most important in determining the dynamics of the biological species of interest, while ignoring those mechanisms that are less important. In this section we derive a model
29
30
2 I MODELS based on the life-cycle characteristics of a particular insect species. However, the model is in fact rather general and it could be applicable to other biological species whose individuals under go similar life-cycle stages during their development. This In our experiments we use strains of the species T. flour beetle species is cannibalistic. Adults feed on eggs, larvae, pupae, and callows (young adults) while larvae eat eggs, small larvae, pupae, and callows. Neither larvae nor adults eat mature adults and larvae do not feed on larvae (except for the smallest of larvae). Our model is built on the hypothesis that the dominant mechanisms driving the dynamics of this species, under the experimental conditions of the project, are the nonlinear interactions caused by these cannibalistic interactions among the various life-cycle stages [26, 142]. Therefore, we are motivated to build a so-called "structured" population model [32, 130]. That is to say, in order to account for cannibalism among life-cycle stages, we build a mathematical model in which individuals are categorized according to their life-cycle stage and the dynamics of these life-cycle stages are described. Population data for biological populations typically come from census counts made at certain discrete m o m e n t s of time, usually at equally spaced intervals (e.g., annually). A deterministic model provides a prediction for the state variables at the next census time from a knowledge of the state variables at the current census time. In keeping with the general theme of this book (recall May’s hypothesis), we wish to investigate low-dimensional models, that is to say, models with as few state variables as possible. This distinguishes the models we will derive and use from population dynamics, other models that have been applied to for example, the high-dimensional models involving partial differential equations or large Leslie matrix models used in [26]. From this point of view, one begins with a low-dimensional model and modifies it to a higher dimensional model only if it fails to provide an accurate description of population data. The simplest (lowest dimensional) model involves only one state variable, say the n u m b e r of adults, whose numerical value is predicted from one census time to the next. In such a model, any interactions among lifecycle stages, such as cannibalism or competition for resources, would not be described by the model, at least not explicitly. The Ricker model (1.2) is an example. A model of minimal dimension that explicitly includes interactions among members of different life-cycle stages would distinguish at least two life-cycle stages, for example adults and juveniles. The model (1.6) is an example. In preliminary studies, we considered both one- and two-state-variable models of these types for the dynamics of
2.1 I The Deterministic LPA Model
as well as more complex formulations. For example, a stochastic onedimensional model accurately portrayed many features of the statistical fluctuations of the adults [26, 39, 41], while an "infinite variable" model (a partial differential equation) model reproduced the propensity of the larval stage to fluctuate periodically [26, 84]. None of these models, however, adequately captured the combined, simultaneous features of stage fluctuations in experimental population cultures. As we will see throughout this book, a model that provides a reasonably complete description of dynamics utilizes three state variables based on the three basic life-cycle stages c o m m o n to most insects, namely, larval, pupal, and adult stages. To construct this three dimensional model, we let L denote the n u m b e r of feeding larvae; P the number of last instar (nonfeeding) larvae, pupae, and callow adults; and A the number of mature adults. We reserve the notation L, P, and A (without "hats") for denoting the stage (numbers divided by the volume of the flour habitat). We call these state variables the L-stage, P-stage, and A-stage, respectively, although for convenience we sometimes refer to the stages as "larvae, .... pupae," and "adults" when the context leaves no chance of confusion. It is important to remark that these state variables can be accurately counted under standard laboratory conditions. Let subscripts t placed on the state variables denote the time at which these stages are counted. Thus, Lt denotes the number of feeding larvae at time t, and so on. Each life-cycle stage is counted at equally spaced census times t = 0, 1, 2, 3 . . . . where we choose the unit of time according to the following considerations. Results of Park ([139], Table 10) suggest that for T. a reasonable estimate for the feeding larva stage is 14 days. Moffa and Costantino estimate the time from egg to adulthood as approximately 27 days ([131], Table 1). This includes 2- to 4-day egg and pupal stages and an additional 2 to 4 days spent as a callow adult. Thus, the durations of the L-stage and P-stage are roughly equal, namely, approximately 14 days under standard laboratory conditions. We exploit this fact by taking the unit of time in the model (and between our census data points) to be 2 weeks. We omit an egg stage from the model. Though eggs can be and are sometimes counted in flour beetle studies, an inordinate amount of time is required to gather egg-count data. Counting just larvae, pupae, and adults allows many more cultures to be maintained in any given experiment. The egg stage is short in duration, namely, approximately 2 to 4 days [168], and consequently most eggs laid within a 2-week period become larvae by the end of the period. Although undoubtedly an improved model would include an egg stage, a three-staged larva, pupa,
31
32
2 I MODELS and adult model turns out, as we will see, to be an accurate predictor of dynamics. The feeding larval stage is the "recruitment" stage in our model. The predicted n u m b e r of larval recruits at the next census time t + 1 is assumed to be proportional to the n u m b e r of adults observed at time t. This assumption potentially introduces some bias in the model’s predictions in that a limited n u m b e r of eggs laid just prior to time t can be present in the larval class at time t + 1, i.e., adults at time t - 1 might make some limited contribution to larval recruitment at time t + 1. However, our hypothesis is that the effect on larval recruitment at time t + 1 is slight compared to the effects of other factors. Thus, the potential n u m b e r of larval recruits at time t + 1 is where b > 0 is the (average) n u m b e r of larvae recruited per adult per unit time (two weeks) in the absence of cannibalism. Before considering the effects of cannibalism, we note that in its absence (and in the absence of any other density effects on reproduction and survival) we have the formulae
Pt+x = (1 -
1 for the numbers of larvae, pupae, and adults at time t + 1. Here the fractions ~l,/Zp, and/za are respectively the larval, pupal, and adult probabilities of dying from causes other than cannibalism. These equations define a Leslie matrix model for our three-staged population, which is often written in the matrix form as Lt+l Pt+l
=
o
o
b
Lt
1-1z/
o
o
Pt
0
1 -/Zp
1 -- ~a
At
0
0
b
1-/z/
0
0
0
1-/zp
1-1Za
At+l
where
is the (Leslie) coefficient matrix [23, 32, 111, 112, 115].
2.1 I The Deterministic LPA Model
33
Thus, in the absence of cannibalism our model for Zt+l Pt+l
--"
becomes
o
o
b
1 -- btl
0
0
/3t
0
1 -/zp
1 -- /.La
At
At+l
9
(2.1)
This model is linear. It therefore (generically) predicts either extinction or unlimited growth at an exponential rate. Extinction occurs, i.e., Lt lim
t---~o~
0
Pt
=
At
0
,
0
if the dominant eigenvalue Z of the coefficient matrix 0
0
b
1 --~l
0
0
0
1 -/zp
1 --/s
is less than 1 in magnitude. (By the dominant eigenvalue Z we mean the eigenvalue of largest magnitude, which turns out to be positive by the Perron-Frobenius theorem [23, 32, 68].) On the other hand, if ~. > 1 then (nonzero, nonnegative) solutions grow exponentially without bound. It turns out ~ > 1 if and only if R0 > 1 where n0=b
(1 - / z / ) ( 1 - / z v) /Za
is the [32, 34]. It is also true that)~ < 1 if and only if R0 < 1. In the case of nonextinction (as is typically observed under standard laboratory conditions), the linear Leslie model (2.1) cannot, of course, predict the long-term (asymptotic) dynamics of flour beetle populations. In order to do that, the model must include the controlling mechanism of cannibalism. Although not biologically complete, an approximation to a particular cannibalistic interaction can be described as follows. Consider the cannibalism of eggs by adults. It has been documented by laboratory experiment that the probability of a contact between an individual adult
34
2 I MODELS
and an egg, during a fixed time interval, is inversely proportional the habitat volume V [29]. If we assume that during a small interval of time At this probability is (approximately) proportional to At (and that u p o n such a contact there is a fixed probability that the egg is eaten), then the probability an egg will survive being eaten by one adult during At time units is approximately 1 -VThe probability it will survive 2At units of time is approximately the product -
1 - --~-A
1 - ---~A
=
1 - --V-A
,
and so on. The probability it will survive a full unit of time is approximately
1 - --~-A
Under the a s s u m p t i o n that encounters between the egg and other adults are i n d e p e n d e n t events, the survival probability of one egg over one unit of time in the presence of two adults is approximately 1
-
1 - --~-A
=
1 - --~-A
In the presence of adults the probability that an egg survives adult cannibalism is approximately
--r Thus, in our which, as At ~ 0, approaches the exponential model we take the probability that an egg is not eaten in the presence of adults to be this exponential term. We refer to the ratio as the coefficient of adult cannibalism on eggs (in a habitat of volume V). In a unit of volume V the adult cannibalism coefficient is Similar exponential terms describe the probabilities of surviving cannibalism that occurs a m o n g the other life-cycle stages. The d o m i n a n t cannibalistic interactions in populations of T. are egg cannibalism by larvae and adult and p u p a cannibalism by adults [128]. Introducing these survival probabilities into the linear Leslie model (2.1) we
2.1 [ The Deterministic LPA Model
35
obtain the nonlinear matrix model
Pt+l
At+l bexp
0
0
o
0
0
(1
~p) e x p ( - ca ^
1
or, componentwise, the difference equations exp - V
- ff
t (2.2)
= (1 At+a--(1-lzp)Ptexp(-~;At~-[-(X-IZa)At.
k
v
7
In these equations and are, respectively, the coefficients of adult and larval cannibalism on eggs and is the coefficient of adult cannibalism on pupae, all of which are in reference to a unit of habitat volume. In some studies it is of interest to have habitat volume V a p p e a r explicitly in the model equations as it does in (2.2) [29, 88, 90]. However, V can be mathematically eliminated from (2.2) by using the life-cycle stage densities in which case the general LPA model equations b e c o m e = (1 -
(2.3)
= (1 -
(1 -
In the experiments studied in this book the volume V is that occupied by 20 grams of flour, which we therefore take as the unit of volume. It turns out that virtually all p u p a e of T. survive to emerge as adults (under laboratory conditions), provided they are uncannibalized. Therefore, for this species offlour beeries we assume/zp -- 0.1 T h r o u g h o u t the book we will refer to the resulting system
= (1 -
(2.4) (1 -
1 This assumption is corroborated by model parameter estimates using data. Therefore, for simplicity we assume a priori that #p = 0.
36
2 I MODELS
as the The mathematical results and statistical methods described and used throughout the book extend straightforwardly to the general LPA model (2.3). We derived the LPA model in the context of a particular insect (flour beetles) and a particular nonlinear interaction (cannibalism). The model is, however, fairly general. With appropriate modifications it could be used to account for the dynamics of many other biological species in which nonlinear interactions occur between three life-cycle stages. Of course, the model equations (2.4) represent a considerable simplification of the biology of T. Whether or not they can nonetheless account for a substantial portion of the population dynamics of this insect is a question we address with data throughout the book. Any of the model systems (2.1), (2.3), or (2.4) predict the demographic triple of larval, pupal, and adult stage numbers from one census time to the next. Mathematically, these so-called "difference equations" (or "recursion equations") define a map, which is said to be threedimensional since at each time tthe state variables form triples that can be plotted using a three-dimensional Cartesian coordinate system. Later it will be necessary for us to modify the model equations in order to account for stochastic deviations from these deterministic predictions. However, it is fundamental to our approach to first understand, as best we can, the dynamics predicted by the deterministic LPA model. That is to say, we want to describe important properties of the orbits arising from equations (2.4). Before doing this in Section 2.3, we pause to discuss briefly the role that flour beetles have played in ecological research.
2.2 God has an inordinate fondness for stars and beetles. m attributed
t o J. B. S. H A L D A N E b y K e n n e t h
Kermack
[75]
What prompted ]. B. S. Haldane to make his famous quip about the special place held by beeries in the eye of the Creator is the fact that the number of species in the insect order Coleoptera is greater than in any other animal or plant group. Experimentalists might also add that two of the 26 different species in the genus a member of the order Coleoptera, family Tenebrionidae, and subfamily Ulominae, namely, T. and T. have turned out to be powerful animal models for the study of the causes of fluctuations in population numbers.
2.2 I The Flour Beetle
The species of are most widely recognized as pests of stored grain products, although the origin of the grain habitat is not known. N. E. Good writes, ’Almostwithout exception, the beeries ofthe sub-family Olominae, ofwhich is a member, occur either as pests of stored products or else under the bark oftrees and in rotting logs" [73]. The ability of the beetle to spend its entire life history successfully in a grain habitat has led to its worldwide distribution as a pest. This same ability also allowed for the development of laboratory culturing techniques consistent with the beetle’s adaptation to a grain habitat and therefore consistent, to a large extent, with its natural habitat. It was Royal N. Chapman who, while attempting to devise preventative measures to protect flour and other cereals from attacks by the beetle, recognized its use as an excellent animal model for scientific studies in population dynamics [24]. Chapman Ten years later, in a paper that appeared in the journal introduced (specifically the species T. as an experimental organism for the study of fluctuations in animal numbers [25]. populations provide a fascinating example of nonlinear demographic dynamics. The beetle is holometabolous, which means it has complete metamorphosis possessing egg, larval, pupal, and adult stages in its life cycle. Laboratory populations maintained under constant environmental conditions usually exhibit dramatic fluctuations in density and age structure. These fluctuations are the result of strong behavioral and physiological interactions among the life s t a g e s ~ t h e most important being cannibalism. Adults eat pupae, and both adults and larvae cannibalize eggs (Fig. 2.1). David Mertz, an outstanding, experienced beetle experimentalist, commented (only slightly in jest) that "cannibalism may be the beginning, middle, and end of ecology" (personal communication, 1986). Interestingly, Alexander Sokoloff points out that a disadvantage in using in genetic research lies "in the propensity of these beetles toward cannibalism" [167]. In the following chapters, experiments with cultures of flour beetle populations will play a central role. The basic laboratory culturing conditions were the same in all of the experiments. In fact a modern beetle laboratory is essentially identical to a laboratory found in the 1920s or 1930s. The equipment used is modest: bottles for housing beetle cultures; small plates on which to place, sort, and count individual organisms; brushes; sifters; ovens; a mixer for medium preparation; and an incubator or two round out the facility. The laboratory culture medium consists of 95% wheat flour and 5% dried brewer’s yeast (by weight). For the experiments reported in this book, the culture containers are half-pint milk bottles (237 mL) and the amount of medium is typically 20 grams. Incubator conditions are highly controlled, most especially for temperature (32~
37
38
2 I MODELS
FIGURE 2.1 [ The life cycle of the flour beetle, showing the dominant cannibalistic interactions between different life-cycle stages.
and relative humidity (55%). Some particular details of the experimental protocols were, by design, unique to each study. We will clearly identify these details when each experiment is individually discussed. Replication is a hallmark of experiments. Many single (or mixed species) cultures can be started with the same initial population numbers and demography and maintained under identical conditions. The ability to obtain replicate cultures is an important asset in conducting studies of population dynamics, an asset all too often not available to researchers. Not surprisingly, however, replicate cultures identically initiated and maintained do not always dynamically evolve in identical ways. Random effects can cause differences m sometimes significant differences ~ among replicates. As we will see in our studies, rather than being an annoying problem such differences can be illuminating and lead to a deeper understanding of a population’s dynamics and their causal mechanisms. The ability to manipulate cultures is also a critical feature of the system when used as an experimental animal model. It is easy to accomplish temporal variations in environmental factors such as
2.2 I The Flour Beetle
temperature, humidity, and the amount and quality ofmedium. Since the cultures are regularly counted, the experimental opportunities to manipulate animal numbers are plentiful. Demographic parameters such as (life-cycle stage-specific) death rates can be altered by the removal or addition of individual organisms at the time of census, genetic perturbations can be imposed by introducing new variants, and so on. Data obtained from this remarkably simple laboratory system can be dazzlingly informative. Recall that one reason it has been difficult to test the idea that complex dynamics can be the outcome of simple deterministic rules is the lack of adequate time series data ~ data that are both accurate in count and sufficient in quantity and length (an essential ingredient of the scientific process). Properly designed and conducted laboratory experiments utilizing flour beetles can provide such data. Separated from the flour by careful screening, all of the animals in a culture can be accurately counted; there is no sampling. After a census is made (e.g., the data triplet (L, P, A) of larva, pupa, and adult numbers is recorded), all animals are placed in fresh medium to await the next census. Individual cultures can be maintained nearly indefinitely. The laboratory-collected time-series data form the backbone of our inquiry. Chapman’s 1928 seminal paper initiated a rich tradition of quantitative and mathematical work using as a research organism. W. C. Allee used flour beetle data in his famous studies of the effects of crowding on organisms [4]. In their well-known textbook, Allee and his coauthors write that the data of Chapman "revealed a more rapid early increase in population density with an initial seeding of 0.125 beetles per gram of flour than at lower (0.062 per gram) or higher densities" [5], a principle that came to be known as the ’Tklleeeffect." In 1931 the wellknown quantitative biologist G. E Gause, then at the Timiriasev Institute for Biological Research in Moscow, incorporated Chapman’s data into his study of the influence of ecological factors on the size of population [69]. His objective was "to express in a mathematical form the experimental data published by Chapman." Gause concluded that flour beetle data supported the idea of"logistic growth" as a fundamental principle ofpopulation growth, a theory energetically promoted at that time by Raymond Pearl [143]. Gause’s analysis was also cited in the highly influential book by Andrewartha and Birch [8]. J. Stanley, a student of Chapman, was another biologist to make use of Chapman’s data, which he used to write several papers (published over the next 34 years) on a mathematical theory of growth of populations of the flour beetle [170]. Stanley was impressed with Volterra’s mathematical work on population growth, but thought that the flour beetle could provide the biological detail so terribly absent (in his view) from the models of Volterra.
39
40
2 [ MODELS
The prominent experimentalist Thomas Park also used Chapman’s data [139], in addition to conducting many flour beetle experiments ofhis own. Park’s career is particularly interesting in that it portrays the special collaborative tradition between experimentalists using the laboratory beetle system and the mathematics/statistics community. Park, who during his long career served as president and chairman of the American Association for the Advancement of Science, president of the Ecological Society of America, and chairman of the National Science Foundation’s Environmental Biology Panel, began his career as a postdoctoral student in Allee’s laboratory. There he extended Allee’s analysis of Chapman’s data with new experiments of his own and initiated his experimental work with flour beetles. From the 1930s to the mid-1970s, Park "addressed many of the central questions of ecology before others were thinking about them, and his work really introduced quantification and statistics. ’’2After a visit in 1948 to the Bureau of Animal Population at Oxford University, Park began a long association with P. H. "George" Leslie. Together they published several key papers on interspecies competition, including studies of the stochastic aspects of competitive exclusion and coexistence [113, 114, 142]. Park’s flour beetle experiments 1940s, 1950s and 1960s were amongst the most influential in shaping ideas about interspecific competition" [16]. Indeed, many of Park’s experiments became classics in population dynamics and his data still often appear in textbooks in ecology. In fact, Park’s data continue to be a marvelous reservoir for research [26, 53]. Other mathematicians and statisticians who studied Park’s experimental data include J. Neyman (with whom Park collaborated) [134], M. S. Bartlett [13, 14], H. D. Landahl [107, 108], B. S. Niven [135, 136], E. L. Scott [134], and N. W. Taylor [175-178]. Since the seminal work of Chapman and Park a large and diverse group of researchers have contributed to an ever-growing body of literature that utilizes flour beetle data for the investigation of a wide variety of issues in population dynamics and ecology. Reviews of the literature can be found in books by Sokoloff [166-169] and Costantino and Deshamais [26] and in review papers by Mertz [129], King and Dawson [102], and Bell [17]. From a historical perspective, the studies presented in the following sections and chapters fall within the collaborative tradition set by Park and his many colleagues. The basic mathematical and statistical framework is that initiated by these early investigators (particularly, Park and Leslie). Furthermore, in the same spirit as that of these pioneers, our r e s e a r c h ~ although firmly based on the biology of the flour b e e f l e ~ 2 Fromthe obituary for Thomas Parkwritten by MichaelWade.
41
2.3 [ Dynamics of the LPA Model
aims to address and investigate general principles and hypotheses in population dynamics and ecology.
2.3 [Y]ou must not confound statics with dynamics, or you will be exposed to grave errors. m JULES VERNE
In this section we summarize some mathematical properties of the deterministic LPA model (2.4) that are fundamental to the following chapters. We will avoid technical details as m u c h as possible; a mathematically inclined reader can find rigorous proofs, relevant theorems, and formal definitions in the cited literature (e.g., in [32]). We assume throughout that 0 1 if and only if R0 > 1 [32, 34]. Thus, this linear model predicts extinction if b < and u n b o u n d e d growth if b > where #a
.
Consider now the nonlinear LPA model (2.4) in which the effects of cannibalistic interactions have been introduced. Unlike the linear LPA model, this nonlinear LPA model does not allow u n b o u n d e d growth. Specifically, it can be shown [32] that there is a three-dimensional box, described by the inequalities 0
0
0
with the property that all orbits--regardless of their initial starting p o i n t s - - lie inside the box after a finite n u m b e r of steps. Because of this, the LPA model is said to be "dissipative." It follows from general theorems global attractor" about dissipative systems that there exists a [79]. Roughly speaking, this means there is a maximal invariant set of points in the box which all orbits approach as t ~ +oo. A major goal in nonlinear dynamics is to describe this attractor. From the model equations (2.4) we see Consider the case when b < that the inequalities 0
(1-
0 0
1
are valid for all orbits. A straightforward induction shows that for every t each individual c o m p o n e n t of the LPA model o r b i t - and is b o u n d e d above by the corresponding c o m p o n e n t of the linear model (2.1). Thus, b < implies all orbits of the LPA model (2.4) tend to the origin (0, 0, 0) as t ~ +o~; that is to say, the global attractor is the origin (0, 0, 0). In this case, the global attractor for the nonlinear LPA model is the same as that for the linear LPA model. In biological terms, this means the
44
2 [ MODELS
population will go extinct when cannibalism is present if it goes extinct when cannibalism is absent. For all values of the parameters, the LPA model has the constant or equilibrium solution (L, P, A) = (0, 0, 0). As we have just seen, this equiwhich, biologically, implies exlibrium is the global attractor when b < tinction. Therefore, as far as long-term survival is concerned, the interesting case is when b > The question is: what is the attractor in this case? First of all, when b > it can be shown that the equilibrium (0, 0, 0) the LPA model (2.4) is is not the attractor. More specifically, when b > "uniformly persistent" with respect to (0, 0, 0) [32]. This means there is a positive distance e > 0 away from (0, 0, 0) which all orbits ultimately attain, i n d e p e n d e n t of their initial conditions (L0, P0, A0) # (0, 0, 0). Thus, whatever the attractor is in this case, it lies a finite distance from the extinction state (0, 0, 0) in state space. Second, an equilibrium satisfies the algebraic equations
P = (1 - / z l ) L
(2.5) (1 -
it can be shown that these equations have a unique nonzero, posIfb > itive solution triple (L, P, A). 4 Thus, when b > there exists one and only one equilibrium (L, P, A) # (0, 0, 0) and this equilibrium is positive. Is this positive equilibrium the global attractor? Computer simulations suggest for certain values of the model parameters that it can be the global attractor, although little has been rigorously proved concerning this question [33, 106]. Some headway can be made, however, ifinstead we askwhether the positive equilibrium can be a "local" attractor. More specifically, we ask whether the equilibrium is (locally asymptotically) stable. By the fundamental theorem of stability [32, 54] an equilibrium of the LPA model is stable if all eigenvalues )~ of the ]acobian matrix \
0 0 0
exp(-cpaA)
b(1
-
0 1
(2.6) 4 TOsee this, solve the third equation for P = The second equation yields which when substituted into the first equation yields a nonlinear equation for namely (after algebraic manipulations and the taking of a logarithm), the equation A consideration of the graphs of the left- and right-hand sides of this equation yields the existence of a unique solution A > 0ifb > The solution determines a positive equilibrium triple for b >
2.3 I Dynamics of the LPA Model
45
evaluated at the equilibrium, satisfy 12~1< 1. On the other hand, if at least one eigenvalue satisfies I~1 > 1 the equilibrium is unstable. Remember that eigenvalues can be complex numbers. Thus, the stability criterion is that all eigenvalues lie inside the unit circle of the complex plane. For example, the Jacobian at the extinction equilibrium (0, 0, 0) is 0
0
b
1-/Zl
0
0
0
1
1 --
. ~a
This is just the Leslie matrix coefficient of the related linear problem (2.1). Thus, (0, 0, 0) is (locally asymptotically) stable if b < and unstable if When b > the positive equilibrium is a function ofall six parameters in (2.4). Suppose all parameters except b are held fixed and we treat the positive equilibrium as a function of b. In [32] one finds a proof that for b sufficiently close to the positive equilibrium is (locally asymptotically) stable. Thus, like the one-dimensional Ricker model discussed in Chapter 1, the LPA model (2.4) has a transcritical bifurcation at the critical value an accompanying exof the larval recruitment rate b, change of stability between the extinction equilibrium and the positive equilibrium. However, the positive equilibrium does not necessarily remain stable for all b > Just as in the Ricker model, it can lose stability as b increases. This occurs if an eigenvalue of the Jacobian evaluated at the equilibrium moves from inside to outside the unit circle in the complex plane. In this case a loss of stability occurs as b is increases past that critical value of bwhere an eigenvalue satisfies I),l = 1, or in other words ), -- exp(i0) for some 0 between 0 and 27r. Unfortunately we have no explicit algebraic formulas for the positive equilibria and therefore it is difficult to perform a general stability analysis for the LPA model. It can be shown that )~ -- 1 is ruled out for this model (which in turns rules out certain kinds of equilibrium bifurcations such as saddle nodes and pitchforks), but there remain the possibilities of a period-doubling bifurcation (when X - -1) or an invariant-loop bifurcation ()~ -- exp(i0), 0 ~ 0, n). It is instructive that a complete stability analysis can be performed in one illustrative special case, namely, w h e n larval cannibalism of eggs is = 0). In this case it has been shown that the posignored (so that itive equilibrium can lose stability by either a period-doubling or an invariant-loop bifurcation [43], depending u p o n the numerical values of the remaining parameters. However, it turns out that larval cannibalism of eggs is not negligible in T. (a fact supported by both biological
46
2 I MODELS observation and by fitting the model to data). As a result this m a t h e m a t i cally tractable special case, while proving that both types of bifurcations can occur in the LPA model, is not directly applicable to this species of flour beetle. C o m p u t e r simulation studies, such as those appearing in Fig. 2.2, show that both types of bifurcations can also occur w h e n In either c a s e ~ a period-doubling or an invariant-loop bifurcation ~ further increases in b can produce more attractor bifurcations and eventually lead to chaos. However, we p o s t p o n e a consideration of chaotic attractors in the LPA model until Chapter 4. Before leaving this section we point out that a parameter other than b can be used as a bifurcation parameter in the LPA model. Indeed, in later chapters we do this, as for example in Chapter 3 where we use the adult death rate tta as a bifurcation parameter. In general, the choice of a bifurcation parameter is dictated by any of a n u m b e r of considerations; in some cases a parameter is chosen for theoretical or mathematical reasons and in other cases one is chosen for biological reasons or experimental convenience.
2.4 The tragedy is that too few researchers realize that both deterministic and stochastic models have important roles to play in the analysis of any particular system. Slavish obedience to one specific approach can lead to disaster. ~ERIC
R E N S H A W [150]
The comparable importance of deterministic and stochastic forces makes ecological dynamics unique. ~O.
N. B J O R N S T A D a n d B. T. G R E N F E L L [19]
Our next goal is to connect the LPA model with data. In order to do this and in order to carry out statistical inferences, we m u s t modify the model so as to account for deviations of the data from the model predictions provided by the equations (2.4). The modification m u s t include a probabilistic portion that specifies in what way variability arises in the data and h o w deviations from the deterministic model predictions occur [40, 42]. Different types of stochastic m e c h a n i s m s produce different patterns of variability. To build a stochastic model in this way, we have to consider carefully the stochastic m e c h a n i s m s affecting the population that give rise to these deviations. Because the approach we describe can also be used with other difference equation models we will treat the a p p r o a c h in some detail. (For alternative approaches, particularly for situations involving observation errors, see [22, 93, 121].)
2.4 [ A Stochastic LPA Model
47
As m e n t i o n e d in Chapter 1 two broad types of stochasticity important to biological populations have been distinguished by ecologists. Environmental noise involves r a n d o m variation in population n u m b e r s due to extrinsic forces that affect all individuals. Demographic stochasticity is that variability caused by i n d e p e n d e n t r a n d o m contributions by individuals to births, deaths, and migrations. There is a growing literature on modeling environmental and demographic stochasticity [6, 63, 64, 74, 100, 101,155, 180, 190, 193]. In particular, stochastic discrete time models of the type in which we are interested are treated in [23, 47]. For us the question is how to modify the LPA model (2.4) in order to incorporate stochastic variability. The choice of an appropriate stochastic model is in general a difficult problem, one that is in need of more research. In what follows, we will simply follow previous work and take as our working assumption, in modeling the experiments considered in this and the next chapter, that the noise structure is due to environmental stochasticity [123]. This is not unreasonable as a starting point, since at the one might expect large sizes found in laboratory cultures of the variability c o m p o n e n t due to environmental fluctuations to outweigh the c o m p o n e n t due to demographic fluctuations [39]. However, in Chapter 4, we reconsider this assumption, and demographic noise will come to the forefront and a different stochastic model will emerge. As an instructive example that illustrates one way to model environmental and demographic stochasticitywe consider a simple survival process in which # denotes the fraction of individuals who die during a unit of time. For pure environmental stochasticity # is a r a n d o m variable. The total n u m b e r of survivors Xt+1
=
(1 - t*) xt
(2.7)
at time t + 1 is then a r a n d o m variable depending on the n u m b e r of individuals present at time t. The m e a n of this r a n d o m variable is Mean(1 where Mean(1 - lz) is the m e a n of the r a n d o m variable (1 - #). The variance of this r a n d o m variable is Var(xt+l) = Var(1 - g)xt2 = VarOz)xt2
(2.8)
where Var(#) is the variance of the r a n d o m variable/~. On the other hand, in the case of pure demographic stochasticity each individual has a r a n d o m chance of dying with probability #. In this case is a binomial r a n d o m variable trials with the n u m b e r of survivors success probability I - ~. The m e a n of this r a n d o m variable is (1 -/~)x~
48
2 [ MODELS
and the variance is (2.9)
Var(xt+l) = (1 -/x)/x&.
A large n u m b e r of statistical techniques is available for stochastic processes of the form (2.10)
nt+l =
where is a normal r a n d o m variable with m e a n 0 and a constant variance az and E0, El, Ez, ... are uncorrelated. Such processes are called nonlinear autoregressive (NLAR) models; the nonlinear function h defines the deterministic "skeleton" the t e r m for the nonlinear model that describes the model dynamics in the absence of noise [181]. The survival processes derived earlier for environmental and d e m o graphic stochasticity do not have the additive form (2.10). However, we can ask in each case w h e t h e r it possible to transform the state variable in such a way that the stochastic process for the resulting r a n d o m variable does have (at least approximately) the additive structure (2.10). Suppose we define a transformation by an invertible function g, i.e., suppose we let n dx
0
define a change to a n e w state variable Taylor polynomial approximation
Consider the first-order
1
1
in which is a r a n d o m variable conditioned on a given value of is the derivative dd-~xX~evaluated at x Then,
and
Var(nt+x) ~ = Var(g’
1
-- Var(xt+l = Var (xt+ 1 ) [gt
If the (conditional) variance of
2 ]2.
is a function of
Var(xt+ then Var(nt+l) ~
2.
so that we can write
2.4 [ A Stochastic LPAModel
49
We wish to find a transformation so that Var(t/t+l) is approximately constant and therefore we need to solve the equation
2 -- CO for where Co is an arbitrary constant. The general solution of this differential equation is (2.11)
f (v-~x)) 1/2
where Cl is another arbitrary constant. For environmental stochasticity we have from (2.8) = Var(/x)x 2 and for demographic stochasticity we have from (2.9) = (1 - #)/zx. These formulas used in (2.11) yield the transformations
CO ) 1/2 g(x)=
Var(#)
In x + c1
for environmental stochasticity and
CO = 2 (1 -/x)/x
) 1/2 ~/x + c1
for demographic stochasticity. In these expressions C1 is another arbitrary constant, but since we need only one transformation for each type ofstochasticitywe can choose Coand c1 in a n y w a y w e wish. Thus, ifwe take Co - Var0z) and c1 - 0 in the environmental case and Co - (1 - / z ) # / 4 and r = 0 in the demographic case, we see that the transformations reduce to log and square-root transformations, respectively. Thus, one way to model environmental and demographic noise for a deterministic model
is to add noise on the logarithm scale in the first case and on the squareroot scale for the second case. For example, an environmental stochastic
50
2 [ MODELS
Ricker model takes the form exp(Et) where is a sequence of random variables, e.g., normal random variables, uncorrelated in time, each with mean 0 and a constant variance a 2. A demographic stochastic Ricker model takes the form 5
For more details on this modeling methodology, as well as models with mixed types of stochasticity, see [42, 47]. Returning to the LPA model (2.4) under the assumption of environmental stochasticity we modify each of the three state variables by adding noise on the log scale. This results in the equations exp(Elt) --
(1
-
(2.12)
exp(E2t) (1 -
exp(E3t)
which we call the "environmental stochastic LPA model." Here
is a random vector which has random variable components and We will assume these components, at any given time t, have means equal to zero and a (symmetric) variance-covariance matrix denoted by ~. Covariances among and at any given time t are given by offdiagonal elements in ~. However, we expect covariances between times to be small by comparison and therefore we assume El, E2, E3, ... are uncorrelated. The stochastic equations (2.12) have a number of statistical advantages. First, on a logarithmic scale the model has the general form Wt+l = h(Wt) + Et
(2.13)
5 In this m o d e l there is a c h a n c e that the expression in the brackets will be negative, in w h i c h case it s h o u l d be set equal to zero.
51
2.4 [ A Stochastic LPA Model
where In W t ---
In In
is the column vector of log-transformed state variables,
h(Wt) -
ln[(1 ln[Pt
(2.14) (1 -
and Et has a multivariate (0, E) normal distribution. A stochastic model of the form (2.13) is a type of multivariate, nonlinear autoregressive (NLAR) model. The development of statistical methods for nonlinear autoregressive models (estimation, testing, evaluation) has received much attention in recent years [181]. Second, the nonlinear map defined by the deterministic model on the logarithmic scale is preserved in the conditional expected values of In In and In given values of In In and In That is to say,
E(ln
-- ln[(1 -
(2.15) (1 -
Thus, the stochastic version retains the essential dynamic properties described in Sections 2.1 and 2.3. Other advantages of the stochastic model (2.12) include the ease of its use for computer simulations and, as we will see, for the computation of model parameter estimates using data. In addition to these mathematical and statistical advantages, the stochastic model (2.12) also has biological advantages. The stochastic model (2.12) allows for covariance of fluctuations in larvae, pupae, and adults in a given time period, as described by the covariance of the elements in E t . A bad or good week for adults is likely related to a bad or good week for larvae, etc. Autocovariances of the noise elements through time, however, are not expected to be important when compared to the covariances among the elements within a time period, provided the underlying dynamics of the deterministic model (2.4) are specified correctly. Also, the different scales of variability for larvae, pupae, and adults are accounted
52
2 I MODELS
for through the parameters on the main diagonal of the variancecovariance matrix E. The stochastic LPA model (2.12) provides an explicit "likelihood function." A likelihood function gives the chance that an outcome of a proposed stochastic mechanism would result in the observed data, relative to all other possible outcomes. Data for a particular population, which take the form of a trivariate time series (10, P0, ao), (/1, Pl, a l ) , . . . , pq, are a realization of the joint stochastic variables and Let In wt =
In
(2.16)
In denote the vector of observations (on the log scale) and let 0 denote the vector of unknown parameters in the function h of Eq. (2.13) (i.e., the parameters in the deterministic model). The likelihood function L (0, E) is given by q
17 i=1
where is the joint transition probability density function (pdf) for Wt conditional on Wt-1 = wt-1 and evaluated atwt. It has a multivariate normal pdf with a mean vector of h(Wt_l) and a variance-covariance matrix E" -- 1~1-1/2(27r) -3/2 e x p [ - ( w t - h ( W t _ l ) ) r ~ - l ( w t -
h(Wt_l))/2]
(where w ~ denotes the transpose of the column vector w). Most statistical calculations utilize the log-likelihood q
In
In i=1
=
32 q l n 2 J r - ~ q In I~1 1
2/~1 (wt -- h ( w t - 1 ) ) r ~ - I ( w t - h ( w t - 1 ) ) .
(2.17)
A likelihood function is a fundamental tool in statistical inference [172] and represents a crucial connection between model and data. We will use it to "calibrate" the LPA model by obtaining estimates of the model parameters from data.
2.5 [ P a r a m e t e r Estimation
53
2.5 The maximum likelihood (ML) estimates of the parameters in 0 and E are those values that jointly maximize the likelihood function L (0, E). This is equivalent to maximizing the log-likelihood function In L (0, E) appearing in (2.17). The ML estimates of the parameters have several desirable statistical properties.As the sample size increases, they are asymptotically efficient (i.e., the variances approach the theoretically lowest bound), unbiased (i.e., the bias approaches zero), and normally distributed (which allows for the construction of approximate confidence intervals) [172]. For nonlinear time series models, the sample size is the number of observations in the time series. Theorems about these and other ML properties generally require the stochastic model to have a stationary distribution [181]. A nonlinear autoregressive model of the form (2.13) typically has a stationary distribution when every trajectory of the underlying deterministic model W t + 1 = h(Wt) has a bounded attractor, which we saw in Section 2.3 is the case for the deterministic LPA model (2.4). No general formulas for ML parameter estimates exist. 6 Neither do such formulas exist for the particular case of the LPA model. Therefore, maximization of the log-likelihood function In L (0, E) must be done numerically. For example, we have found the Nelder-Mead simplex algorithm [138, 148] convenient, reliable, and easy to program for this purpose. There is, of course, uncertainty associated with ML parameter estimates. However, with likelihood methods it is straightforward to compute confidence intervals of the individual parameters and joint confidence regions for sets of parameters (we use 95% confidence intervals and regions). Among the methods we use for constructing confidence intervals and regions is the method based on the "profile likelihood." Profile likelihood intervals require a great deal of computation, but can be applied to many different types of statistical models [127, 189]. They are only approximate in that their coverage frequencies only asymptotically converge to 95% as the sample size (time series length) increases without bound. The intervals are usually asymmetric and typically have better small-sample coverage frequencies than do symmetric confidence intervals arising from the matrix of second derivatives of the log-likelihood function. Profile likelihood intervals and regions are calculated as follows. Suppose 3 is a parameter, or a vector of r/>_ i parameters, of interest in the 6 However, the ML estimates of the p a r a m e t e r s in the matrix ~2 can be written in terms of the ML estimates of the p a r a m e t e r s in 0. Specifically, ~2 RR ~/q where R = [el, e2 . . . . . eq] is a matrix with the residual vectors (2.21) as c o l u m n s a n d R r is its transpose. A
54
2 I MODELS TABLE 2.1 I ML Parameter Estimates for the LPA Model Calculated from Control Cultures of the Desharnais Experiment, with (Profile Likelihood) 95% Confidence Intervals. a
t~a
vector 0 and let r denote the vector of remaining parameters. Let r and E~ denote the values of r and E obtained by maximizing the likelihood function In L (8, E) = In L (~, r E) for a particular fixed value of/3. Then L (/3, r E~) taken as a function of~ is the "profile likelihood." Evaluating this function requires a separate maximization for each value of/3. The 95% profile likelihood interval (or region) is the set of all values of ~ for which -2[ln L(/3, r
~]~) - In
~])] -< X20.05(rl)
where X0.05 2 (~) is the 95th percentile of a chi-square distribution with n degrees of freedom. For example, X0.05 2 (~) ~ 3.843 if/3 is just one parameter. (This interval is the set of all/3 values for which a likelihood ratio test on would not reject the null hypothesis.) Am alternative method for confidence intervals, based on "bootstrapping," is described in Chapter 4. As an example we apply these methods, with the environmental stochastic LPA model, to the data from the "Desharnais experiment" reported in [48, 49, 50].7 A description of the experiment and the data from the four control replicates in that experiment appear in the Appendix. From these data we obtain the ML estimates and confidence intervals in Table 2.1 [43]. 7 For an example of this parameterization m e t h o d using the environmental stochastic Ricker model see [32, 95].
2.5 I Parameter Estimation
55
Experimental data typically come from replicated laboratory cultures, as is the case for the data used to calculate the estimates in Table 2.1. One test of a dynamic model is that replicated populations are expected to have the same parameter values. We can statistically test whether or not the parameter estimates obtained from separate replicates could have arisen from one c o m m o n model with identical parameters using a likelihood ratio test [172]. If r replicate populations are cultured and counted, r multivariate time series would result. (It is not necessary that each time series be of the same length of time.) The log-likelihood for the j t h replicate, denoted In ~j), is given by (2.17), except with and h(wj, t-1) substituted for q, wt, and h(wt-1). Here is the sample size ofthe j t h replicate, is the vector of observed (logarithmic) population sizes for the j t h replicate at time t, and h(wj, t-1) is the vector of conditional expected values for g i v e n Wj, t-1 = (Eqs. (2.15)). The joint log-likelihood for all r replicates, provided they are independent replicates, is the sum of the individual log-likelihoods: ...,
In
...,
In
(2.18)
j=l
Here and Ej contain the parameters for the j t h replicate. The null hypothesis of the test is that the r replicates are trajectories from the stochastic LPA model (2.12) with identical parameters: n0: 0 j -- 0,
~-~j = X
for all j = 1, 2 , . . . , r.
The test requires ML parameter estimates under both null and alternative hypotheses. For the alternative hypothesis, the model is fitted individuallyto each replicate (ln given by (2.17) is maximized), to obtain the ML estimates 0j and ~ j for each j = 1, 2 . . . . , r. Substitution of these estimates into (2.18) produces the maximized log-likelihood under the alternative hypothesis: In
= In
....
...,
For the null hypothesis, we substitute 0 for 01 . . . . , 0 and ~3 for ~-’]1,- . . , Y]r in the log-likelihood (2.18). The maximized log-likelihood under the null hypothesis is then = l n L ( O , . . . , O, ~ , . . . , ~).
56
2 I MODELS TABLE 2.2l ML Parameter Estimates for Each Individual Replicate from the Desharnais Experimental Data Given in Appendix A.
If the null hypothesis is true, the likelihood ratio test statistic given by G2 = - 2 ( l n
In
(2.19)
will have an approximate chi-square distribution with 12(r - 1) degrees of freedom (the n u m b e r of parameters estimated under the alternative hypothesis minus the n u m b e r of parameters estimated under the null hypothesis), a The conditions for the chi-square approximation to hold are the same as the conditions for the asymptotic efficiency of the ML estimates, namely, a stationary distribution, large sample size, and appropriateness of the model itself. For example, the data from each of the individual four replicates that we used to construct the parameter estimates in Table 2.1 should, theoretically, arise from the same model with the same parameter values. To test the null hypothesis H0 that the parameters are identical for all that the parameter four replicates versus the alternative hypothesis values are different among the replicates we calculate the ML parameter estimates for each separate replicate. The results appear in Table 2.2. If we use the likelihood ratio statistic (2.19) for the test, the result is that we fail to reject the null hypothesis at the 0.05 significance level = 36, P = 0.065). (G 2 = 49.6, The statistical properties of ML estimates do not hold if the model is a poor description of the underlying stochastic mechanisms producing the data. In particular, if the noise vector Et in (2.13) does not have a multivariate normal distribution, or is correlated through time, then the ML estimates could be biased. Since we aim to identify dynamic behavior 8 The number 12 is the number of parameters in 0 plus the number in ]C, which in both cases is 6.
2.5 I P a r a m e t e r
57
Estimation
by estimating the parameters in the LPA models, we will also use an alternative estimation m e t h o d - - o n e that yields more robust parameter e s t i m a t e s - - a s a check on the ML estimates. Conditional least squares (CLS) estimates relax most distributional assumptions about Et [103, 181]. If the normality assumptions about Et are reasonable, both ML and CLS parameter estimates are consistent (i.e., converge to the true parameters as sample size increases) and thus they should be approximately equal. However, CLS estimates remain consistent even if Et is nonnormal and autocorrelated, provided the stochastic model has a stationary distribution [103, 181]. CLS parameter estimates for multivariate time series models have not received much study. The estimates are typically described only for univariate models [181]. Fortunately, for the LPA model the CLS estimates reduce to three univariate cases. This is because each parameter appears in no more than one model equation. The CLS parameter estimates are calculated from the sum of squared differences between the value of a variable observed at time t and its expected (i.e., one-step forecast) value, given the observed state of the system at the previous time t - 1. For the LPA model, we have three conditional sums of squares: q -
2
~--~{ln i=1 q
Qq(02)
--
ff-~{ln
ln[(1
-
(2.20)
2
i=1 q =
E{lnat
-
ln[pt_l
(1 -
2.
i=1
Here
01
,
a
are the parameter vectors from each of the individual equations in the LPA model (2.4). These conditional sums of squares are calculated on the log scale because that is the scale on which we assume the noise is added in (2.12). The conditional one-step expected values appearing in (2.20) are from Eqs. (2.15). The CLS estimates minimize each of these sums of squares. Thus, three separate numerical minimizations are required.
58
2 I MODELS
TABLE 2.3l CLS Parameter Estimates for Each Individual Replicate from the Desharnais Experimental Data Given in Appendix A.
#/
(The Nelder-Mead simplex algorithm is again a convenient algorithm to carry out these minimizations. Alternatively, one can use standard nonlinear regression algorithms.) The CLS estimates for the LPA model using the data from the Desharnais experiment appear in Table 2.3. These estimates are similar to the ML estimates in Table 2.2, which suggests that the distributional assumptions about the noise vector E t in the stochastic LPA model (2.12) are reasonable.
2.6 Evaluation procedures for the parameterized, environmental LPA model (2.12) center on the log-scale residuals. These are defined as the differences between the logarithmic state variables and their one-step estimated expected values: h
et
(2.21)
= wt - h(wt-1). i
Here wt is the log-transformed data vector (2.16) and h denotes the vector (2.14) with parameters given by the ML estimates in Table 2.1, evaluated at these data points. If the model "fits," then the residuals e~, e2, ..., eq (calculated using the data used to estimate the parameters) should behave approximately like uncorrelated observations from a trivariate normal distribution. Another evaluation of the model is to consider residuals calculated from data that were used in the parameter estimation (in other words, to see how well the model predicts other data).
2.6 [ Model Validation
59
2.6.1 Unlike the original noise vectors Et, the residual vectors et are correlated and their normality is approximate, with the quality of the approximation varying among different nonlinear time series models. Thus, autocorrelation tests and normality tests should be used only as rough guides to potential areas in which the model is not adequate. The residuals for each individual state variable should have small autocorrelations and approximate univariate normal distributions. In addition to standard normal probability plots, statistical tests for (univariate) normality such as the Lin-Mudholkar test are useful [117, 181]. Standard univariate autocorrelation tests are informative as well. A useful scan for outliers from multivariate normality is to calculate the quadratic form
(2.22) for each residual vector (where denotes the transpose of a vector). If et is indeed an observation from a multivariate normal (0, E) distribution, then is an observation from a chi-square distribution with three degrees of freedom [30, 163]. We use the ML estimate of E in (2.22) and therefore the chi-square distribution is only approximate. Figure 2.3 shows times series plots of the data from replicate A (see Appendix A), together with the one-step LPA model predictions for each life-cycle stage. The predictions use the deterministic LPA model (2.4) with the parameter estimates given in Table 2.1. This plot, together with similar plots of the data from the other three replicates, shows, visually at least, that the one-step model predictions are reasonably accurate. Another visual way to inspect the residuals is to plot, for each stage, the differences between the logarithms of the observed and predicted numbers at time t + I against the numbers at time t of those stages on which they depend according to the LPA model (2.4). This is done in Fig. 2.4. We see from these plots that overall the residuals are not large in magnitude and, furthermore, that they do not seem to vary systematically with the sizes of the state variables. Table 2.4 displays the results of a univariate normality analysis of the residuals. The residuals were calculated using data, and the LPA model (2.4) with ML parameter estimates in Table 2.1 (which we recall were obtained from these same data). Shown are first- and second-order autocorrelations and the Lin-Mudholkar normality statistic for each state variable in each of the four replicates. Only replicate B reveals some slight autocorrelation (and only second-order). Departure from normality is displayed only by the pupae of replicates A and C and the adults of
60
2 [ MODELS
400
300
g
0 300
a. lOO
o 15o
< 50
0
|
|
|
|
|
!
|
FIGURE 2.3 [ Time series data (solid circles) and one-step forecasts (open circles) for Replicate A from the Desharnais experimental data given in AppendixA. Forecasts are calculated from the deterministic LPA model (2.4) using 2.1. (From B. Dennis, R. A. Desharnais, 1. M. Cushing, and R. F. Costantino, Nonlinear demographic dynamics: mathematical models, statistical methods and biological experiments, 65, No. 3 (1995), 261-281. Reprinted with permission from The Ecological Society of America.)
a
2.6.2 2.1
a
2.6
61
0
Lt
400 =
3
0 B (week 34)
0
_
-3 ~)
400
Pt
400 ~
o
~
J 2.4 1 a a 2.5. 65,
a 2.1.
3
a
62
2 I MODELS 2.4 I
(2.4) with P a r a m e t e r Values A p p e n d i x A. a
a 2.1
27 2.4
10
a
2.7
BLAISE PASCAL
2.1
a 2.1
a
2.7
63
0iii I&.
x
LH
3 2 1 0 -1
17
2.5 2.1.
J. 65,
3
2.7.1 2.1, bc/" --"
0.2275 2.3, a
64
2 1MODELS equations (2.5) with the ML parameter estimates from Table 2.1, is L
124.0
P
~
60.40
A
.
96.34
This equilibrium is unstable. This can be seen from the eigenvalues )~1 = -1.152,
~.2
--
0.6944, )~3 = 0.007877
ofthe Jacobian (2.6) evaluated at the equilibrium, namely, the 3 x 3 matrix -1.149
0
0o8 1
0
-0.
317 /
One eigenvalue (namely,)~1) has magnitude greater than 1 while the remaining two eigenvalues have magnitudes less than 1. When eigenvalues of magnitudes both greater and less than 1 occur (i.e., there exist both "stable" and "unstable" eigenvalues), the unstable equilibrium is called a "saddle." It is characteristic of a saddle equilibrium that even though it is unstable there nonetheless exist orbits that approach it asymptotically. The collection of orbits in state space that approach a saddle is called its "stable manifold." Globally, a stable manifold can be geometrically very complicated. However, near the equilibrium it is well approximated by (i.e., is tangent to) the line or plane spanned by the eigenvectors associated with the "stable" eigenvalues. In the case just given, the local stable manifold is two-dimensional and is approximated by a plane. 9 All other orbits not lying on the stable manifold eventually move away from the equilibrium. However, those orbits initially near the stable manifold will initially approach the equilibrium, only eventually to move away from it. We call this motion in state space a "saddle flyby." The parameterized deterministic LPA model for the Desharnais experiment predicts persistence, but no stable equilibria. What, then, are the asymptotic dynamics ofthis model? Notice that the "unstable" eigenvalue )~1is close to - 1. This suggests that the system is close to a period-doubling bifurcation and that the attractor might therefore be a 2-cycle at the 9 The tangent plane to the stablemanifoldis obtained fromthe span ofthe unit eigenvectors
V2 ~
0.02782
-0.9988
,
V3 ~
0.9687
-0.2476
(associated with X2and )~3,respectively) translated to the equilibrium point.
2.7 I Predicted Dynamics
65
FIGURE 2.6 [ The bifurcation diagram for the deterministic LPAmodel (2.4) shows a period8.737. Plotted vertically is the total population size on the doubling bifurcation at b attractor. All other parameters fixed at the ML estimates in Table 2.1. The shaded region shows the confidence interval 6.2 < b < 22.2 for the ML estimate b = 11.68 in Table 2.1. estimated p a r a m e t e r values. This in fact turns out to be the case. The bifurcation d i a g r a m in Fig. 2.6 shows, for the e s t i m a t e d value of b = 11.68, that the deterministic LPA m o d e l does i n d e e d have a stable 2-cycle. This 2-cycle, w h i c h can be calculated numerically, consists of the two stage vectors L1 PI A1
L2
18.2 ~
158.5 106.4
,
P2 A2
325.3 ~
8.9
(2.23)
118.5
visited at alternate time steps. The 2-cycle has two possible phases, one of which a s s u m e s the first stage vector in (2.23) at even times steps t = 0, 2, 4, . . . , while the other a s s u m e s this stage vector at o d d times steps t = 1, 3, 5, .... For the D e s h a m a i s e x p e r i m e n t the a s y m p t o t i c prediction of the parameterized deterministic LPA m o d e l (2.4) is the 2-cycle (2.23). We have argued in this c h a p t e r that this m o d e l accurately describes the data from that experiment. On these g r o u n d s we conclude that the beetle cultures in the e x p e r i m e n t should, in the long run, exhibit an oscillation of period 2. How does this prediction actually c o m p a r e with the data?
66
2 I MODELS
Since the experimental data does not start on the predicted 2-cycle, but instead starts at the stage vector Lo Po Ao
10 =
35
,
(2.24)
64
there are also model predicted transient dynamics that occur before the data comes near the 2-cycle. See Fig. 2.7. For data not initiated on or 400 300 o’)
_5
FIGURE 2.7 I The solid circles are the deterministic LPA model predicted L, P, and A-stage numbers starting from the initial stage vector (2.24) and using the estimated ML parameters in Table 2.1. A time unit equals 2 weeks.
67
2.7 I Predicted Dynamics
! ~
........ .+-+’"
o
i
~
"+~"’"’"’"" ’ - .
I
~. 100
!
m
lOO 0
’I
FIGURE 2.8 [ The open circles are the data from replicate A in the Desharnais experiment for t >__10 (20 weeks). The solid circles are the model predicted 2-cycle (2.23). We point out that the model 2-cycle comes in two phases.
very near the predicted 2-cycle we expect, therefore, to find evidence for (or against) the 2-cycle only near the end of a time series of data, after necessary transients due to the initial condition have elapsed. Figure 2.8 shows a comparison of replicate A in the experiment with one of the phases of the predicted 2-cycle attractor (2.23) from week 20 until the end of the experiment at week 36. The accuracy of the model in both these time series plots and the state space plots is striking. It is interesting to note in Fig. 2.7 that the LPA model time series starting at the experimental initial stage vector (2.24) approaches a different phase of the 2-cycle from that approached by the data shown in Fig. 2.8. The explanation of this peculiarity (as well as other peculiarities in this and the other replicates of the Desharnais experiment) is to be found in the effects of stochasticity, to which we now turn our attention.
2.7.2 Based on our analysis using the LPA model we can reasonably conclude that the beetle populations in the Desharnais experiment tend to a stable oscillation of period 2. This conclusion is based on an asymptotic property of the parameterized, deterministic LPA model,
68
2 I MODELS
namely, that this model has a 2-cycle attractor (2.23). However, population data deviate from deterministic model predictions because of stochastic effects. Therefore, we expect differences to occur between the deterministic model predictions and any of the experimental replicates, with regard to both the long term dynamics (the predicted 2-cycle state) and the transient path data orbits take to this attractor. For this same reason, we also expect differences to occur among the replicates. In this section we investigate some of these deviations and show they possess patterns accounted for by the stochastic version (2.12) of the LPA model. If noise of sufficiently large magnitude is imposed on a deterministic process, any deterministic pattern will be totally obscured. We are interested, however, in processes in which both deterministic and stochastic effects are observed. For example, the larval and pupal data do not exactly match the deterministic model predicted 2-cycle in Fig. 2.8. Nevertheless, the noise (i.e., the deviations from the deterministic model equations) is not of sufficient magnitude to erase the "influence" of the deterministic 2-cycle in the larvae and pupae numbers. ("Of sufficient magnitude" here means relative to the amplitude of the oscillations exhibited by these two components.) Nor is the noise large enough to erase the 2-cycle influence on the entire three-dimensional demographic stage vector, as is evidenced bythe state space plot in Fig. 2.8. (In this case, the relative scale is set by the distance separating the two points of the cycle in state space.) However, the amplitude of the adult component of the predicted 2-cycle is considerably less than that of the either the larval or pupal components. This makes adult numbers susceptible to noise at a magnitude that does not affect the two immature stages. Indeed, the adult data in Fig. 2.8 do not exhibit a discernible 2-cycle oscillation. T h e stochastic LPA equations (2.12) represent a model that attempts to account for random deviations from the deterministic model. The estimated matrix E in Table 2.1 approximates the nature and magnitude of the noise in the experiment. Howwell does this stochastic model compare with the data? A stochastic model, such as (2.12), does not produce a unique predicted time series or orbit starting from given initial conditions, as does the deterministic model (2.4). Equations (2.12) generate time series as follows. The three random components of the random vector Et with the necessary statistical prol~erties are simulated using, say, a computer. For each time t, the vector Et must be a trivariate, normally distributed random variable with mean 0 and must have the variances and covariance estimated in Table 2.1. Then Eqs. (2.12), together with initial conditions (L0, P0, A0) and the parameter estimates in Table 2.1, determine a unique time series. One example of a time series generated this way appears
2.7 J Predicted Dynamics
50
FIGURE 2.9 I The time series data (open circles) for replicate A in the Desharnais experiment is shown together with one simulated times series (closed circles) obtained from the stochastic LPA model (2.12). Parameter estimates are from Table 2.1.
in Fig. 2.9. This particular simulated time series bears a close resemblance to the data from replicate A in the experiment. However, simulated time series arising from other realizations of the random vector Et will be different (with high probability), even though the initial conditions are the same. The times series in Figs. 2.10 and 2.11 illustrate two ways in which simulations of the stochastic model (2.12), starting from the same initial condition, can differ. First, Fig. 2.10 shows that two stochastically simulated,
69
70
2 I MODELS
g -
& 50
0
4
8
12
16
20
24
28
32
36
FIGURE 2.10 [ Shown are two simulated time series obtained from the stochastic LPAmodel (2.12) with the same initial conditions a n d using p a r a m e t e r s from Table 2.1. Note the phase difference b e t w e e n the oscillations that starts at time week 10 (t = 5).
oscillatory solutions can become out-of-phase with one another. This shift in the phase has the following model-based explanation [89]. In the three-dimensional (L, P, A)-state space the stable 2-cycle consists of two distinct points. Mathematically these points are fixed points (equilibria) of the so-called first composite of the model (which defines the map that takes a point ahead two steps into the future). Each of these two equilibria of the composite has a basin of attraction, defined to be the set of points (L0, P0, Ao) whose orbit approaches the equilibrium
2.7
71