Chaos in Ecology, Volume 1: Experimental Nonlinear Dynamics (Theoretical Ecology Series)

Chaos in Ecology

Experimental Nonlinear Dynamics

A volume in the Academic Press I THEORETICAL ECOLOGY SERIES Editor in Chief Alan Hastings, University of California, Davis, USA

Editorial Board 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

Published Books in the Series Cushing, Costantino, Dennis, Desharnais, Henson, Chaos in Ecology, 2003

C H A O S IN E C O L O G Y

Experimental Nonlinear Dynamics

J. M. Cushing

University of Arizona R. E Costantino

University of Arizona Brian Dennis

University of Idaho Robert A. Desharnais

California State University, Los Angeles Shandelle M. Henson

Andrews University

~

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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]

This Page Intentionally Left Blank

CONTENTS Foreword Preface

ix rdi

1 ] INTRODUCTION 1.1 WhatIs Chaos? 1.2

1

4

Bifurcations a n d C h a o s

1.3 The Hunt for Chaos 1.4

6

17

I MODELS

27

2.1

The Deterministic LPA Model

2.2

The Flour Beetle

2.3

Dynamics of the LPA Model

2.5

Parameter Estimation

2.6

Model Validation

58

2.6.1

Model Fit

59

2.6.2

Prediction Fit

2.8

Predicted Dynamics

41

46 53

60 62

2.7.1

Deterministic Dynamics

2.7.2

Stochastic Dynamics

Concluding Remarks

3 I BIFURCATIONS 3.1

29

36

2.4 A Stochastic LPA Model

2.7

21

Mathematical Models and Data

63

67

77

81

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

98

84

91 vii

viii

Contents

4 I CHAOS

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

143

5 [ PATTERNS IN CHAOS 5.1

Sensitivity to Initial Conditions

5.2 Temporal Patterns 5.3 Lattice Effects 5.4

147

153

159

Concluding Remarks

171

6 I WHATWE LEARNED BIBLIOGRAPHY APPENDIX

173

183

195

A The Desharnais Experiment

195

B The Bifurcation Experiment

196

C The Chaos Experiment Index

223

213

149

138

F o r e w o r d : An I n o r d i n a t e Fondness for Beetles 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 Science and Nature. 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 per capita 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 should 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 all the 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 hoc 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 overly ambitious, 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, Tribolium castaneurn. 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 both 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

PREFACE

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

eeo

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

Department of Mathematics, Program in Applied Mathematics University of Arizona R. E Costantino

Department of Ecology and Evolutionary Biology University of Arizona Brian Dennis

Department ofFish and Wildlife Resources, Division of Statistics University of Idaho Robert A. Desharnais

Department of Biological Sciences California State University Los Angeles Shandelle M. Henson

Department of Mathematics Andrews University

1 [ INTRODUCTION 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 predictions 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 fairly realistic model, 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 S A N T A Y A N A (Dominations and Powers)

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 is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight a m o u n t of randomness, but does not look deterministic." In his popular 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 approximately. If that enabled us to predict the succeeding situation with the same approximation, 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 trajectory is chaotic if it is bounded in magnitude, is neither periodic nor approaches a periodic state, and is sensitive to initial conditions. 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 [ BIFURCATIONS AND CHAOS If the Lord Almighty had consulted me before embarking on creation I should have recommended something simpler. m attributed to ALPHONSO X (the Wise) 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 x t 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 Xt+l -- bxt(1

-

xt)

(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 p o i n t s - - x o = x ~ , X l = x ~ , x z = x ~ , x 3 = x 1*, . . . - - a n d is called a periodic "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 bxt(1 - xt) for population abundance xt+l at time t + 1 to the formula b x ( t ) ( 1 - x(t)) for the rate of change of population abundance at time t found in the famous logistic differential equation d x / d t = bx(1 - x). 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 xt = x(t) satisfies the so-called Beverton-Holt equation Xt+l - - 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 Xt+l = bxt exp(--cxt),

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 I Bifurcations and Chaos

9

f 1400 1200 1000 800 600 400 200

5

10

15

20

25

30

35

b

1000

800

X

600

400

200

20

22

24

b

26

28

30

FIGURE 1.1 [ The top plot shows a bifurcation diagram for the Ricker equation (1.2) with c - 0.01. Points on the attractor are plotted above the value of b. Thus, when b = 5 the attractor is a single point (an equilibrium), at b = 10 it consists two points (a 2-cycle), and at b = 13 it consists of four points (a 4-cycle). For larger values of b the attractor consists of a large number of points (in some cases infinitely many points). The lower graph shows some detail of a period locking window that occurs from approximately b -- 22 to 25. In this graph we see other period-locking windows and, within the large window, a period-doubling cascade of cycles.

be but one possible route-to-chaos for a dynamical system, although it is a typical route for one-dimensional systems defined by a broad class of difference equations, such as the Ricker and discrete logistic equations. The attainment of chaos through a sequence of attractor bifurcations is fundamental to the approach taken in this book. However, the dynamical systems with which we will deal are not "one-dimensional" like the

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 Xt+l -" f ( X t )

(1.3)

the simplest attractors are stable equilibria. Equilibria are constant solutions xt = x* and therefore correspond to the roots of the algebraic equation x*-- f ( 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 xt+~ = )~xt in which ~ is the derivative d f ( x ) / d x evaluated at x = x*. All solutions xt = ~.txo 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 f(x) = bx e x p ( - c x ) a n d

dfCx) dx

= b(1 - c x ) e x p ( - c x ) .

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 * = f ( f(x*)). Thus, a 2-cycle is an equilibrium of the "first composite" equation Xt+ l =

f ( f (xt) )

and we can apply the same stability analysis we apply to equilibria, but now with X equal to the derivative d f ( f ( x ) ) / d x 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 (xt, Yt) = (x*, y*) of a two-dimensional dynamical system defined by the pair of difference equations

xt+ l = f (xt, Yt) Y t + l --"

g(xt, Yt)

correspond to the roots of the two simultaneous algebraic equations

x* = f (x*, y*) y* = g(x*, y*). An equilibrium is stable if its linearization is stable, i.e., if the equilibrium (xt, Yt) = (0, 0) of the associated pair of equations Xt+ l =

axt + byt

Yt+ ~ = cxt + dyt 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 x* :/: f ( x * ) .

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 all 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 (xt, Yt, zt) 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 f

Plotting Numbers in the Complex Plane

r

a

~.~

eiO

The Unit Circle in the Complex Plane

,~

FIGURE 1.2 [ Complex numbers have the form ~ = a + bi, where a and b are real numbers and i satisfies i 2 = -1 (and is sometimes written i = V~l). One can also write a complex number in the form X r e i ~ and represent it geometrically as in this figure. The real number 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 ~ e i ~ . =

=

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 " i n v a r i a n t loop" b i f u r c a t i o n 7 is n o t chaotic, however. It d o e s n o t p o s s e s s 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

]t+l = bAt exp(-ClAt) At+l

=

(1 -

#j)]t exp(-c2]t) +

(1 -

t-ta)At

(1.6)

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 I INTRODUCTION f

80

b=9 b=7

"~ 60

g

g 40 ~- 20

i

,

,

0

~

|

,

i

i

,

5

i

,

10

,

i

,

i

i

,

15

b

i

i

,

,

i

20

4o r 30

~ 20 10

20

60

40

80

Time

100

120

140

160

5 b=9

4

g3

~2 1

0

10

20

J-stage

30

40

J

FIGURE 1.3 I The top plot shows a bifurcation diagram for the juvenile-adult model (1.6) with parameter values c] = 0.01, c2 = 0.10,/zj = 0, #a = 0.90.As in Fig. 1.1, points from the attractor are plotted above the value of b. An invariant loop bifurcation occurs at approximately b = 7.6. The middle graph shows time series plots of the attractors before and after the bifurcation. In the lower graph these attractors are shown in state space.

cause new bifurcations to occur. Unlike the one-dimensional case where there is a typical bifurcation sequence (period doubling), in higher dimensional systems there can be a variety of different types of bifurcation sequences and cascades that eventually result in chaos. Researchers have classified and studied several types of bifurcation sequences, but a complete catalog has yet to be made. One possibility is for an invariant loop attractor to bifurcate into a double loop, in a kind of period-doubling bifurcation. In other cases the loop can become twisted and convoluted, even break into separate pieces as the parameter changes. A periodlocking window might "open" and the resulting cycles undergo a perioddoubling sequence of bifurcations, or even invariant loop bifurcations, toward chaos. Chaotic attractors frequently occur abruptly upon the closing of a period-locking window, in what are termed "crises."

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 I THE HUNT FOR CHAOS 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 etal. [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 et al. 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 etal. 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 et al., 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 "on 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 "the 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 I MATHEMATICAL MODELS AND DATA 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 Tribolium, 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 unstable 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 "Is this biological system chaotic?" or "Is 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?", "Is it c o m m o n or rare?", and "Do 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 I MODELS 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. G A L I L E 0 G A L I L E I (The Sidereal Messenger)

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 p r e d i c t s 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 Tribolium. 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 [ THE DETERMINISTIC LPA MODEL 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

K A N T (Critiqueof Pure Reason)

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. In our experiments we use strains of the species T. castaneum. This 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 other models that have been applied to Tribolium population dynamics, 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 Tribolium,

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 Tribolium 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 densities (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. castaneum 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 Tribolium 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 of At_l 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 bat 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

Lt+l = bat Pt+x = (1 - lZl)Lt

At+ 1 = (1 - Izp)fit + (1 -/za)At 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 Tribolium becomes o

o

b

Lt

1 -- btl

0

0

/3t

0

1 -/zp

1 -- /.La

At

Zt+l Pt+l

--"

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 net reproductive number [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 -VCea The probability it will survive 2At units of time is approximately the product -

At.

( 1 - --~-A CeaO( 1 - ---~A CeaO(=

Cea02,

1 - --V-A

and so on. The probability it will survive a full unit of time is approximately

Cea t)l/At

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

-

Ceat)1~At= ( 1 - --~-A Ceat)2~At Cea 1~At( 1 - --~-A At

In the presence of adults the probability that an egg survives adult cannibalism is approximately

Cea 1----~-A

t)At//xt

--r ~ Thus, in our which, as At ~ 0, approaches the exponential exp( -cea3 model we take the probability that an egg is not eaten in the presence of At 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

Cea.

castaneum

Cea/V

2.1 [ The Deterministic LPA Model

35

obtain the nonlinear matrix model

I Lt+l Pt+l

At+l 0 --

1--/Zl

0

0

bexp ( - ~ L t

0

o

(1 - ~p) e x p ( - --if-At) ca ^

1

-

-- "-~-1a-t]Cea~ ~ t~a

Lt

At

or, componentwise, the difference equations

( c e I L t c e a- A f) f

Lt+l = bat exp - V

t

Pt+l = (1 - lZl) Lt

(2.2)

At+a--(1-lzp)Ptexp(-~;At~-[-(X-IZa)At.

k

v

7

In these equations Ceaand Cel are, respectively, the coefficients of adult and larval cannibalism on eggs and Cpa 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 Lt - Lt/ v, Pt = fit/v, At = A t / v , in which case the general LPA model equations b e c o m e

Lt+l = bAt exp(-CelLt

-- ceaAt)

Pt+l = (1 - lzl)Lt

(2.3)

At+l = (1 - tzp)Ptexp(-cpaAt) + (1 - tza)At. 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. castaneum 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

Lt+l = bAt exp(-CelLt - ceaAt) Pt+l = (1 - lzl)Lt At+l = Pt exp(-cpaAt) -+- (1 - tza)At

(2.4)

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 deterministic LPA model. 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. castaneum. 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 (Lt, Pt, At) 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 (Lt, Pt, At) 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 [ THE FLOUR BEETLE 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 Tribolium, a m e m b e r of the order Coleoptera, family Tenebrionidae, and subfamily Ulominae, namely, T. castaneum and T. confusum, 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 Tribolium 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 Tribolium 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]. Ten years later, in a paper that appeared in the journal Ecology, Chapman introduced Tribolium (specifically the species T. confusum) as an experimental organism for the study of fluctuations in animal numbers [25]. Tribolium 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 Tribolium ecology" (personal communication, 1986). Interestingly, Alexander Sokoloff points out that a disadvantage in using Tribolium 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 Tribolium 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 Tribolium 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 Tribolium 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 "in the 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 I DYNAMICS OF THE LPA MODEL [Y]ou must not confound statics with dynamics, or you will be exposed to grave errors. m JULES VERNE

(Twenty Thousand Leagues under the Sea)

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__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 I Stochastic Dynamics

Based on our analysis using the deterministic 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

69

2.7 J Predicted Dynamics

f

5OO 4OO 300 _l

200 100

250 200 150

I

I

I

I

I

0

’

4

’

&

!

I

I

l

I

I

I

I

13_ 100 50

150

100

50

0

’ ~;

’ ~6’

~0

’ 24

’ ~’8’

3;

’ 36

Week 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,

70

2 I MODELS

~’~

800

g

600 400200 0

300 -,..-, 200 !

13_ 100

150

o’) 100

&

50

I

i

0

k,...

i

!

4

!

!

8

!

!

12

!

!

I

16

!

20

!

i

24

i

i

28

!

!

32

!

!

36

Week

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 (Lt, Pt, At) 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 [ Predicted Dynamics

71

800 ~600 -J 400

Distance to the Saddle Equilibrium

200

700

800

500

~600 "~ & 400

300

200

1O0

0

c~
(0.28) (0.31) 0.000154]

\(-0.02) (-0.33)

-oo o / 0.0206 ]

In Tables 3.1 and 3.2 appear the results ofusing the Nelder-Mead simplex algorithm [148] to maximize numerically the log-likelihood function for each strain separately. The calculations were numerically well-behaved, and repeated maximizations from many different starting points converged to the same parameter values. Confidence intervals for each unknown parameter also appear in these tables (as calculated from the deterministic skeleton (3.1) using the profile likelihood method as described in Section 2.5).

3.1 [ A Bifurcation Experiment

The estimated parameters appearing in Tables 3.1 and 3.2 for the SS and R R genetic strains differ somewhat from the estimates in Table 2.1 for the cos strain of T. c a s t a n e u m obtained in Chapter 2. These new parameterizations (provided by the "estimation" subset of the data) yield two modified bifurcation diagrams, one for each experiment. As Figs. 3.2 and 3.3 show, however, the new model predicted bifurcations are qualitatively and quantitatively similar to those in Fig. 3.1 on which we based the original experimental design. Both bifurcation diagrams in Figs. 3.2 and 3.3 have, at intermediate values of the adult death rate/Za, the "2-cycle bubble" that results from two period-doubling bifurcations. A notable

FIGURE 3.2 [ The top graph shows the equilibrium stability boundaries in the #a and cpa parameter plane for the LPAmodel (3.1) when the remaining model parameters are set at the estimated values for the SS strain in Table 3.1. The horizontal line is the path followed in the bifurcation diagram and the solid circles indicate the control and adult mortality treatments enforced in the bifurcation experiment (with cpa confidence intervals). The bottom graph shows the resulting bifurcation diagram. Reprinted from [44].

87

3 I BIFURCATIONS

88

F

Cpa

0.010

~. 2-cycles

ilibria

o.oos ;i---~--- ..... ~ ............ ,;_----........ ~

...... .

\ eou,,oo\ 0.000

0’.1 0’.2 0’.3 0’.4 0’.5 0’.6 0’.7 0’.8 0’.9 1’.0

600

< 500 +

~...~

13_ +

,, 4oo

9~

..

c 300

.o

’:

~

,, :-

X

-

lOO

1

! J i

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

F I G U R E 3.3 I T h e s a m e g r a p h s as in Fig. 3.2 are s h o w n u s i n g t h e RR strain p a r a m e t e r estim a t e s in Table 3.2.

difference is that the invariant-loop bifurcations now occur at m u c h higher values of/Za. Before comparing the experimental data to these model predictions, we pause to analyze the statistical connection between the model and the data.

3.1.3 I Model Evaluation

To test model goodness-of-fit, we calculated residuals on the logarithmic scale. This was done for each state variable from each treatment population in the "estimation" data sets by substituting the ML estimates into the conditional expected values (2.15). Differences of the logstate variables from their estimated conditional expected values are the

3.1 [ A Bifurcation Experiment

log-residuals. We subjected these log-residuals to various diagnostic tests for autocorrelation and normality, as described in Section 2.6. The results of the residual analyses for those SS and RR data sets designated for parameter estimation only appear in Appendix B. The residuals are the estimated outcomes of the noise variables Eit in the stochastic LPA model (2.12) and should display approximately the assumed staffstical properties of these noise variables. In constructing these tables of statistics, each individual time series of residuals, representing one state variable (L, P, or A) from one population, was tested separately for departure from normality and autocorrelation (first- and second-order). Significant first-order autocorrelation is present in none of the SS series and in just one (3%) of the 36 RR residual times series. We find significant second-order autocorrelation in just two (6%) of the SS series and two of the RR series. Departure from a normal distribution is seen in six (17%) of the SS series and 12 (33%) of the RR series. These normality departures can be traced to a small n u m b e r of oufliers; most of the observations in each series are well described by a normal distribution. Notice that these results compare favorably with the same analyses performed in Chapter 2 on the cos strain of T. c a s t a n e u m (in which, for the twelve time series in that data, there are no first-order autocorrelated series (0%), two secondorder autocorrelated series (17%), and three nonnormal series (25%)). To test the predictive capability of the parameterized model we also performed a "prediction fit" using the data sets that we set aside and did not use in either the parameter estimation or the goodness-of-fit tests. That is to say, we repeat the diagnostic residual tests for autocorrelation and normality on the validation data sets using the s a m e p a r a m e t e r estim a t e s as a p p e a r in Tables 3.1 a n d 3.2. The results appear in Appendix B. The prediction errors analyzed in Tables A.16 andA. 17 are standardized (centered at their m e a n and divided by their standard deviation) in order to separate autocorrelation and normality evaluation from prediction bias evaluation. As is seen in these tables, no significant first-order or second-order autocorrelation is present in any of the times series in the validation data sets. Departure from a normal distribution is seen in just 8 (22%) of the SS series and 14 (39%) of the 36 RR series. Once again, a small n u m b e r of identifiable outliers cause these departures. These results compare favorably with the same analyses performed in Chapter 2 on the cos strain of T. c a s t a n e u m (in which, for the 27 time series in that data, there is one first-order autocorrelated series (4%), two second-order autocorrelated series (7%), and 10 nonnormal series (37%)). It turns out that prediction bias is negligibly small. A prediction bias is indicated if the prediction errors are centered at a nonzero value. A systematic tendency for the model to underpredict or overpredict would

89

90

3 [ BIFURCATIONS

be the cause. We tested each series of prediction errors for whether or not it arises from a distribution with a m e a n of zero (t-test). The result is that only two of the 36 SS time series show significantly nonzero means (in replicate 17, test statistic T --- -2.83 and probability value P --- 0.02 for adults and in replicate 18, T --- -3.03 and P = 0.01 for adults). For the 36 RR time series only three show significantly nonzero means (in replicate 1, T = -2.46 and P -- 0.03 for pupae; in replicate 8, T = 2.40 and P = 0.04 for larvae; and in replicate 11, T - 3.07 and P -- 0.01 for pupae). On the original scale the prediction biases of the two SS series are only about three adults and seven adults, respectively, while those for the three RR series are about two pupae, 13 larvae, and six pupae, respectively. Our residual analyses suggest that most of the systematic, predictable variability in t h e data is accounted for by the deterministic LPA model (3.1) and that the stochastic LPA model (2.12) adequately describes the remaining unpredictable variability. Plots of the both the estimation and validation data time series together with each one-step model prediction visually support this conclusion. Such plots of all life-cycle stages for all treatments in both SS and RR experiments are given in [44]. A sample of these plots appear in Figs. 3.4 and 3.5.

600

Replicate 4

Replicate 10

Replicate 16

Replicate 22

450

300 150 0 600 450

150 0 600

r

450

"~

300 150

0 1,2,1,6~2,0,2~4,2,8’,3~,3~ 12 16 20 24 28 32 36

....

Week

Week

12 16 20 24 28 32 36

Week

1~’16’2b’2~4’2~’3~’3~

Week

J

FIGURE 3.4 I Census data (open circles connected by lines) and one-step predictions (solid circles) for all four replicates of the/Za = 0.5 treatment for the SS strain in the bifurcation experiment. The control treatments for the adult death rate began at week 12.

91

3.2 [ The Experimental Results

(f

g 45~I .5

3ooi-

Replicate 4 ~

Replicate 10

Replicate16

Replicate22

I

9

150 0

g~

" ~

..

450 t 3001-

v

I

& 150 0 450 ~

300 150 0

~

,

.

.

.

.

.

.

,

,

,

,

,

,

2 16 20 24 28 32 36 12 16 20 24 28 32 36 12 16 20 24 28 32 36 12 16 20 24 28 32 36 Week

Week

Week

Week

J

FIGURE3.5 [ Census data (open circles connected bylines) and one-step predictions (solid circles) for all four replicates of the #a = 0.5 treatment for the RR strain in the bifurcation experiment. The control treatments for the adult death rate began at week 12. The statistical analyses described here confirm the accuracy of the LPA model with regard to its description of the data obtained from the bifurcation experiment. On this basis we conclude that the beetle populations do display the asymptotic dynamics predicted by the deterministic LPA model as the adult death rate/za is changed. In the next section we take a closer look at these dynamics and the predicted sequence of bifurcations.

3.2 [ THE FXPERIMENTAL RESULTS Prediction is difficult, especially of the future. m MARK TWAIN (also attributed to Niels Bohr) In this section we examine the data obtained from the bifurcation exp e r i m e n t in light of the predicted bifurcation sequence of the parameterized and validated LPA model (3.1) appearing in Figs. 3.2 and 3.3. According to these figures we expect the long-term dynamics (attractors) of both the SS and RR strains to undergo a 2-cycle bifurcation, following by a reequilibration due to a reverse period-doubling bifurcation, as the

92

3 [ BIFURCATIONS

TABLE3.3l

Components of the Equilibria and 2-Cycles of the LPA Model (3.1) for the Experimental Values of the Adult Death Rate/Za in the SS and RR Bifurcation Experiments. a

/~a

Equilibrium (L, P, A)

2-cycle

/(LI’ P1, A1)

! (L2, P2, A2)

SS strain

0.04

Stable

(61.44, 45.03, 316.9)

0.27

Unstable

(93.79, 68.75, 141.7)

Stable

0.50

Unstable

(94.89, 69.56, 94.19)

Stable

0.73

Stable

(91.48, 67.05, 69.03)

0.96

Unstable

(86.45, 63.37, 53.01)

0.04

Stable

(47.62, 39.94, 288.6)

0.27

Unstable

(78.92, 66.19, 136.4)

0.50

Unstable

(82.68, 69.34, 92.98)

0.73

Stable

(81.53, 68.38, 69.30)

0.96

Stable

(78.46, 65.81,54.28)

[ (239.7, 9.638, 205.4) (13.15, 175.7, 154.0) (12.40, 186.81 91.28) (254.9, 0.086, 173.7)

RR strain

Stable

{ (191.3, 10.62, 190.9) (12.66, 160.5, 144.0)

Stable / (234.2,6.322,177.7) (7.538,196.5,91.77)

a The remaining model parameters are those in Table 3.1 and 3.2.

adult death rate/Za assumes the values 0.04, 0.27, 0.5, and 0.73 used in the experiment. The numerical values of the predicted equilibria and 2-cycles at these values of/Za appear in Table 3.3. We can graphically view the data obtained from the bifurcation experiment in three-dimensional state-space plots and in time-series plots of the three life-cycle stage components. We can visually compare the experimental data with the model predicted attractors by plotting both in the same graph. Figure 3.6 shows state-space plots of data (with initial transients removed) from the five experimental treatments for both strains and model attractors. Equilibria appear as points and 2-cycles appear as two points in these graphs (see Table 3.3). The invariant loop attractor appears as a close curve. Figure 3.7 shows time-series plots of each lifecycle stage component for all treatments. For the SS strain the clusters of data for the predicted equilibrium c a s e s (JZa 0.04 and JZa -- 0.73, Fig. 3.6) are clearly different from the two separated clusters of data in the predicted 2-cycle cases (/Za = 0.27 and//,a = 0 . 5 0 , Fig. 3.6). In each case the data cluster near the model

3.2 I The Experimental Results

93

f

SS Strain ~t a = 0 . 2 7

ILl,a = 0 . 0 4 _ .

.

.

-..

.

.

_

.

,

..

a~I.., o ~oL

~ o

. . . . . .

,,

0%:

.

~a = 0 . 5 0 .

_

-.

,

g

_

~

.

a~ ".. o :

~.

. . . . . .

’, ,

"" o

.

ILl,a = 0 . 9 6

ILta = 0 . 7 3

RR Strain ~1,a = 0 . 2 7

ILl,a = 0 . 0 4

~"r/

9

’

:

~a = 0 . 7 3 ~

~a = 0 . 5 0

~a = 0 . 9 6 J

FIGURE 3.61 Data from the bifurcation experiment are shown as open circles. Data are from weeks 18 through 24 of the experiment. The first 16 weeks after the A-stage death-rate manipulations commenced have been removed as transients. The solid circles represent the LPA model-predicted attractors.

94

3 I BIFURCATIONS

f

SS Strain P-stage

L-stage "r 0 0

A-stage

400

II :~. 200

I".. 400 C’kl 0 II 200

d

400

~

II t~ 200 :=L

]~ , ~ 1 ~ . . ,

High

- ~

Low

Low

. . . . . .

otD

AI A,

High

High

S L, tw-

Low

High -

Low

co 400 II

200 l

o Y ’~lm’~

fF

9

IV’ v" ~ V ~

=~2oof

0 - , , , , , , ,

__

\

. . . . . . .

,,

. . . . . . .

,

.

.

.

12 16 20 24 28 32 36

12 16 20 24 28 32 36

Week

Week

.

.

.

.

.

,

. . . . . . . . .

12 16 20 24 28 32 36 Week J

FIGURE 3.7 I Time-series data of all four replicates are shown for each life-cycle stage and each A-stage death-rate treatment ofthe bifurcation experiment. The control treatments for the adult death rate/Za began at week 12. The horizontal lines are the LPA model-predicted equilibrium levels or, in the case of a 2-cycle attractor, are the maximum and minimum of the cycle (Table 3.3). (From R. E Costantino, J. M. Cushing, B. Dennis, and R. A. Desharnais, Experimentally induced transitions in the dynamics behavior of insect populations, Nature 375 (1995), 227-230. Reprinted with permission from Nature.)

predicted attractors. In particular, notice how the data for the treatment /Za = 0.96 lie closely around the invariant loop. In these plots the SS strain data appear remarkably consistent with the model predicted attractors and their bifurcations as shown in Fig. 3.2. This conclusion is also supported by the time series plots in Fig. 3.7. The same observations and conclusions hold for the RR strain data shown in Figs. 3.6 and 3 . 7 ~ w i t h one notable exception. The data in the predicted equilibrium case/Za = 0.73 exhibit two distinct clusters,

95

3.2 [ The Experimental Results

f ,,,r 0

RR Strain P-stage

L-stage

400

A-stage

d

II

2o0

0 400 b~

^ AA^,A High

0

II 200

0

.

.

.

.

.

.

.

Low

.

.

.

.

i’22 .

.

.

400

High

o II 200

g

i High

Low 400 o

II

200

0

4OO o

II

High

LOW

.

0

m.

v / =~~N( j ,

,-,v,],,-

"~’~

L

^A’Hioh

Low

AAAAAJ,AAAAA /VVVVV

200

0

,,,,,,,,,,,,,, 12 16 20 24 28 32 36 Week

,,,,,,,,,,,,,, 12 16 20 24 28 32:36 Week

,,,,,,,1,,,,,1 12 16 20 24 28 32 36

Week

Y

FIGURE 3.7 [ (Continued)

rather than a single cluster near the predicted equilibrium point. In this treatment the data are not consistent with an equilibrium, as predicted by the bifurcation diagram in Fig. 3.3, but are instead consistent with a 2-cycle (also see Fig. 3.7). This treatment is not as anomalous as it first appears, however. The experiment was not of long duration and transient dynamics can be significant. Notice in Fig. 3.3 that the manipulated death rate lZa - - 0.73 is very near the 2-cycle bifurcation point (which occurs at approximately/Za = 0.7). In fact, one of the three eigenvalues of the ]acobian evaluated at the stable equilibrium point in Table 3.3 is very n e a r - 1. (These eigenvalues are -0.9662, -0.4672, and 0.3585.) This means the stability of the equilibrium is weak and long 2-cycle-like transient approaches to the equilibrium are to be expected. Moreover, well within the confidence intervals of the estimated parameters used in the

96

3 I BIFURCATIONS

A

B

.

0

"’-.

0

)

"-.

~

j

FIGURE 3.81 Both plots A and B show simulations of treatment/Za----0.73 using the stochastic LPA model (2.12) with (deterministic and stochastic) parameter estimates from Table 3.2. Each plot shows the final four data points taken from four "replicate" simulations for t = 0 to 24. The four simulations shown in A produce a cluster around the deterministically predicted stable equilibrium points. The four simulations shown in B produce two separated (2-cycle-like) clusters. Note the similarity of these clusters to those of the experimental data for this treatment. c o n s t r u c t i o n of the bifurcation d i a g r a m are values for the p a r a m e t e r s t h a t predict a stable 2-cycle for the case/Z 0.73, instead of a stable equilibrium. For example, if the s a m e p a r a m e t e r s in Table 3.1 are u s e d except that b is c h a n g e d from 7.876 to a value satisfying b >_ 8.2 (but within the confidence interval, i.e., 8.2 _< b < 10.3), t h e n the LPA m o d e l predicts a stable 2-cycle. As we p o i n t e d out in Section 2.7.2 (see Figure 2.15), it w o u l d be difficult u n d e r these c i r c u m s t a n c e s to distinguish in data (or stochastic simulations) a 2-cycle-like t r a n s i e n t a p p r o a c h to a stable equilibrium from a sustained 2-cycle oscillation. In fact, simulations of the e x p e r i m e n t a l t r e a t m e n t s using the stochastic LPA m o d e l (2.12) can produce data r e m a r k a b l y similar to the e x p e r i m e n t a l data (Fig. 3.8B). As a final observation it is interesting to notice that even in the relatively short time series of these two e x p e r i m e n t s the stochastic phen o m e n a of phase shifting a n d saddle flybys discussed in Section 2.7.2 occur. For example, in Fig. 3.7 we see in the SS strain time series for the 20.50 that two of the replicates b e c o m e out of p h a s e cycle treatment/Za with the other two replicates. However, in the RR strain 2-cycle treatment/Za 0.50 all replicates r e m a i n in p h a s e during the d u r a t i o n of the experiment. The phase shift can be attributed to a saddle flyby, w h i c h disrupts the oscillation for a period of time after w h i c h it r e s u m e s with the opposite phase. Also in the 2-cycle t r e a t m e n t / Z a = 0.50 a saddle flyby occurs before the data attain the predicted 2-cycle oscillations; see Fig. 3.9. a

- -

-

-

-

- -

3.2 [ The Experimental Results

97

f

(a) Replicate 16 600

~400 ~

500 400

J 200

300

6

Lag metric (saddle equilibrium)

200

0

0

0

~

100

& 400

o

Lag metrics (2-cycle)

200 5OO

0

600[

400 200

200 1

100 ............. 12 16 20 24 28 32 36

0 ! ............. 12 16 20 24 28 32 36 Week

Week

(b) Replicate 10

r & 6~176

Lag metric (saddle equilibrium)

500

~176I *~

400

2

300 200

600[

0

1O0

4oo[

o- 2

Lag metrics (2-cycle)

500 600 [

400

400]-

300

< 200

/~

200

[

1oo

0 ! ............. 12 16 20 24 28 32 36 Week

12 16 20 24 28 32 36 Week

FIGURE 3.9 [ (a) The time series plots of replicate 16 of the treatment/Za 0.50 for the SS strain cultures exhibit a flyby of the model’s (unstable) saddle equilibrium point (L, P, A) = (94.89, 69.56, 94.19) during the early part of the experiment. The replicate begins an initial move toward one phase of the model’s stable 2-cycle oscillation (characterized by L-stage highs occurring at even time steps). This trend is interrupted, however, by the saddle flyby. By the end of the experiment the data have returned to the stable 2-cycle, but with a phase shift (in which L-stage highs occur at odd time steps). The phase shift is indicated by the crossing of the lag metric plots for the two phases of the 2-cycle. Note that this crossing occurs during the saddle flyby. This phase shift places replicate 16 out of phase with the other replicates. (b) Replicate 10 does not experience a saddle flyby. (c) The saddle flyby in replicate 16 is also visible in state space. Three state-space plots show replicate 16 broken into early, middle, and late temporal sequences. The individual stage equilibria are denoted by dashed lines in the time-series plots; in state space the saddle equilibrium is denoted by a solid square. =

98

3 I BIFURCATIONS f

(c) Replicate 16

........... i .............

............

.......................

~_~l~.

~ /

. ............ ~.............

::;

o’... .... - ....

--

......... ~l~

o ~

. .;-~-... . . . .... "

,~ /

"’--~::::

........

9 sA~:

.

’.

_~I 9~ ’ " 1 0 J "’"’"’" ~L

~ 1 2 ~

81

. . . . . . . . . . . . FIGURE 3.9 I (Continued}

3.3 I CONCLUDING REMARKS IT]he advantages of the laboratory can.., be brought to bear. Whole series of experiments can be performed yielding sequences of dynamical states whereby one regime gives way to the next in response to varying a control parameter .... The upshot is that while any one of the individual data sets might fail to stand up to the rigors of statistical analysis, the cumulative effect of all the data is compelling. - - - W I L L I A M S C H A F F E R [144]

In this chapter we considered a bifurcation sequence predicted by the deterministic LPA model as the adult death rate/~a is varied. This sequence included two bifurcations: a period-doubling bifurcation and a reverse period-doubling bifurcation. These transitions in dynamics motivated the implementation of experiments designed to d o c u m e n t their occurrence in laboratory cultures of flour beetles. The statistical analyses ofthe resulting data showed the accuracy ofthe model’s description and prediction ofthe data. Graphs ofthe data, in both time series and in three-dimensional state space, also show quite convincingly that the beetle population numbers did undergo the predicted bifurcations w h e n their adult death rates are manipulated experimentally, with only one exception. In the/~a = 0.73 treatment of the R R strain experiment the model prediction was an equilibrium while data exhibit an oscillation very m u c h like a 2-cycle. We argued, however, that even in this exceptional case the data are well accounted for by the LPA model, in that the

3.3 [ Concluding Remarks

treatment value/Za " - 0.73 lies near a period-doubling bifurcation value o f / Z a and a 2-cycle is predicted over a significant portion of the parameter estimate confidence intervals. The fact that parameter estimates come with confidence intervals is often overlooked. If the same type of attractor (e.g., an equilibrium) is predicted throughout parameter confidence intervals, then one expects that type of attractor to be more easily observed in data. In that case, one can assign that robust attractor type to the biological population with a certain degree of confidence. However, in the case of more complicated dynamics ~ s u c h as those often associated with bifurcations and chaos more than one type of attractor can exist within the confidence intervals. Indeed, it can h a p p e n that m a n y types of attractors are intricately interspersed throughout confidence intervals. In such cases an assignment of a specific attractor type to the dynamics of the population must be appropriately qualified. Although the bifurcation experiment was designed around the attractors predicted by the deterministic LPA model, we also saw the effects of stochasticity in the results. These effects are not just a "fuzzing out" of the attractors; they induce phase shifting of oscillations and quiescent episodes caused by saddle flybys. Stochasticity also means that each replicate of a treatment has its own unique history, which is explained by a mix of deterministic entities (transients as well as stable and unstable invariant sets) and stochasticity. Notice that the c o m m o n practice of averaging over replicates is not necessarily a helpful way to deal with stochasticity. Indeed, averaging over replicates might erase information contained in these unique histories, can obscure the effects of the deterministic model, and can lead to erroneous identification of dynamic patterns. The success of the bifurcation e x p e r i m e n t ~ d o c u m e n t i n g modelpredicted dynamic changes by means of controlled and replicated laboratory experiments ~ gives us confidence to investigate other, more complicated model-predicted bifurcation sequences, including sequences involving chaotic attractors. This we do in the next chapter.

99

This Page Intentionally Left Blank

4 [ CHAOS We are marching straight to chaos.

- - S T EN DA H L (TheRed and the Black) The Greeks said that Alexander went as far as Chaos; Goethe went, only the other day, as far; and one step farther he hazarded, and brought himself safe back. --

RALPH

WALDO

EMERSON

(RepresentatiueMen)

In Chapter 2 we derived and parameterized a model for the dynamics of a population with three life-cycle stages. In Chapter 3 we showed that this low-dimensional, nonlinear model can accurately describe and predict the dynamics of a real biological population. Specifically, we saw that the equations Lt+l = bAt exp

(Cel

Cea)

- - ~ Lt - --~-At

Pt+l = (1 - lzl)Lt

(Cpa)

At+l = Pt exp - - - ~ At

(4.1) + (1 - lZa) At

can predict the dynamics of flour beetle cultures observed in controlled, replicated laboratory experiments. This mathematical model, together with a stochastic version that accounts for deviations from its deterministic predictions, accurately accounted for the demographic triple (Lt, Pt, At) of larval-, pupal-, and adult-stage numbers from one census time to the next. The model predicted characteristics of the long-term dynamics (attractor) of this triple and the changes in these characteristics when adult mortality rates were deliberately changed. In addition, the model accounted for responses (transient dynamics) that result when deviations from the long-term attractors occurred. The success of the LPA model (4.1) in describing and predicting flour beetle dynamics is not restricted to the bifurcation experiment in

101

102

41 CHAOS Chapter 3. Several other studies of an entirely different kind also support the accuracy of the LPA model and its ability to predict and explain dynamic patterns observed in flour beetle populations. For example, in 1980 ]illson [97] reported an unexpected increase in biomass resulting from a periodically fluctuating flour volume. This unusual result is in fact predicted by the LPA model when V is periodically changed [29, 88]. In other studies multiple attractors predicted by the model have explained unexpected differences among replicated cultures in historical data sets. Moreover, the model-predicted multiple attractors have been documented in controlled experiments [90, 91]. Saddle flybys and phaseshifting occurrences in many data sets have been found to correspond very accurately to model predictions [35, 36, 89, 90]. The model has also been successfully used for other species of Tribolium. Using the LPA model, Benoit et al. [18] studied the effects that the different interstage cannibalism interactions have on population stability and cycling in populations of T. confusum. It is noteworthy that in these studies the accurate predictions of the LPA model result not from fitting the model to the data set involved, but byuse ofparameter estimates obtained from other, independent data sets. More accurate model descriptions undoubtedly could have been obtained in these studies by a refitting of the model to each data set under consideration; however, an important point is that the LPA model has the capability of accurate (sometimes astonishingly accurate) a priori predictions in a variety of circumstances and with regard to a considerable number of different nonlinear phenomena. In Chapter 3 we studied a sequence of dynamic bifurcations predicted by the parameterized LPA model to occur in cultures of T castaneum as the A-stage death rate/Za is changed. The bifurcation sequence in that experiment contains no chaotic dynamics. In this chapter we consider a different bifurcation sequence predicted by the LPA m o d e l ~ one that does result in chaotic dynamics ~ and we examine results of experiments based on this prediction. One outcome of this study will be a demonstration, by means of this bifurcation theory approach, that a biological population can indeed exhibit chaotic dynamics, although we will need to clarify what we mean by "exhibit chaotic dynamics." The route-to-chaos involved in the experiment will not be the famous period-doubling cascade (Fig. 1.1, Chapter 1). Instead it will entail invariant loop bifurcations and quasiperiodic oscillations, as described in the following section.

4.1IA ROUTE-TO-CHAOS [T]here is something to be said for giving the reader an inkling of the labor involved in studies such as these. This is advanced, not boastingly, but as a

4.1 [ ARoute-to-Chaos caution that population ecology, apart from its exciting theoretical aspects, contains much that is best described as unadulterated drudgery. m THOMAS PARK [140] We u s e d the bifurcation diagram in Fig. 3.3, o b t a i n e d from the deterministic LPA m o d e l with p a r a m e t e r estimates from Table 3.2 (corresponding to V = 1), to design the bifurcation e x p e r i m e n t studied in Chapter 3. The p a r a m e t e r m a p in Fig. 3.3 shows the p a t h taken in the (lZa, Cpa)p a r a m e t e r plane that p r o d u c e s this bifurcation diagram. Along this horizontal, straight-line p a t h in the (lZa, Cpa)-plane the adult d e a t h rate/Za varies from 0 to 1 a n d the c a n n i b a l i s m coefficient Cpa r e m a i n s fixed at its e s t i m a t e d value in Table 3.2. One t r e a t m e n t in that e x p e r i m e n t was located at the point on the line where/Za = 0.96, i.e., the adult death rate was m a n i p u l a t e d to be 96% (per two week period). Figure 4.1 shows the bifurcation d i a g r a m that results from following a different p a t h in the (lZa, Cp~)-parameter plane, n a m e l y a vertical, straight-line p a t h along which tZa is fixed at 0.96 a n d Cpa varies from 0 to 1. This bifurcation diag r a m shows a c o m p l e x bifurcation s e q u e n c e of attractors (Fig. 4.2). For very small values Of Cpathe attractor is an equilibrium a n d for values Of Cpa larger t h a n a p p r o x i m a t e l y 0.41 the attractor is a distinctive 3-cycle. For Cpa b e t w e e n these two values there is a c o m p l e x collection of attractors. At a quite small value Of Cpa (approximately 0.0059) the stable equilibria in Fig. 4.1 u n d e r g o an invariant-loop bifurcation leading to quasiperiodic oscillations. The s u b s e q u e n t Cpa interval of n o n e q u i l i b r i u m d y n a m i c s consists of aperiodic m o t i o n s interspersed with regions of "period locking" (in w h i c h the attractor is a periodic cycle, usually of high period). The m o s t notable period-locking w i n d o w in this region occurs b e t w e e n (approximately) r - - 0.088 a n d 0.18. At Cpa ~, 0.18 the period-locking w i n d o w a b r u p t l y "closes" in w h a t is t e r m e d a " b o u n d a r y crisis." ~Another b o u n d a r y crisis occurs, in reverse, at a value of Cpa slightly greater t h a n 0.4, after w h i c h stable 3-cycles occur. Between these two crises, i.e., for Cpa b e t w e e n a p p r o x i m a t e l y 0.18 a n d 0.42, there are chaotic orbits (Fig. 4.3). 2 1 As a parameter is changed a boundary crises occurs when the attractor collides with the boundary of its basin of attraction. 2 Mathematical proofs for the existence of chaotic attractors are difficult and (therefore) rare. There are no rigorous proofs of the existence of chaotic attractors for the LPAmodel. However, numerical approximations of Lyapunov exponents indicate that chaotic attractors exist for many values of Cpasatisfying 0.18 < Cpa < 0.42. For example, the attractor at Cpa= 0.35 in these figures has positive dominant Lyapunov exponent ~. ~ 0.0976. Another fact about this chaotic attractor is that its "fractal" box dimension 1.26. Attractors with fractal dimension are called "strange" attractors. (We thank Dr. ]effery Edmunds and Professor Marek Rychlik for the calculation of this box dimension.)

103

104

4 I CHAOS

FIGURE 4.1 ] The upper graph is a parameter plane map showing the boundaries defined by different types of attractors of the LPA model for differing values of #a and Cpa. The remaining parameters are fixed at the estimates in Table 3.2, Chapter 3. As the vertical (dashed line) path on this map is traversed we obtain the bifurcation diagram in the lower graph. The bifurcation diagram shows a plot of the total population size L + P + A for the attractor against the parameter Cpa. In Fig. 4.2 are plots of the attractors located at 4 of the vertical dashed lines in the bifurcation diagram. Reprinted from ]ournal of Theoretical Biology, Vol. 194, No. 1, ]. M. Cushing, R. E Costantino, B. Dennis, R. A. Desharnais, and S. M. Henson, Nonlinear population dynamics; models, experiments and data, pp. 1-9, 1998, by permission of the publisher Academic Press/Elsevier Science.

A notable feature of the bifurcation sequence is the existence of 3-cycles for large values of Cpa.Their occurrence provides an added opportunity for experimental documentation of the bifurcation sequence, since relatively uncomplicated (and therefore, one hopes, easily observable) attractors appear at both ends of the bifurcation sequence.

4.1 I A Route-to-Chaos

105

Cpa = 0 . 0 5

Cpa = 0 . 0 0

~

o

~ ~ , . ~ ,

’

2

i

.-

.-

,~

’,

,

~SL

~

"’’--24

::

Cpa = 0 . 3 5

i

Cpa = 0 . 5 0

= 0.00

Cpa

0

Cpa= 0.05

,~g2~176 I

_5 100 f ......................................... 0 /

,

~

-,

,

,-

~

9

,

, -,

,

200

g _5

i v

0

-r

"r

10

-i-

-i-

20

v,-

r

30

|

l-

40

|-- -|

50

|--r--|---l--r-|-

0

10

20

Cpa = 0.35

-.-

Time

Time Cpa

30

9 -|-

40

-,- I

50

= 0.50

FIGURE 4.21 The three-dimensional, state-space graphs show plots of the attractors located at 4 of the dashed lines in the bifurcation diagram in Fig. 4.1. Parameter values in the LPA model (4.1) are the estimates from Table 3.2, Chapter 3, except that/s = 0.96 in all cases and the value of Cpa is changed to that indicated. The attractors are: an equilibrium (Cpa = 0.00); an invariant loop (Cpa -- 0.05); a chaotic (strange) attractor (Cpa= 0.35); and a 3-cycle (Cpa= 0.50). The time-series plots show the larval component of the corresponding attractor. Figure 4.1 p r e d i c t s a d i s t i n c t i v e s e q u e n c e of d y n a m i c b i f u r c a t i o n s that i n c l u d e s c h a o t i c attractors w i t h i n a large interval of Cpa values. Can this route-to-chaos be documented w i t h c u l t u r e s of flour b e e t l e s ?

by m e a n s of m a n i p u l a t i v e e x p e r i m e n t s

106

4 I CHAOS ..-7 ..............

.-~ .............

~

~"~ ’ ~ ~ ~

I

-~.

’

FIGURE 4.3 [ Blow-ups of the corners of the chaotic attractor at Cpa -- 0.35 in Fig. 4.2 show s o m e of the complicated g e o m e t r y of the chaotic attractor in Fig. 4.2 located at Cpa - 0.35.

In November 1995, we implemented an experiment designed to test the predictions contained in the bifurcation diagram in Fig. 4.1. In this "routeto-chaos experiment" we used a protocol similar to that used in the bifurcation experiment described in Section 3.1. Twenty-four cultures of the RR strain of T. castaneum were initiated with 250 L-stage individuals, five P-stage individuals, and 100 A-stage sexually mature young adults and were maintained in a half-pint (237 ml) milk bottle with 20 g of standard media placed in a dark incubator at 32~ Every 2 weeks the individuals in the L-, P-, and A-stages were counted and returned to fresh medium. Dead adults were also counted and removed. We designated three cultures as unmanipulated controls. From the remaining 21 cultures we randomly assigned three populations to each of seven treatments. In each treatment we manipulated A-stage mortality #a by removing or adding adults at the time of census in such a way as to make the fraction of individuals that died between census counts equal to 0.96 (see Section 3.1). However, in this e x p e r i m e n t - - u n l i k e the bifurcation experiment in Chapter 3 - we also manipulated the cultures so that a desired value of Cpa was obtained. This was done at census time t + 1 by manipulating adult recruitment to equal the n u m b e r Pt exp(-cpaAt), with values of Cpa

4.2 I DemographicVariability set at one of the seven values 0.00, 0.05, 0.10, 0.25, 0.35, 0.50, and 1.00. (The value c m = 1.00 is not shown in Fig. 4.1, but the predicted attractor is a 3-cycle.) We controlled recruitment into the A-stage by removing or adding young adults at the time of census so as to make the number of new adult recruits consistent with the treatment value of Cpa. To counter the possibility of genetic changes in life-history characteristics, at every other census the A-stage individuals returned to the population after the census came from separate stock cultures maintained under standard laboratory conditions. The experiment was carried out for 80 weeks. 3 The resulting census data appear in Appendix C. For plots of the data see Figs. 4.6 and 4.8. The next step is to perform some statistical analyses of this data. However, before we to this we need to discuss demographic stochasticity.

4.2 ! DEMOGRAPHIC VARIABILITY No one supposes that all the individuals of the same species are cast in the same actual mold. CHARLES DARWIN (The Origin of Species) The stochastic LPA model (2.12) we used in Chapters 2 and 3 is a model for environmental noise. Environmental noise involves the chance variation arising from extrinsic sources that affects many individuals in the population. Demographic stochasticity, on the other hand, is the variability in population size caused by independent random contributions of births, deaths, and migrations of individual population members. Generally, demographic stochasticity becomes important when the number (or density) ofindividuals becomes low. While overall total numbers (from all life stages) do not become low in the route-to-chaos experiment, the numbers of individuals in each life-cycle stage often do become small. Therefore, it is natural to consider whether a demographic stochastic version of the LPA model might be more appropriate for this experiment. To model demographic stochasficitywe take each individual’s net contribution to the population in a unit of time to be the outcome of an independent, discrete random variable. A simple example is survival. An individual contributes zero to the population in the next time unit with probability #, where # is the probability of death during the time unit. The individual contributes one to the population in the next time unit with probability 1 - tz. 3 The predicted "chaotic" cultures at C p a " - 0.35, however,were not terminated at 80 weeks. As of this writing, data have been collected for 398 weeks.

107

108

4 I CHAOS If m a n y individuals are m e m b e r s of a h o m o g e n e o u s cohort, t h e n it is often the case that d e m o g r a p h i c events can be aggregated analytically at the cohort level into a known probability distribution for cohort size at the next time unit. For instance, suppose each m e m b e r of a cohort of nt individuals at time t has the same unit-time survival probability. If we denote this probability by 1 - #, a deterministic model for the n u m b e r of survivors at time t + 1 is nt+~ - (1 - lz)nt. On the other hand, if each cohort m e m b e r were instead subjected ind e p e n d e n t l y to the simple r a n d o m survival process described earlier, the cohort size at time t + 1 would be a r a n d o m variable Nt+~ with a binomial distribution

Pr(Nt+l=nlnt)=(nt) (1-#)nlzn’-n’n Note that this binomial distribution for the r a n d o m variable Nt+~ is conditional on the attained value nt of the previous cohort size. We denote this binomial r a n d o m variable by binomial (nt, 1 - tz). The conditional expectation of the binomial r a n d o m variable Nt+~ is E(Nt+llnt) = (1 - #)nt. This expectation gives rise to the deterministic model prediction

nt+l = (1 - #)nt for the (mean) n u m b e r of individuals at the next time unit. The conditional variance of Nt+~ is/~(1 - I~)nt and is proportional to nt. In passing, we also note that the m o d e of the binomial r a n d o m variable N~+a also yields a deterministic prediction, namely,

nt+l

=

floor((1

-

#)nt + 1 - tx).

This deterministic prediction 4 is integer valued, unlike the m e a n expected value. With these ideas in m i n d consider the construction of a stochastic version of the three life-cycle stage LPA model (4.1) based on d e m o g r a p h i c variability. In the stochastic model developed here we will use binomial and Poisson distributions to describe the aggregation of d e m o g r a p h i c events within the life stages. We begin with the simplest stage in the LPA model, the P-stage. Under the assumptions of the LPA model, the n u m b e r of p u p a e at t + 1 is conditional only on the n u m b e r of larvae at time t and their survival probability 1 - # 1 over one unit oftime. Reasoning as before, we d e t e r m i n e 4 The formula floor (pn + p) gives the mode of n binomial trials with success probability p. In the event of a nonunique mode this formula chooses the larger.

4.2 [ DemographicVariability

109

that the n u m b e r of pupae Pt+1 at time t + 1 is a binomial random variable

Pr(Pt+l-pllt)=(lt)(1-txl)plzllt-P P with expectation

E(Pt+lllt) = (1 - #l)It. This provides the deterministic prediction of pupae numbers at time t + 1 given in the LPA model (4.1). Consider now the A-stage. Under the assumptions of the LPA model the number At+l of A-stage individuals at time t + 1 is the sum of two, independent random variables: the number Rt+l of emerging pupae (adult recruits) and the number St+x of surviving adults from time t. The demographic survivorship model we are utilizing assumes that both of these random variables have binomial distributions. In the case of adult survivorship the random variable St+x has a constant, per-unit-time survival probability fla. The expected number of survivors is then

E(St+l[at) = (1 - lza)at.

(4.2)

In the case of emerging pupae (adult recruits), however, survival probability is conditional on the number of adults at at time t. This is because of the cannibalism of pupae by adults. Assuming all noncannibalized pupae survive to emergence, this probability in the LPA model is given by the exponential formula exp(- f-~ at). Thus, the expected number of adult recruits is

E(Rt+I[ pt, at)=pt exp ( - ~-~at).

(4.3)

Finally, the random variable At+l is the sum of the two, independent binomial random variables Rt+l and St+x. The expected n u m b e r of adults at time t + 1 is therefore the sum of the expectations (4.2) and (4.3)

E(At+llPt, at) = E(Rt+llPt, at) + E(St+llat), which leads to the deterministic prediction of adult numbers at+l at time t + i given in the LPA model (4.1), namely,

E(At+llPt,at)=ptexp(Cpa) ----~-at + (1 - lza)at. To complete the stochastic LPA model we must consider the L-stage component. The n u m b e r of L-stage individuals at time t + 1 is a random variable composed from a compound process. A random number of potential recruits is produced (with conditional mean bat) and each

4 [ CHAOS

110

potential recruit subsequently undergoes a survival process. In the LPA model this survival process is given bythe conditional survival probability e x p ( - ~ l t - -VC~at),which depends on the system state variables. Suppose we assume the n u m b e r of potential recruits has a Poisson distribution with m e a n bat. Then the larval recruitment process is a comp o u n d e d Poisson and binomial process. It turns out that such a r a n d o m variable also has a Poisson distribution. Specifically, if a r a n d o m variable Y has a binomial distribution Pr(Y = y) = Y

pY(1 - p)N-y.

where N is a r a n d o m variable with a Poisson distribution with m e a n ~. Pr(N = n) = n!

exp(-,~),

then the resulting distribution of Y has a Poisson distribution with m e a n ,~p. See [201. In our case, we have a s s u m e d that the r a n d o m n u m b e r Lt+x of potential recruits at time t + 1 has a Poisson distribution with m e a n = bat. Their survival is a binomial process with probability p = exp(-~/t Ceafn’l The expectation of the Poisson r a n d o m variable Lt+l -V,~tj. provides the deterministic prediction of L-stage n u m b e r s at time t + 1 given in the LPA model (4.1), namely, -

Cel Cea E(Lt+l[ pt, at) = )~p = bat exp - - ~ l t - -vat).

(4.4)

In summary, a model for demographic stochasticity is the Poissonbinomial LPA model

( ( r

tea))

(a) Lt+l "~ Poisson bat exp ---~lt - -vat (b) Pt+l "~ binomial(/t, 1 - / z l )

(C) At+, ~" binomial(pt, e x p ( - C--~-at))qpa binomial(at, l

(4.5)

/Za)

where "-~" means "is a r a n d o m variable distributed as." The m e a n s of these r a n d o m variables yield the deterministic LPA model (4.1). For m a n y statistical inferences, it is convenient to make a modification of the Poisson-binomial LPA model (4.5). Namely, we transform the observed variables so that the model is approximated by a nonlinear autoregressive (NLAR) model of the form

Xt+l = h(Xt) q- Et

(4.6)

where Xt is a state variable at time t, the function h is the deterministic skeleton (the underlying deterministic equation for Xt [181]), Et is a

4.2 [ Demographic Variability

111

normally distributed random variable with mean zero and variance 0 - 2 , and El, E2,... are uncorrelated. NLAR models were discussed in Chapter 2 (see Section 2.4). These kinds of stochastic models have some advantages and conveniences for our analyses. First, the conditional least-squares parameter estimates we use in the present chapter retain a theoretical robustness to departures of the data from distributional assumptions (see Section 4.3). Second, the numerically intensive calculations and simulations that we need involve straightforward algorithms for least-squares and normal random variable generation. Finally, NLAR model evaluation can be based on familiar diagnostic methods using residuals [43]. Following the procedure outlined in Chapter 2, we use the variance stabilizing transformation given by (2.11). The demographic noise solution there turned out to be the square-root transformation. Indeed, it is a wellknown result from statistics that the square-root transformation stabilizes the variance of a Poisson distribution. Specifically, if a variable Nt+l has a Poisson distribution with mean ~, then Xt+ 1 --" ~ / N t + 1 has approximately a normal distribution with mean Vr~ and a constant variance that does not depend on the value of ~. [149]. Thus, a square-root transformation stabilizes the variance and normalizes the one-step distribution for the L-stage variable (4.5a). Also, the binomial distributions describing the P-stage and A-stage variables in (4.5b) and (4.5c) are well approximated by Poisson distributions [151], and consequently the square-root transformation approximately stabilizes and normalizes those variables as well. Thus, we obtain a "demographic" NLAR approximation (4.6) for a variable Nt by transforming the deterministic equation for Nt to the squareroot scale-- producing the transformed map given by Xt+l - h(Xt) ~ and adding normally distributed noise. The LPA model (4.1) has three state variables. The demographic NLAR approximation applied to these variables produces a multivariate NLAR process. In (4.6), Xt becomes a vector of transformed state variables, h becomes a vector-valued function, and Et becomes a vector of noise variables. This yields the stochastic d e m o g r a p h i c LPA m o d e l 5.

Cel -vAt Cea ) 1+2Elt ( ---~LtLt+l -- [ ~ b Atexp Pt+l = [v/(1’ - . l ) t t "+"E2t]2

At+l =

[~p:

( exp

(4.7)

)

---~ CpaAt

]2

q- (1 -

tza)Atq- Eat

5 If, at any time t, a term within a bracket is negative, then it is set equal to O.

112

4 1CHAOS The Et - (Elt, E2t, E3t) is a r a n d o m noise vector assumed to have a joint normal probability distribution with a m e a n vector ofzeros and a diagonal variance-covariance matrix denoted by E. The demographic nature of the stochasticity would stipulate that the noise variables are uncorrelated with each other within a time unit (off-diagonal elements of the matrix E; are zero) as well as uncorrelated ~ r o u g h time. The deterministic skeleton ofthe m o d e l m that is to say, the LPA model (4.1)mis obtained by setting E - 0, or equivalently, by letting Elt, E2t, and E3t equal zero. As written, the demographic NLAR model with normal noise (4.6) can produce negative values for the (transformed) state variables. This is because the normal noise variables have a nonzero chance of being negative and of large magnitude. Such instances have negligible probabilities under the model parameter values associated with our data. In the rare instances in our simulations when noise produced a negative value of a state variable, the state variable was set to zero. The demographic NLAR model eliminates an awkwardness in data analysis that was present using the logarithmic transform for environmental stochasticity: the problem of zeros in the actual data. With the square root transform, zeros are used in the analysis just like any other observations. 6 We have compared the accuracy of the demographic (square root) and environmental (logarithmic) transformations using the data from the bifurcation experiment. That analysis suggests that the model for demographic stochasticity is statistically superior. This is not surprising. The bifurcation experiment produced time series with occasional low (singledigit) values for larvae and pupae, due to the experimental manipulations of the treatment populations. Demographic variability is known to be more important for low population sizes [164]. Therefore, despite the fact that we used environmental noise to design the bifurcation experiment, the analyses in the following sections are based on the stochastic demographic LPA model (4.7). Nonetheless, we will see that the predictions of the fitted models (cycles, chaos, etc.) for both formulations were very similar.

4.3 I ANALYSIS OF THE EXPERIMENT No human investigation can be called real science if it cannot be demonstrated mathematically. m LEONARDO DA VINCI 6 For the logarithmic transform under environmental variability we set zero values in observations of a state variable equal to a small but positive value (i.e., 1/2).

113

4.3 I Analysis of the E x p e r i m e n t

In any particular theory, there is only as much real science as there is mathematics. -- IMMANUEL

KANT

(Critique of Pure Reason)

In this section we obtain parameter estimates from the 80 weeks of experimental data (given in Appendix C) using the demographic stochastic versions of the LPA model discussed in the previous section. We follow this with an evaluation of the reparameterized model. Since the cultures in this experiment were all placed in 20 g of flour medium, V = 1 throughout this section.

4.3.1 I Parameter Estimation To refit the LPA model using the demographic stochastic version (4.7) we use the m e t h o d of conditional least squares (CLS) for the estimation of the parameters. CLS estimation was discussed in Chapter 2 (Section 2.5); there we noted that CLS methods relax m a n y distributional assumptions about the noise variables in the vector Et [103, 181]. Also, recall that CLS estimates are consistent (converge to the true parameters as sample size increases), even if Et is nonnormal and autocorrelated, provided the stochastic model (4.6) has a stationary distribution. (For comparison purposes we will also calculate m a x i m u m likelihood (ML) estimates for the LPA parameters in the Poisson-binomial model (4.5).) CLS estimates are based on the sum of squared differences between the value of a variable observed at time t and its expected (or one-step forecast) value, given the observed state of the system at time t - 1. For fitting t h e LPA model to a single time series, there are three such conditional sums of squares:

Q1 (b, Cel, Cea) -- ~

q

t=l

(~t-

v/bat-1

exp(-Ce, lt_l - Ceaat-1)) 2

(4.8)

q

Q2(/zt) = ~ ( 4 ~ - ; - v/( 1 - ~,)/t-1) 2 t=l

Qa(cpa)

.__

q

~ (~t t=l

__

v/Pt-1 exp(-cpaat_l) + (1 - lza)at-1

(4.9) )2

9(4.10)

Here lt, Pt, and at, for t = 0, 1 , . . . , q, are the observed census counts for the three life stages. Recall that in Chapter 2 the conditional sums of squares (2.20) were constructed using the logarithmic transform. The conditional sums of squares are here constructed on the square-root scale because that is the scale on which we assume noise is additive in (4.7).

114

4 I CHAOS We estimate the parameter tz~ directly for the control cultures as a binomial probability from the counts of dead adults and the number of adults at risk of mortality at each census interval. To estimate Cpa in the control cultures, we use (4.10) with the value of #a fixed at the binomial estimate. We do not use Eq. (4.10) for the treatment cultures because we experimentally fixed Cpa and # a in those cultures. CLS estimates for one population minimize the conditional sums of squares (4.8), (4.9), and (4.10). The entire data set consists of 24 populations: three populations in each of eight treatments, with each population initiated at time t = 0 and halted at time t = 40. If we denote the conditional sum of squares for state variable i in the jth treatment of the kth population by Qijk, then the total conditional sum of squares for the ith state variable is 8

3

Q, =CZo,, j=l k=]

In this way we obtain one set of CLS parameter estimates for the whole experiment. Three separate numerical minimizations are required, one for each Qi. For these minimizations we use the Nelder-Mead simplex algorithm [148]. Mternatively, one can use a standard nonlinear regression package. The results appear in Table 4.1. (Since/za and Cpawere fixed

Conditional Least Squares (CLS) Estimates of LPA Model Parameters and Their Bootstrapped Confidence Intervals Obtained from the 80-Week, Route-to-Chaos Exp e r i m e n t Data Given in Appendix C a .

T A B L E 4.1 ] 95%

Parameter

b

CLS estimate

10.45

#/

0.2000

/~a

0.007629

95% confidence interval

(10.04, 10.77) (0.1931,0.2068) (0.006769, 0.008489)

Cel

0.01731

Cea

0.01310

(0.01285, 0.01340)

c*pa

0.004619

(0.004446, 0.004792)

(0.01611,0.01759)

a An asterisk indicates an estimate for the control cultures only. The CLS estimates of the entries in

the variance-covariances matrix E = (~ij) are o11 - - 1 . 6 2 1 , O’12-- O’21 " - - 0 . 1 3 3 6 ,

O’13 "" O’31 = - 0 . 0 1 3 3 9 ,

~22 -’- 0.7375, ~23 "-- ~32 "- -0.0009612, o’33 --" 0.01212 for the controls. For the treatments where adult

mortality and recruitment were manipulated the estimates 0.2374, ~13 =o’31 = ~ 2 3 =O’32 = O ’ 3 3 - " 0 .

are o11 ---2.332,

&2 = &21 = 0.007097, o’22 =

(From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle

M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological

Society of America.)

4.3 [ Analysis of the Experiment

115

in the manipulated treatments, we obtain the point estimates and confidence intervals for these two parameters from the data on the controls.) Also appearing in Table 4.1 are estimates of the noise variances (the diagonal elements in the variance-covariance matrix ~ of Eqs. (4.7)) and the covariances. We estimated the covariances in order to check the assumption under demographic variability that the random departures from the model skeleton are uncorrelated (that is, the off-diagonal elements of I3 are zero). We obtained these estimates by the conditional residuals. Let xt denote the vector (vq~t, ~ , 4 ~ ) of observed, transformed stage abundances in a population at time t. Thus, x0, Xl, x2, ... is a time series of vector observations arising from one of the experimental populations. Also, let h(xt) denote the vector of functions in the model skeleton (4.6) evaluated using the CLS parameter estimates. Each population in the experiment yields a series of residual vectors defined as J~

A

et = x t - h (xt-1)

(4.11)

for t = 1, 2 , . . . , 40. We estimated the noise variances and covariances for the three control populations from the sum of squares and crossproducts matrix calculated with the 120 residual vectors. 7 The control residual vectors were 3 x 1; however, the treatment residual vectors were 2 x 1 (L and P stages only) because the experimental manipulations rendered the adult populations deterministic. The estimates in Table 4.1 show little or no correlation among the noise fluctuations of the state variables. The estimated covariances translate into the following correlations: 0.10, 0.096, and 0.010 for the unmanipulated controls, and 0.0030 for the manipulated treatments; despite the large sample sizes, none of these correlations are statistically different from zero. If the fluctuations were environmental in origin, we might expect that the departures of model and data would be related among the life stages (a good time interval for the L stage, say, would be related to whether or not the interval was good for the other stages). The lack of such correlation suggests that the noise is mostly demographic in nature. Table 4.1 also contains 95% "bootstrap" confidence intervals. These were calculated as follows [42]. Suppose we ask the questions: if we were to repeat the experiment, how much variability in the results might be expected? If the fitted LPA model is chaotic for a particular treatment, would it reliably be chaotic again? We deem it crucial to obtain an estimate of 7 A separate variance-covariance matrix was estimated for the 21 treatment populations, to allow for the possibility that the adult manipulations might alter the stochastic variability in the L and P stages, particularly the variability in L-stage recruitment.

116

4 [ CHAOS

how variable the estimated dynamic behaviors might be under stochastic repetitions of the experiment. This variability is sometimes termed "sampling variability," although it must be kept in mind that what is sampled are realizations of the stochastic population process. We use a bootstrapping technique for time series models as follows. 8 Recall that the residual vectors (4.11) from the 24 populations are in two groups, control and treatment. For each group of residual vectors, we calculate the sample m e a n vector and subtract the resulting vector from each residual in the group. The resulting centered residuals are those we use in our bootstrapping procedure. This centering is theoretically important because the collection of centered residual vectors provide a statistically consistent estimate of the (possibly multivariate) noise distribution [109]. 9 To construct a bootstrapped time series for a single population, we sample 40 residual vectors with replacement from the appropriate group (control or treatment) of centered residuals. Then we construct the bootstrapped time series by using the formula x t - h (xt_ 1 ) -’[- e t where x~ = x0, the actual initial value, e~ is the tth sampled residual vector, and h(xt_ ~) is the vector of functions in the model skeleton (4.6) evaluated using the CLS parameter estimates and relevant values of Cpa and/Za. The resulting time series of vectors x~, x~, x~ . . . . . x~0 is a kind of simulated trajectory oflife stages (square-root scale) for a population in a given treatment under the estimated model. In this way, we construct an entire bootstrapped data set of 24 populations in the form of the original data. We refit the model to this bootstrapped data to obtain a set of bootstrapped CLS parameter estimates. We repeat this process of generating a bootstrapped experiment (using the original CLS estimates) and calculating bootstrapped CLS estimates 2000 times. This yields 2000 sets of bootstrapped CLS parameter estimates. This collection forms a statistically consistent estimate of the multivariate sampling distribution of the CLS parameter estimates. The collection thus provides confidence intervals for parameters (or functions of parameters, such as Lyapunov exponents). For each LPA model parameter, we use the 2.5th and 97.5th sample percentiles of the bootstrapped estimates as a 95% confidence interval (which appears in Table 4.1). 8 A version of the technique is used in [65, 66]. We incorporate a centering modification recommended in the statistics literature. 9 In our data, it turns out that the sample means of the residuals are near zero, and the centering is of little consequence.

117

4.3 [ Analysis of the Experiment

f

4

A

o uj F-

2

a

1

uJ 1 0 ,,, 0 n x -1 uJ -2

o LU

0 n x -1 uJ -2

2

-3

-3 -4

B

4 3

;

3

-Z

-3 -2 -1

0

1

2

3

-4

4

-4 -3 -2 -1

OBSERVED

0

1

2

3

4

OBSERVED

4

4

3

3

~2

2

~0

~ 0

-2

-2

-3

9

-3

-4

D 4

-Z

9

-4

I

-4-3

~

I

I

'

I

I

3

4

4

i::., ....... -

-3-2-1

0

1 2 3 4

OBSERVED J

FIGURE 4.41 Normal quantile-quantile plots of the (standardized) bootstrap parameter estimates. Units on both axes are standard deviations. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monograph 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

The confidence intervals in Table 4.1 indicate that the parameters are estimated with good precision; the half-widths of the confidence intervals ranged from around 2% to 11% of the CLS point estimates. The bootstrapped parameter sets also provide additional information on the distributional properties ofthe parameter estimates. The quantilequantile plots appearing in Fig. 4.4 compare the observed quantiles of the standardized bootstrapped parameters to those of the standard normal distributio n. The lack o f significant departure fro m straight lines indicates that the estimates are well described with normal probability distributions. Figure 4.5 shows a scatter-plot matrix of the bootstrapped parameter vectors. The histograms along the diagonal of the matrix also support

118

4 I CHAOS

FIGURE 4.5 [ Scatter-plot matrix of the bootstrap parameter estimates. (From B. Dennis, R.A. Desharnais, J. M. Cushing, ShandeUe M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monograph 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

the hypothesis that the parameter estimates are normally distributed. The lack of any strong correlations among the parameters, as indicated by the scatter plots, suggests that the model is not overparameterized. These results taken together suggest the experimental data provide good information about the model parameters. If the model and data were a poor match, we would expect to see evidence of poorly estimated parameters. We can also calculate maximum likelihood (ML) estimates for the LPA parameters in the Poisson-binomial model (4.5). The purpose for doing this is to compare these additional estimates with the CLS estimates from the NLAR approximation (4.7). This serves as a check on how closely the square-root model (4.7) approximates the Poisson-binomial model (4.5). With the Poisson-binomial model, the likelihood function for each state variable is a product of one-step Poisson or binomial probabilities. For illustration, we focus on the time series emerging from one population.

4.3 I Analysis of the Experiment

119

The likelihood function for the parameters in the L-stage model is

L l(b, Cel, Cea) q

1

H exp[bat_l exp(-Cellt-1 - Ceaat-1)l~tl.[bat-1 exp(-Cellt-1 - Ceaat-1)] It t=l

This is a product of Poisson probabilities for the values It (t = 1, 2 , . . . , q), with means as indicated by (4.4). The likelihood functions for the parameters in the models for the other state variables are products of binomial probabilities:

q(lt-1) Pt

L2(/x/) -- H t=l

(1

q( )

La(cpa) = H t=l

Pt-1

rt

-- ]J~l) p t l t - l - V t

#l

[exp(_cpaat_l)]rt[1 _ e x p ( _ c v a a t _ l ) ] P t _ l - r t

o

L4(ga) -- H t=l

st

(1

- -

’ ’

(4.12)

~St,tat-i--St

Here st is the n u m b e r of adults surviving from time t - 1 (at_l minus the n u m b e r of dead adults counted at time t) and rt - at - st. The ML estimates are the values of the unknown parameters b, Ce/, Cea, lZl (and, in the case of the control cultures, Cpa and #a) that jointly maximize the likelihood functions. As with the CLS estimates, we co mb ine all the time series from the experiment to obtain one set of ML parameter estimates. So, the actual likelihood functions we use are products of functions like equations (4.12) taken over all the experimental populations. The ML estimates appearing in Table 4.2 were calculated by numerical maximization. 1~ Notice how close these estimates are to the CLS point estimates in Table 4.1. The ML estimates of b, IZl, Cel, and Cea are well within the 95% CLS confidence intervals. The least similar ML estimate, that of Cpa in the control cultures, differs from the CLS estimate by only 10.5%. The binomial ML estimate for/Za in the control cultures was taken as fixed in the CLS estimation routines, and so is identical in both tables. Overall, the close similarity of the CLS and ML estimates suggests that the NLAR model (4.7) provides a good approximation to the Poissonbinomial model (4.5). 10 The first and second equations in (4.12) also admit simple closed-form solutions for the ML estimates.

120

4 I CHAOS TABLE 4.2 I Maximum Likelihood (ML) Estimates of the Parameters in the PoissonBinomial Model (4.5) Using the 80-Week, Route-to-Chaos Experimental Data Given in Appendix C. Parameter

b

ML estimate

10.67

#/

0.1955

#a

0.007629

Cea

0.01647

Ce/

0.01313

c*pa

0.004135

(From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. R Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

4.3.2 ] Model Evaluation In the two previous studies described in Chapters 2 and 3 we withheld at least half of the data from the parameter-estimation process and used the withheld data in model prediction/validation analyses. By contrast, in this chapter we used all the data for parameter estimation in the routeto-chaos experiment. We did this in order to concentrate all the available information into confidence statements about dynamics. The analyses described here are therefore to be interpreted as goodness-of-fit analyses. Nonetheless, because the model has so few parameters, the data are so abundant, the experimental manipulations are so distinct, and the model-predicted responses to those manipulations are so sharp, evidence that the LPA model fits this experiment serves as a fairly severe test of the model. We begin by calculating a generalized R 2 statistic to quantify the overall influence of the LPA model skeleton. The generalized R 2 is constructed by analogy to the well-known R 2 measure in ordinary linear regression analysis. In regression, R 2 compares the variability (SSE) remaining after the regression model has been fitted with the variability ( S S T ) remaining when just the sample m e a n is used as a predictive model. The value of R 2 ss~. and represents the proportion of variability in is given by R 2 = 1 - s-~ the response variable that is accounted for by the regression model. First we calculate the conditional sum of squares QCj,for state variable i in the j t h experimental treatment, by using the CLS parameter estimates in (4.8)-(4.10). Then we calculate a second conditional sum of squares Sq using the sample mean of the (square-root transformed) state-variable values as the one-step predicted value.

4.3 [ Analysis of the Experiment

121

For example, for the L stage, using subscripts to denote the kth population in the j t h treatment we have 3

40

Slj -- Z Z ( l~t- mlj) 2 k=l t=l

where

mlj

1 ~4kV~k t

= 120 k=~ t=l

is the sample m e a n of the square root transformed L-stage abundances in the j t h treatment. We define the generalized R 2 value for state variable i in the j t h treatment as

Qij Sij A

R~-- 1 An overall

R2

A

~

for the ith state variable is

E :I S i j

"

Note that each treatment retains its own m e a n in the overall formula (as opposed to a c o m m o n m e a n obtained by pooling observations across treatments). A table of the calculated generalized R 2 appears in Appendix C. From this table we see that the generalized R 2 values for the experimental populations were very high. Twelve of the L-stage R 2 values were above 0.9, 20 of them were above 0.75, and three were between 0.4 and 0.5. For the P-stage, only two R 2 values were below 0.9 (both were for control populations). The lowest R 2 value for the P-stage in all of the treatment populations was 0.96 (Cpa= 0.00, replicate 15). Many treatment R 2 values for the P-stage were above 0.99. The L-stage values were generally lower than those for the P-stage. The greater variability of the L-stage is easy to interpret in that recruits arise from the intrinsically variable processes of egg-laying and egg survival. P-stage recruits, by contrast, arise through a simpler survival process (see the equation for P in (4.1)). We do not report R 2 values for the A-stage in the treatment cultures because the experimental manipulations render the treatment A-stage populations essentially deterministic. Note that in the control populations the L-stages and P-stages are more variable (lower R 2 values, on

122

4 1CHAOS average) than those in the treatments. The additional variability stems from the unmanipulated adults in the controls. In summary, the overall R 2 values indicate that the fitted LPA skeleton (4.1) accounts for 89.9% of the L-stage variability and 98.5% of the P-stage variability in the experimental populations. We further evaluate the model using residual analyses. If the deterministic skeleton (4.1) captures the essential dynamic behavior, the leftover variability on some appropriate scale should be just noise. Additionally, the demographic scale we selected for the noise, as described by Eqs. (4.7), should turn out to be an appropriate one. In order to examine these assumptions, we subject the residuals for each stage in each culture (4.11) to the diagnostic procedures described in Chapters 2 and 3, i.e., we calculate first-order and second-order autocorrelations of the residuals and the Lin-Mudholkar statistic. Autocorrelation of residuals indicates a relationship between successive prediction errors, and thus might suggest a systematic lack of fit between model and data. The Lin-Mudholkar statistic, designed for power against asymmetrical alternatives, tests for alack offit to a normal distribution [117, 181]. 11 The residual test statistics are summarized inAppendix C. They indicate that the stochastic portion of the stochastic LPA model (4.7) is a good representation of the noise in the data. Significant first-order autocorrelation was detected in just five (10%) of the 51 time series for which there were residuals (L-, P-, and A-stages for 3 control populations, L- and P-stages for 21 treatment populations). Significant second-order autocorrelation was detected in just 7 (14%) of the time series. Departure from a normal distribution was detected in 17 (33%) of the time series. These results are similar to results for 36 time series of the R R flour beetle strain in the bifurcation experiment of Chapter 3. Given that the series are long enough (q - 40) for the normality test to have reasonable power [117], the magnitude and frequency of departure from normality is not severe or widespread enough to warrant the a b a n d o n m e n t of this convenient assumption. That the estimated model skeleton (4.1) accounts for a high degree of variability in the populations can be seen in time series plots which display one-step model predictions. Representative samples appear in Fig. 4.6. (See [45] for many more such plots.) 11 We also screened the residuals using standard diagnostic plots, including normal quantile-quantile plots.Asymmetryis a main feature that is visuallydetectable in normal quantile-quantile plots. Wereport onlythe Lin-Mudholkarstatistic in order to condense the information from many figures. Although CLS estimates are robust to autocorrelation and departure fromnormality,the diagnosticscreeninghelps determinewhether the demographic noise model adequatelyportrays the stochastic variabilityin the system.

4.3 I Analysis of the Experiment f

Control

123 Cpa=

0.00

Cpa= 1.00

Cpa = 0 . 3 5

~

300 Q m

200

9

9

.

100

oo

,

,

,

,

,

,

0

20

,

,

40

60

,

,

,

0.

. 20 .

,

,

,

,

80

0

20

,

,

,

40

60

80

300

200

& 100

, 0

, 2O

, 40

Week

, 6O

,

80

Week

80

40 .

Week

. 60

Week

J

FIGURE 4.6 I Time series data (open circles) and one-step predictions from the fitted model

(solid circles) for some representative cultures from the control and from the three experimental treatments, cpa = 0.00, 0.35, and 1.00. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

The inferences we have made about parameters are based on CLS estimation that makes no particular assumptions concerning the distributional form of the noise. The diagnostic tests and plots mainly suggest there are no major systematic or symptomatic patterns in the departures of model and data. This allows us to have a high level of confidence in inferences obtained from the model. On that note, we now turn our attention to the long-term dynamics and the attractors predicted by the model.

4.3.3 I Population Dynamics

We divide our analysis of the population dynamics of the various treatments from the route-to-chaos experiment into two parts. First we compare the data to the deterministic and stochastic model predicted attractors at the point estimates of the parameter values. Then we study the variability in the dynamics due to uncertainty in these point estimates.

124

4 1 CHAOS

The dynamics predicted by the demographic LPA model (4.7) with the CLS point estimates in Table 4.1 are very s i m i l a r ~ but not entirely identical~ to those in the bifurcation diagram of Fig. 4.1 obtained from separate data using the point estimates of the environmental stochastic model in Chapter 3. The new bifurcation diagram appears in Fig. 4.7. The similarity between the bifurcation diagrams is a reflection of the fact that

FIGURE 4.7 I (A) The bifurcation diagram generated by the LPA model (4.1) using the de0.96 is similar to that in Fig. 4.1. mographic CLS point estimates in Table 4.1 with/Za The total population size L + P + A of the attractor is plotted against the parameter Cpa. (B) and (C) show the deterministic Lyapunov exponents and stochastic Lyapunov exponents as functions of Cpa. Arrows indicate experimentally fixed values of c pa. (From B. Dennis, R. A. Desharnais, J. M. Cushing, ShandeUe M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.) =

125

4.3 I Analysis of the Experiment TABLE 4.3 I Deterministic Attractors Predicted by the LPA Model (4.1) a. Treatment

Attractor

Control

Equilibrium

Deterministic LE

-0.046 (-0.047, -0.045)

Stochastic LE

-0.044 (-0.045, -0.044)

Cpa -- 0.00

Invariant loop

0.000 (-0.036, 0.000)

0.009 (-0.007, 0.014)

Cpa = 0.05

Chaotic

0.029 (-0.227, 0.029)

0.045 (0.042, 0.046)

Cpa = 0.10

26-cycle

-0.109 (-0.115, -0.0004)

0.060 (0.056, 0.062) 0.068 (0.066, 0.071)

Cpa = 0.25

8-cycle

-0.076 (-0.090, -0.071)

Cpa = 0.35

Chaotic

0.096 (-0.066, 0.100)

0.053 (0.049, 0.055)

Cpa = 0.50

3-cycle

-0.089 (-0.101, -0.080)

0.019 (0.013, 0.025)

Cpa = 1.00

6-cycle

-0.003 (-0.025, -0.0003)

a

-0.075 (-0.087, -0.063)

For the control treatment the parameters are the point estimates in Table 4.1. For the other treatments the point estimates in Table 4.1 are also used except for #a = 0.96 and the value of Cpa displayed. Also shown are the deterministic and stochastic Lyapunov exponents (LE) and their 95% bootstrapped confidence intervals (in parentheses). (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

the parameter estimates in Tables 4.1 and 3.2 are similar. In fact, all the parameter estimates from the environmental noise analysis in Table 3.2 (except that of b) lie within the confidence intervals from the demographic noise analysis appearing in Table 4.1. Table 4.3 summarizes the attractor types predicted by the LPA model skeleton (4.1) for each experimental treatment and the demographic point estimates in from Table 4.1. The prediction for the control treatment is still a stable equilibrium. That for the treatment Cpa = 0.0 is an invariant loop, although upon close examination it turns out that the attractor consists of two small, separated invariant loops (arising from a bifurcation caused by the destabilization of a 2-cycle). As a result, the dynamics are quite similar to those of a 2-cycle. In addition to the treatment Cpa = 0.35, the treatment Cpa = 0.05 also has a chaotic attractor. Treatments Cpa = 0.10 and 0.25 are stable, long-period cycles that occur in period-locked windows nestled among more complex regions of the bifurcation diagram. The treatment Cpa -- 0.50 is still a stable 3-cycle. The treatment Cpa = 1.00 is now a 6-cycle, although it is very close to a period’doubling bifurcation of the 3-cycle. The (few) differences among the attractors from the preliminary bifurcation diagram in Fig. 4.1 used to design the experiment and those in the refitted bifurcation diagram in Fig. 4.7 are not surprising. The attractors for Cpa less than 0.50 are not robust on large parameter intervals in these bifurcation diagrams (as they were in the bifurcation experiment in Chapter 3). Instead they can come from parameter intervals containing

126

4 [ CHAOS

complicated mixtures of periodicity, aperiodicity, and chaos (Figs. 4.1 and 4.7). Therefore, changes in attractor type with small changes in the point estimates of the parameters are inevitable. This raises an important question about attractor identification, in circumstances such as these, given that parameter estimates necessarily come with confidence intervals. More will be said about this later. Rigorous mathematical proofs that orbits of a dynamical system are chaotic are notoriously difficult to obtain. This is the case for the LPA model (4.1), for which no proof of chaotic dynamics is available. To study chaos one has little recourse but to rely on evidence gathered from computer studies. One approach is to estimate Lyapunov exponents. Recall from Chapter 1 (Section 1.1) that the dominant Lyapunov exponent measures sensitivity to initial conditions and that a positive value of this number is evidence for chaos. The dominant Lyapunov exponent is defined as follows. In threedimensional state space, consider the sphere of unit radius centered at the initial point (L0, P0, A0) ofan orbit ofthe LPA model. When each point in this sphere is mapped to an image point by the formulas (4.1), the entire sphere is mapped, approximately, to an ellipsoid. This is because near the point (L0, P0, A0) the action of the map is approximated by a linear map defined by the Jacobian matrix (2.6) evaluated at (L0, P0, A0) and because a linear map takes spheres into ellipsoids [7]. Consequently, this mapping contracts the unit sphere in certain directions and expands it in others. Specifically, the sphere is contracted in the directions of those axes of the image ellipse whose lengths are less than 1. In these regions nearby orbits of the LPA model converge. Conversely, the mapping expands the unit sphere in those directions determined by the axes with lengths greater than 1 and in these regions nearby orbits diverge. We can repeat this procedure along the orbit by applying the LPA map repeatedly. For example, by applying the LPA map twice we obtain a mapping that takes the initial point 2 time units into the future. Under the application of this iteration the sphere will again evolve into an ellipsoid that can be approximated by action of a Jacobian matrix, but this time the Jacobian involved is that of the iterated LPA model (mathematically speaking, the "first composite" map). Similarly, we can approximate the ellipsoid obtained after t iterations of the LPA map by calculating the Jacobian matrix of the tth composite of the LPA map, a mapping that takes the initial point t time steps into the future. The length rt of the longest axis of this ellipse is the maximum factor of growth and In rt is its exponential growth rate over the time period t. The per-unit-time average of this exponential growth rate (namely, t -1 In rt), in the limit as time increases, is the

4.3 [ Analysis of the Experiment

127

dominant Lyapunov exponent LE, i.e., LE = lim -1 ln rt. t~c~ t

(4.13)

If the dominant Lyapunov exponent of an orbit is positive, then on average along the orbit there is divergence of (at least some) nearby orbits and hence sensitivity to initial conditions. This is the case if the orbit is chaotic. ~2 One way to calculate the Lyapunov exponent is to compute rt for large values of t. In principle this is straightforward since the lengths of the axes of the image ellipse at step t are the square roots of the eigenvalues of the matrix I t ] I , where It is the Jacobian matrix (2.6) evaluated at the tth point of the orbit. However, many other methods of calculation, designed for more efficient numerical computation, appear in the literature. ~3 In the formula (4.13), we use points from an orbit of the deterministic LPA model (4.1). However, one can also use this formula with other sequences of points (Lt, Pt, At) since all that is required for the calculation is an evaluation of the Jacobian matrix (2.6) at each point. One could, for example, use a sequence of real population data, or a sequence obtained from a stochastic version of the LPA model (e.g., (4.7)). In the latter case, we obtain a "stochastic Lyapunov exponent" (SLE). The resulting number is a measure of the tendency of nearby stochastic orbits to diverge or converge [31, 57, 60]. A positive value of SLE indicates a kind of sensitivity to initial conditions associated with the stochastic model, although this sensitivity might arise from causes totally unrelated to deterministic chaos. Indeed, a positive SLE does not imply that the deterministic skeleton has a chaotic attractor, as we will see in later examples (also see [51]). 12 Chaotic attractors typically have in them a chaotic orbit that recurrently comes arbitrarily close to every point of the attractor. Because of this, we can m e a n i n g h ~ y associate the Lyapunov exponent with the attractor itself (although not every orbit on a chaotic attractor is necessarily chaotic!). 13 We used a method based on the formula l i m t _ ~ 71 In IlJtJt-1 "" ]2]111for the dominant Lyapunov exponent. Here II"IIdenotes a matrix norm, for example, IIMII = (~-_,i~., j m2j) 1/2 for matrix M = (mq). This formula arises from the "chain rule" of differentiation applied to the Jacobian of the tth composite map, which results in the product of the Jr. Using an orbit the LPA model (4.1), one can, in principle, calculate the t -1 In [[]t]t-1... ]2]11[ for large values of t until the results "converge" (i.e., stabilize to a specified number of digits). There are, however, several practical problems associated with this method. For example, the matrix product is numerically unstable for large t and, unfortunately, extremely large values of t are usually needed for numerical convergence. See [45] for more details on the method used to obtain the estimates in Table 4.3.

128

4 I CHAOS Nonetheless, stochastic Lyapunov exponents are often estimated in studies of chaos, and we include them here for completeness and comparison. Both the deterministic Lyapunov exponent (LE) and the stochastic Lyapunov exponent (SLE) for each experimental treatment are functions of LPA model parameters. Using the CLS parameter estimates of b, Cel, Cea, and #l, and the fixed values of # a and Cpa for each manipulated treatment (or the relevant CLS estimated values for the controls), we can calculate point estimates of the LE for each treatment. In addition we can calculate point estimates of the SLE for each treatment by using instead the stochastic model (4.7) and the estimates of the noise variances cr~ obtained from the residuals of the fitted model. The results appear in Table 4.3. The point estimates of the deterministic Lyapunov exponents are all negative (or zero) except for the "chaotic treatments" Cpa - 0.05 and 0.35 whose exponents are positive. The point estimates in Table 4.3 come with confidence intervals calculated from the 2000 bootstrapped parameter vectors generated in the parameter estimation process described in Section 4.3.1. The 2.5 and 97.5 percentiles of the 2000 bootstrapped LE (and SLE) values at each treatment provide 95% confidence intervals for the LE (and the SLE) of that treatment. These intervals appear in Table 4.3. The LE confidence intervals for the control and the Cpa - 0.10, 0.25, 0.50, and 1.00 treatments are all strictly negative. However, for Cpa = 0.10 most of the bootstrapped parameter vectors predicted high-period cycles with weak stability and LEs that are negative but near zero; the 97.5th percentile was just below zero. Similarly, for Cpa - 1.00 the 97.5th percentile is negative but close to zero. The Cpa = 0.00 treatment has an LE confidence interval with zero included as the upper bound. This reflects an area of parameter space in which most parameter vectors yield stable cycles (LE < 0), but a measurable proportion ofvectors result in invariant loops (LE = 0). The LE confidence intervals for the Cpa = 0.05 and 0.35 treatments include positive and negative values. The LE confidence interval for the Cpa - 0.35 treatment, however, lies mostlyin the positive range, withjust a small portion overlapping into negative values. We will see hereafter that these negative values correspond mostly to high-period cycles of unusual periods (e.g., 19-cycles). Thus, we confidently claim that the attractor for the Cpa = 0.35 treatment is either chaotic or "aperiodic." The Cpa = 0.05 treatment is unusual in that the LE point estimate is on the upper edge (to two decimal places) of the confidence interval. We show later how this stems from extreme skewness of the sampling distribution of LE estimates for the Cpa = 0.05 treatment.

4.3 [ Analysis of the Experiment

The confidence intervals for the stochastic Lyapunov exponents are more straightforward. Two of the treatments, the control and the Cpa -1.00 treatments, have SLE confidence intervals that are strictly negative. Five of the treatments, Cpa = 0.05, 0.10, 0.25, 0.35, 0.50, have SLE confidence intervals that are strictly positive. The Cpa = 0.00 treatment has an SLE confidence interval that includes both positive and negative values. Although one can debate about exactly what it is that the SLE measures, estimating the SLE appears more stable than estimating the LE. We will also see later that the sampling distributions of the SLE estimates for all the treatments are fairly well behaved. However, the estimated stochastic Lyapunov exponents and their confidence intervals suggest the influence of chaos in all treatments of the experiment except Cpa - 1.00 and are inadequate for diagnosing the underlying deterministic dynamics arrayed across the bifurcation diagram. The agreement of the model and the data is well visualized in state space. Figure 4.8 shows the estimated attractors of the deterministic LPA model (listed in Table 4.3) plotted in three-dimensional state space along with the data triples (after initial transient observations are removed). The model attractors capture the transitions in dynamic behavior of the data from treatment to treatment extremely well. Note the tight cloud ofpoints around the stable point equilibrium for the control treatment. Note the two clouds of points around the small invariant loops for the Cpa = 0.00 treatment (a fine distinction not captured by the predicted equilibrium attractor in the original bifurcation diagram of Fig. 4.1). Particularly striking are the attractor and data matches for the treatments Cpa = 0.50 and C p a - - 1.00.14 O f particular interest, of course, are those treatments in the region of complex dynamics. Here, too, we see the irregular, triangular clouds of data points that accompany the three-pointed cyclic or chaotic attractor of the 0.05, 0.10, 0.25, and 0.35 treatments. Clearly the data in Fig. 4.8 are strongly influenced by the deterministic forces present in the model. There are, of course, clear departures of the data from the deterministic model predictions. Such departures should be accounted for, to a large extent, by the stochastic version of the model. So we ask: does the variability predicted by the s t o c h a s t i c model resemble the variability in the data? Also appearing in Fig. 4.8 are plots of stochastic orbits obtained from simulations using the estimated stochastic model (4.7) with parameter values given by Table 4.1 (also with initial transients removed). For the 14 The 6-cycle attractor consists of three groupings of nearby clusters of two points. This is because the 6-cycle is very near a period-doubling bifurcation of a 3-cycle. Thus, we cannot expect to distinguish the 6-cycle from a 3-cycle in data.

129

130

4 1CHAOS purpose of these simulations, we assumed the noise vector Et has a multivariate normal distribution (recall that multivariate normality was not assumed for the estimation methods). In the figure, each plot is matched with that of the data from the corresponding treatment. The resemblance of these pairs of plots, for all treatments, is striking. The stochastic model produced cloud patterns of observations around the estimated attractors that can hardly be distinguished visually from the experimental data. In ecology, such concordance of model and data across experimental manipulations is rare. So far, we have compared the data to model predictions at point estimates for the model parameters. Estimated parameters come, however, with confidence intervals. We now consider the variability in the

Experimental control

Treatments

Cpa= 0.05

Cpa = 0.00

::.7

..--"

|

..

Cpa= 0.10 ...---’"

’: "-... ~ :.7

:: ’",,, ~ ,,

o_

o""

o~

i

""" :

Stochastic

I

,,

,,

~’~) !

Simulations

"’"~ :

"-.

i

.--’’’’’’’"’i"". ";- - :

-.

~

"~

i ..........

FIGURE 4.8 [ Data from the control and seven treatments of the route-to-chaos experiment are plotted in three-dimensional state space (open circles). In each case all three replicates are plotted, with some initial transients removed. The predicted deterministic attractors (solid circles or lines) are included in the plots. Below each data plot appears a simulation of that experimental treatment using the demographic stochastic LPA model (4.7) with parameters from Table 4.1. The simulations consist of three "replicates" starting from the experimental initial conditions and lasting for 80 weeks. (From B. Dennis, R. A. Desharnais, ]. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

4.3 I Analysis of the Experiment

131

r

Experimental Treatments Cpa 0.25

Cpa 0.35

-

Cpa 0.50

=

:2.

’

Cpa = 1.00

=

"’--.

,,

-.

_,

Stochastic Simulations

:,[

~ "-..

FIGURE 4.8 I (Continued)

estimated dynamics within the estimated confidence intervals. Bootstrap methods (such as those described in Section 4.3.1) provide a means for doing this. The confidence intervals for the parameters (Table 4.1) can be regarded as a (hyper-) rectangular region of parameter space, within which any parameter "vector" is plausible under the experimental evidence. ~5The region generously indicates how much variability might be expected under repeated sampling. A fascinating picture of the variability in the estimates of the skeleton attractor results from plotting the Cpa bifurcation diagram for different parameter vectors in the region. The bifurcation diagrams in Fig. 4.9a are for nine combinations of b and C~l values, with the other parameters held fixed at the CLS estimates. The three rows represent, respectively, the upper 95% confidence limit, the CLS point estimate, and the lower 95% ]5 The region defined by the individual 95% confidence intervals in Table 4.1 is not a joint 95% confidence region. Under Bonferroni’s inequality [151], the six 95% confidence intervals together form a conservative 70% joint confidence region. The confidence intervals for the four estimated parameters b, #l, Cel, and Cea in the treatment populations form a conservative 80% joint confidence region.

FIGURE 4.9 I (a) Bifurcation diagrams plotted for all combinations of the upper and lower 95% confidence limits of parameters b and Cel for Cpa values ranging from 0 to 1. (b) Bifurcation diagrams plotted for all combinations of the upper and lower 95% confidence limits of parameters b and Celfor Cpavalues ranging from 0.34 to 0.36. (From B. Dennis, R. A. Desharnais, I. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

133

4.3 I Analysis ofthe Experiment

confidence limit of the parameter b. The columns represent the same limits for the parameter Cel.The center diagram in Fig. 4.9 is, for reference, the CLS point estimate of the bifurcation diagram as in Fig. 4.7. There is a strong qualitative similarity in the nine diagrams. For values of Cpaless than about 0.5 the diagrams show a region of complex behavior, with a large window of periodic behavior centered at approximately Cpa 0.2. At a value ofjust above 0.5, the complex behavior gives way abruptly to an extended region of stable 3-cycles. At particular values of Cpa, however, there is considerable local variation in the diagrams. For instance, at Cpa -- 1.00 (which is one of our experimental treatment values) the lower row of diagrams, corresponding to the smallest value of b, indicates that the attractor is estimated to be a 3-cycle, whereas in the upper two rows, corresponding to the largest value of b, the estimated attractor has locally bifurcated into a 6-cycle. Similarly, around the treatment values of Cpa = 0.05 and Cpa = 0.10, the bifurcation diagram is locally "wispy"; the dynamics of the attractors vary between cycles and chaos in intricate lacelike patterns. The Cpa --- 0.35 treatment lies within the dark, complex behavior region in all nine bifurcation diagrams of Fig. 4.9. The strange complexities of the attractors are revealed by zooming in for a close-up view. In Fig. 4.9b, the interval from Cpa = 0.34 to 0.36 in the nine bifurcation diagrams is magnified. Between dark areas of aperiodic chaotic behavior, thin windows appear in which stable, high-period cycles undergo perioddoubling bifurcations. Note the considerable variation in the locations of the windows. Similar features would be revealed, fractal-like, upon further magnification. Changing a parameter value a small a m o u n t in such regions can change the behavior of the attractor considerably, from stable cycles to chaos, or the reverse. The parameter values giving rise to periodic behavior (LE < 0) in such regions are sets of positive measure (they have positive volume in parameter space). It is not clear that this is the case for chaotic attractors, but it appears that it may be so. Histograms of the bootstrapped LE values show how variability in parameter estimates becomes propagated into t h e LE estimates (Fig. 4.10). The bootstrapped LE values for the control treatment form a conventional bell-shaped histogram. The LE values represent estimated strengths of local stability of the stable point equilibrium of the skeleton. The LE histograms for the Cpa --- 0.25 and Cpa = 0.50 treatments are also reasonably bell-shaped. The LE values for these treatments represent the estimated strengths of local stability of the cyclic attractors. The Cpa - 0.25 histogram actually extends into the region of positive LE values; 6 out of 2000 bootstrapped LE’s were positive (not visible in the resolution of Fig. 4.10). =

134

4 I CHAOS

FIGURE 4.10 I Histograms of the 2000 bootstrap estimates of the deterministic Lyapunov exponents for each experimental treatment. Not visible on the Cpa 0.25 histogram are six positive Lyapunov exponents. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.) -

-

The LE histograms for the other treatments are highly skewed and peculiar (Fig. 4.10). For the Cpa --- 0.00 treatment, most of the bootstrapped LE values are negative, but there is a spike of frequency at 0. The sampling distribution of the LE estimates here is apparently a mixed discrete and continuous distribution, with the positive probability at 0 giving the proportion of times the dynamic behavior is estimated to be aperiodic cycling on an invariant loop. The Cpa = 0.10 treatment has a left-skewed, rampshaped histogram of LE values in the negative region, with a small but visible tail extending into the positive region (38 out of 2000 bootstrapped

4.3 I Analysis of the Experiment

LEs were positive). The r = 1.00 treatment has a left-skewed, J-shaped histogram of LE values. The LE values for r -~- 1.00 were all strictly negative, although the histogram peak is near 0; the estimated cycles are weakly stable. Though the LE histograms for the Cpa ~-- 0.05 and r --" 0.35 treatments have negative portions, they extend considerably into the positive region (Fig. 4.10). For the r = 0.05 treatment, the skeleton attractor is highly variable in this region of parameter space (recall Fig. 4.9). A variety of stable cycles (1218 negative estimated LE values out of the 2000), as well as chaos (782 positive estimated LE values out of the 2000), are plausible behaviors estimated for the attractor, according to the bootstrapped LE values for Cpa "-" 0.05. The bulk (1670 out of the 2000) of the bootstrapped LE values for the Cpa = 0.35 treatment are positive. The remaining negative portion of the LE values for that treatment correspond to small windows of high-period stable cycles in an otherwise complex bifurcation diagram (recall Fig. 4.9). The histograms for the bootstrapped SLE values are bell-shaped for all the treatments (Fig. 4.11). 16 The control and r = 1.00 treatments have SLE histograms that are unambiguously within the negative region. The r = 0 . 0 5 , 0 . 1 0 , 0 . 2 5 , 0 . 3 5 , and 0.5 treatments have SLE histograms that are unambiguously positive. The r -- 0.00 treatment has an SLE histogram that straddled 0. The normal-like shape of the histograms suggests that the SLE is a stable function of the model parameters for which the conventional asymptotic normality theory of maximum likelihood estimates might apply [110]. For each treatment the 2000 bootstrapped parameter set provides 2000 estimated attractors. We summarize the frequencies of different dynamic behaviors of those attractors in a series of pie diagrams in Fig. 4.12. Recall that we have no formal, rigorous proof for the existence of chaotic attractors for any parameter values. An attractor is termed "chaotic" for the construction of these pie diagrams if its estimated LE is positive. The control and the Cpa -’- 0.50 treatment are both robust in the sense that each shows entirely one type of attractor in Fig. 4.12, namely, a stable point equilibrium for the control and stable 3-cycles for the Cpa = 0.50 treatment. The remaining treatments display more than one attractor type, although the r --" 0.25 treatment is also reasonably robust in that it displays stable 8-cycles almost entirely, with only a tiny portion (0.3%) of the attractors being chaotic. These three treatments have ordinary bellshaped histograms of bootstrapped LE values (Fig. 4.10). 16 The calculation of the bootstrapped SLEs for these histograms with appropriate numerical precision and safeguards required weeks of running time on a contemporary Pentium computer.

135

136

4 I CHAOS

FIGURE 4.11 I Histograms of the 2000 bootstrap estimates of the stochastic Lyapunov exponents for each experimental treatment. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

4.3 I Analysis of the Experiment

FIGURE 4.12 I Pie charts showing the frequency of predicted deterministic attractors at each treatment for the 2000 bootstrap parameter estimates. (From B. Dennis, R. A. Desharnais, I. M. Cushing, Shandelle M. Henson, and R. E Costantino, Estimating chaos and complex dynamics in an insect population, EcologicalMonographs71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

137

138

4 I CHAOS 1.00 treatment is split roughly evenly between estimates The Cpa of 3-cycles and 6-cycles (Fig. 4.12). The basic uncertainty arises in this treatment because the value Cpa = 1.00 lies near a bifurcation point where a 3-cycle period doubles to a 6-cycle (Fig. 4.7). For the Cpa = 0.00, 0.05, 0.10, and 0.35 treatments we see combinations of more complex behaviors in Fig. 4.12. At Cpa = 0.00, the spike of bootstrapped LE values at 0 in the histogram (Fig. 4.10) corresponds to the invariant loops (27%) indicated in the pie diagram (Fig. 4.12), while the negative LE values for that treatment (Fig. 4.10) are all stable 2-cycles (Fig. 4.12). Recall that the invariant loops in the point estimate of the attractor consisted of two tiny separate loops, producing orbits resembling 2-cycles. For the Cpa = 0.05 treatment, 39.1% of the bootstrapped parameter sets give chaotic attractors, and the remaining sets produce unusual stable cycles (Fig. 4.12). The Cpa = 0.10 treatment shows mostly stable cycles of unusual periods (e.g., 13-cycles). The Cpa = 0.35 treatment is our showcase for chaos. For that treatment, 83.5% of the bootstrapped parameter sets produce chaotic attractors, 7.1% produced stable 19-cycles, a n d t h e rest produced other stable cycles of higher periods (Fig. 4.12). Thus, in the event that the behavior of the Cpa = 0.35 treatment is a stable cycle, it would most likely have a high period relative to the length of the time series and occur in a small periodic window of parameter space (Fig. 4.9). -

-

4.3.4 I Transients and Effects of Stochasticity

In the design and analysis of the route-to-chaos experiment we have focused attention on the attractors predicted by the LPA model and on the long-term behavior of data in relation to those attractors. However, as we have seen from previous studies in Chapters 2 and 3, it is also important to consider transient behavior. The behavior of orbits when they are not on or near an attractor can play a role in a c c o u n t i n g - - b y means of a dynamic m o d e l - - for patterns seen in data. The underlying cause of such transient p h e n o m e n a might be due to a population being initially far from the attractor, or it might be due to a stochastic perturbation that displaces the population away from the attractor at any point in time during the course of the experiment. We find occurrences of both types of transients in the route-to-chaos experiment. In Fig. 4.8 data from the early part of the experiment are excluded from the plots. The rationale for doing this is that the experimental cultures were not started on the predicted attractors and that some transient time needs to elapse before the data are compared to the predicted attractors. However, the deterministic LPA model predicts not only the attractor, but

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also the particular transient path taken to the attractor. In several of the experimental treatments it turns out that the LPA model predicts initial transient dynamics that involve a saddle equilibrium flyby. In each of these cases, the observed population data closely resemble the predicted saddle flyby early in the experiment. An example taken from one of the experimental treatments appears in Fig. 4.13. Because of stochastic perturbations, transient b e h a v i o r ~ and in particular transient behavior influenced by an unstable e q u i l i b r i u m ~ can also occur at later times during the experiment. At any point in time a random event can place a data point near the equilibrium (or, if the equilibrium is a saddle, near its stable manifold). This is possible even after the transients due to the initial placement of the population have subsided and the data have reached the vicinity of the attractor. After such a perturbation, the data will likely linger in the vicinity of the unstable equilibrium for an observable length of time and exhibit geometric characteristics of a saddle flyby in state space. An example from the treatment Cpa = 0.05 appears in Fig. 4.14. While the LPA model predicts a very moderate initial transient movement

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FIGURE 4.141 (a) For the treatment Cpa ~- 0.05, and the experimental initial conditions (L0, P0, A0) = (250, 5, 100), the LPA model does not predict a flyby of the saddle equilibrium. (b) However, data from one from replicate (7 in Appendix C) for this treatment shows a significant saddle flyby. A random perturbation occurs at week 16 that places the population near the saddle equilibrium. (c) The saddle flyby in the data is strikingly apparent in state space. It is also interesting to observe the "starlike" rotational motion in state space that occurs during weeks 24 through 34 as the data orbit leaves the vicinity of the saddle equilibrium. This geometrically distinctive path is in fact predicted by the deterministic LPA model. The linearization at the equilibrium has a conjugate pair of complex eigenvalues re ii~ of magnitude r ,~ 1.265 > 1 and polar angle 0 ~, 2.576 (and a third real positive eigenvalue ~ ~, 0.3945 < 1). This complex eigenvalue implies a rotational motion away from the saddle of approximately 2Jr/0 ,~ 2.439 radians (139.8 ~ per step, the motion occurring approximately in a plane parallel to that spanned by the eigenvectors (L, P, A) (1, - 1.166, 0.4860) and (1, -0.3526, -0.2817). Reprinted from Journal of Theoretical Biology, Vol. 194, No. 1, 1. M. Cushing, R. E Costantino, B. Dennis, R. A. Desharnais, and S. M. Henson, Nonlinear population dynamics: models, experiments and data, pp. 1-9, 1998, by permission of the publisher Academic Press/Elsevier Science.

4.3 I Analysis of the Experiment

toward the saddle for this treatment, it does not predict the dramatic saddle flyby that occurs during weeks 16 to 34 in the replicate shown in this figure. At week 16 a random event placed the population in this particular replicate quite near the saddle equilibrium, after which the population lingered near this unstable equilibrium for 18 weeks! Moreover, the subsequent return of the data to the chaotic attractor follows a path with geometric and temporal characteristics predicted by the LPA model. Specifically, the linearization of the model at the saddle equilibrium predicts the distinctive rotational motion in state space observed in the data as they return to the vicinity of the (chaotic) attractor. See Fig. 4.14b. This remarkable observation is yet another testimony to the ability of the deterministic LPA model to describe the dynamics of Tribolium with an unusual amount of accuracy and subtle detail. Stochastically induced saddle flybys, such as the one shown in Fig. 4.14, can occur more than once in a data time series and might even occur often. If such random perturbations occur too frequently, the data might exhibit few characteristics of the model predicted attractor. An example occurs in the treatment Cpa = 0.10. One replicate, shown in Fig. 4.15, undergoes three separate saddle flybys during the course of the experiment and as a result does not spend a significant amount of time near the model predicted attractor. If, on these grounds, that replicate is deleted from the data plot in Fig. 4.8, leaving only the remaining two replicates, the match between data and attractor is highly i m p r o v e d ~ even to the point where the data clusters around the individual points of the predicted 26-cycle! See Fig. 4.16. Deterministically predicted and stochastically induced saddle flybys also appear in replicates from several treatments in the route-to-chaos experiment. The details are slightly different in each case and the flybys vary in their geometry and frequency of occurrence. On the other hand, in some cases few if any saddle flybys happen to occur. For example, in the Cpa = 0.35 (chaos) treatment there is only one saddle flyby in only one replicate (in Appendix C, see replicate 22 during weeks 16 through 28). Thus, the saddle equilibrium had little effect in this particular treatment. For those treatments with periodic fluctuations another notable effect that can result from random perturbations is a shift in phase. Such a phase shift might (or might not) be associated with a saddle flyby, in which a regular oscillatory pattern is temporarily disrupted during the flyby and subsequently returns with a changed temporal phase. When phase shifts occur in replicated experiments, it can happen that replicates

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143

4.4 I Concluding Remarks

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u l t i m a t e l y oscillate o u t of phase. For example, in t r e a t m e n t Cpa = 1.0 the d e t e r m i n i s t i c a t t r a c t o r is a 6-cycle. In Fig. 4.17 we see t h a t the L-stage c o m p o n e n t s f r o m all three replicates in this t r e a t m e n t r e s e m b l e the periodic 6-cycle b u t are all, b y t h e e n d o f t h e e x p e r i m e n t , o u t o f p h a s e w i t h o n e another. Notice, as we h a v e p o i n t e d o u t before, h o w averaging over replicates w o u l d d e s t r o y the d e t e r m i n i s t i c signal! (Phase differences a m o n g replicates are also p r e s e n t in t r e a t m e n t Cpa - 0.50.)

4.4 I CONCLUDING REMARKS I am in a state of chaos. m L A O T Z U (Tao Te Ching)

The p r i n c i p a l a i m of the r o u t e - t o - c h a o s e x p e r i m e n t w a s to d o c u m e n t the o c c u r r e n c e of a (deterministic) m o d e l p r e d i c t e d s e q u e n c e of dyn a m i c t r a n s i t i o n s ~ a r o u t e - t o - c h a o s e x p e r i m e n t ~ in a real biological

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FIGURE 4.17 I The attractor in experimental treatment Cpa 1.00 is a 6-cycle whose Lstage component appears in the top graph. The remaining three graphs show that all three replicates from this treatment resemble the predicted transients and the 6-cycle attractor through week 14. At week 10 each begins a pattern in phase with the model 6-cycle. However, by the end of the experiment all replicates are out of phase with one another. Replicate 23 sustains an oscillatory pattern similar to the model 6-cycle to the end of the experiment. Replicate 3 suffers a random perturbation at week 16 that shifts its phase by one time unit. Replicate 18 experiences a "mild" saddle flyby during weeks 18 to 34 after which its phase is different from either of the other two replicates. =

population. The experiment was designed to place cultures of flour beetles at selected locations along this predicted sequence, including locations where chaotic dynamics are predicted to occur. The experimental protocol required a straightforward manipulation of certain demographic parameters of the beetle populations, namely the death and recruitment rates of the adult life-cycle stage; other life-cycle stages were left unmanipulated. To document the responses of the beetle populations to these selected manipulations we collected census data from replicate and control cultures for 80 weeks.

4.4 [ ConcludingRemarks Comparison of the data with the a priori predicted deterministic attractors indicates convincingly that the populations did undergo the predicted dynamic changes. Particularly striking are the two treatments at the ends of the bifurcation sequence where the attractors are simple, distinctively different, and hence readily apparent in the data. The close match between prediction and data is also strongly supported by statistical analysis. In particular, for the two deterministically predicted chaotic attractors at the treatments cpa - 0.05 and 0.35 the model statistically accounts for 92.5% of the variability in the transformed L-stage abundances and 98.8% of the P-stage abundances in the six cultures allotted to these treatments (according to the generalized R2 value calculated for these populations). Furthermore, bootstrap analyses show that chaos is highly prevalent for these two treatments within the estimated confidence regions for the estimated parameters. Because the dynamics have a stochastic component we cannot, in a strict mathematical sense, label the populations in the chaotic treatments as chaotic (a property of deterministic dynamical systems). Nonetheless it is reasonable to conclude that their dynamics are "highly influenced" by deterministic chaos. By this we mean that their dynamic properties are closely connected to the model predicted attractors and chaos is prevalent throughout the confidence intervals for the parameter estimates. In the next chapter, we explore in more detail the influence that the chaotic attractor has on these populations.

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51 PATTERNS IN CHAOS Chaotic solutions of simple systems of equations are n o t e d for their freq u e n t approximate but not exact repetitions. Sometimes more than one "theme" will be repeated . . . . EDWARD LORENZ [120]

In the previous chapter the approach taken to the issue of chaos in a biological population is based on how a population’s dynamics respond to perturbations. The fundamental goal in this approach is to determine whether the responses that result from changes in certain demographic parameters conform to the bifurcations in a model-predicted route-to-chaos. From this point of view, all the treatments performed at various locations in the bifurcation diagram play a role in determining whether chaos is present in the experiment. Because the experimental populations underwent the dynamic changes indicated by the bifurcation diagram and because the deterministic LPA model predicts chaos specifically for the treatments designated by Cpa = 0.05 and 0.35, we have some confidence in asserting that the beetle cultures in these treatments are c h a o t i c ~ or, as we would rather say, that their dynamics are highly influenced by a chaotic attractor. In the present chapter we focus our attention on one ofthe chaotic treatments in the route-to-chaos experiment (located at Cpa = 0.35 in Fig. 4.1). We wish to see if certain characteristics and properties of the model predicted chaotic attractor are observable in the experimental data. In so doing, we will gather further evidence for the chaotic nature of these particular cultures and gain some new insights into several issues involved in the detection of chaos in biological populations and ecology in general. A study of complex d y n a m i c s ~ i n particular, chaotic d y n a m i c s ~ requires a sufficiently long time series of data. With this in mind we continued the three replicates of the chaos treatment Cpa = 0.35 (and the control cultures) beyond the 80 weeks reported in Chapter 4. In Fig. 4.8

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of that chapter we see how the data from this treatment (with some transients removed) cluster in state space near the predicted chaotic attractor for 80 weeks. In Fig. 5.1A we see a similar figure for data gathered over a m u c h longer time period, namely, 372 weeks. 1 Rigorous comparisons of 1 At the time of writing, we are still maintaining these cultures in our laboratory.

5.1 [ Sensitivity to Initial Conditions

149

the data distribution in state space and the deterministic chaotic attractor, or the stationary distribution predicted by the stochastic LPA model appearing in Fig. 5.1B, are technically difficult. However, even without such statistical analyses one can see from the plots in Fig. 5.1 that the experimental data remained near the predicted deterministic chaotic attractor for a considerable length of time (the 372 weeks of data encompass more than 90 generations, in each ofthe three replicates). Also notice that the distribution of the data in state space is very similar to the predicted stochastic stationary distribution. These comparisons in state space show a clear relationship between the chaotic attractor and the population data over an extended length of time. However, such state-space plots do not allow comparisons of the temporal characteristics of the dynamics. In what follows we consider some distinctive dynamic properties of the deterministic chaotic attractor and see to what extent they are exhibited by the beetle populations. We begin in Section 5.1 with a look at sensitivity to initial conditions. An experiment based on the "hot spot" of sensitivity located on the chaotic attractor not only documents the presence of this hallmark property of chaos in the dynamics of the beetle populations, but also illustrates how one can utilize this feature to control chaotic outbreaks of population numbers. In the rest of the chapter we consider recurrent temporal patterns. As the quote at the beginning of this chapter indicates, chaotic dynamics typically contain such patterns, and we will find that near-periodic motion on the chaotic attractor is in fact observable in the experimental data.

5.1 [ SENSITIVITY TO INITIAL CONDITIONS More individuals are born than can possibly survive. A grain in the balance will determine which individual shall live and which shall die, which variety or species shall increase in number, and which shall decrease, or finally become extinct. m CHARLES

DARWIN

(On the Origin of Species)

A characteristic trait of chaos is the property of sensitivity to initial conditions. Recall that this refers to the tendency for nearby states to diverge rapidly and over time to become very different from one another. Do the beetle populations in the chaos treatment of the route-to-chaos experiment exhibit sensitivity to initial conditions? In Chapter 1 we pointed out the difficulties involved in determining sensitivity to initial conditions in d a t a ~ difficulties related to insufficient amount of data, the presence of noise, and the influences of deterministic

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5 [ PATTERNS IN CHAOS

entities other than the chaotic attractor (transients, unstable invariant sets, etc.). Another approach is to conduct experimental tests for sensitivity to initial conditions or for predicted consequences of this distinctive property of chaos. An experiment to test for the presence of sensitivity to initial conditions in the beetle populations of the route-to-chaos experiment is reported in [52]. The design of that study was based on the location in state space where sensitivity to initial conditions is greatest, as predicted by the deterministic LPA model for the chaos treatment case Cpa = 0.35. As one can see in Fig. 5.2, the most sensitive region on the attractor turns out to be near the "corner" where A-stage and L-stage numbers are low and P-stage numbers are high. Points in state space on or near this part of the chaotic attractor diverge rapidly when m a p p e d forward in time by the LPA model. Computer simulations using the deterministic and stochastic LPA models reveal an interesting prediction when orbits enter this high-sensitivity

FIGURE 5.2 I Color coding depicts the degree of sensitivity to initial conditions at points on the chaotic attractor in the treatment Cpa = 0.35 from the route-to-chaos experiment. The coloring scheme is based on the logarithm of largest moduli of the three eigenvalues of the Jacobian matrix of the LPA model evaluated at the point (using the parameter estimates in Table 4.1). On the attractor these logarithms range from -1.03 to 3.95. The colors range from yellow (for - 1.03) to red (for 3.95). Orbits converge near yellowish points and (at least some) orbits diverge near reddish points. Thus, red regions are "hot spots" where orbits are sensitive to slight perturbations.

5.1 [ Sensitivity to Initial Conditions

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FIGURE 5.3 [ The first row of plots shows the effects on L-stage numbers of applying the in-box and out-box perturbation protocols described in the text to a chaotic attractor of the deterministic LPA model. Parameter values are those in Table 4.1 with Cpa = 0.35 and /Za = 0.96. Notice the suppressed amplitudes caused by the in-box perturbations applied from week 134 to week 212. The dashes above the plots during the perturbation period indicate those weeks at which three adults were added to the population. The out-box procedure, on the other hand, does not suppress the oscillatory amplitudes (although it does regularize the chaotic fluctuations by producing a periodic orbit). Notice the return to chaotic oscillations when both procedures are terminated. The second row of plots shows the result of applying the perturbation schemes when using the demographic stochastic LPA model (4.7). These stochastic orbits can be considered simulations of the experiment and indeed they are very similar to the data shown in the third row of plots. These data are from one replicate in an experiment in which the same perturbation schemes were applied to laboratory cultures of T. c a s t a n e u m .

region or "hot spot" of state space. If an orbit is slightly perturbed in a certain way when it enters the region (specifically, if only three adults are added to the culture), then the amplitude of the chaotic oscillations in the L-stage are significantly dampened. See Fig. 5.3. This predicted reduction in oscillatory amplitude led to the following experiment. We maintained nine populations of T. c a s t a n e u r n according to the experimental protocol for the chaos treatment described in Section 4.1 for 132 weeks, after which we chose six cultures at random and divided

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them into two treatment groups of three cultures each. We maintained the remaining three cultures as controls. Starting at week 134, we subjected the two treatment groups to perturbations as follows. The region in state space where L _< 150 and A _