Complex Population Dynamics: A Theoretical/Empirical Synthesis

by Peter Turchin

Overview

Why do organisms become extremely abundant one year and then seem to disappear a few years later? Why do population outbreaks in particular species happen more or less regularly in certain locations, but only irregularly (or never at all) in other locations? Complex population dynamics have fascinated biologists for decades. By bringing together mathematical models, statistical analyses, and field experiments, this book offers a comprehensive new synthesis of the theory of population oscillations.


Peter Turchin first reviews the conceptual tools that ecologists use to investigate population oscillations, introducing population modeling and the statistical analysis of time series data. He then provides an in-depth discussion of several case studies--including the larch budmoth, southern pine beetle, red grouse, voles and lemmings, snowshoe hare, and ungulates--to develop a new analysis of the mechanisms that drive population oscillations in nature. Through such work, the author argues, ecologists can develop general laws of population dynamics that will help turn ecology into a truly quantitative and predictive science.



Complex Population Dynamics integrates theoretical and empirical studies into a major new synthesis of current knowledge about population dynamics. It is also a pioneering work that sets the course for ecology's future as a predictive science.

Product Details

ISBN-13:	9781400847280
Publisher:	Princeton University Press
Publication date:	02/15/2013
Series:	Monographs in Population Biology , #35
Sold by:	Barnes & Noble
Format:	eBook
Pages:	472
File size:	4 MB

About the Author

Peter Turchin is Professor of Ecology and Evolutionary Biology at the University of Connecticut. He is the author of Quantitative Analysis of Movement and more than sixty scientific articles, including several in Nature and Science.

Read an Excerpt

Complex Population Dynamics

A Theoretical/Empirical Synthesis

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By Peter Turchin

Princeton University Press

Peter Turchin
All right reserved.
ISBN: 0691090203

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Chapter One

Introduction

1.1 AT THE SOURCES

Population dynamics is the study of how and why population numbers change in time and space. Thus, population dynamicists document the empirical patterns of population change and attempt to determine the mechanisms explaining the observed patterns. Temporal population dynamics is not the only subject that population ecologists study. Among other things, they are also interested in statics (what sets the level around which populations fluctuate) and population structure (e.g., age distribution). More recently, there has been a lot of progress in spatiotemporal dynamics of populations. Nevertheless, population dynamics in time has been at the core of population ecology ever since the origins of the discipline during the 1920s (Kingsland 1995), largely as a result of efforts of Charles Elton, Alfred Lotka, Vito Volterra, and A. J. Nicholson.

1.1.1 The Puzzle of Population Cycles

Abrupt and seemingly inexplicable changes in population numbers have fascinated and puzzled humanity from prehistoric times. The Bible records the effects of locust swarms and mice "plagues" on humans. Hunters and trappers surely knew about periodic changes in populations of furbearing mammals and game birds.Norwegians have long been aware of mysterious invasions by lemmings (Stenseth and Ims 1993a). Nordic folklore has provided the basis of the modern myth of lemmings marching off to the sea to commit mass suicide, as popularized by Walt Disney's White Wilderness.

The scientific study of population oscillations begins with the work of Charles Elton (Stenseth and Ims 1993a; Lindström et al. 2000). In 1923 the young Elton passed through the Norwegian town of Tromsø on his way back from a zoological expedition to the Spitsbergen. In a Tromsø bookstore, he noticed Norges Pattedyr (Norwegian mammals) by Robert Collett. Although Elton could not read Norwegian, he noticed a very curious-apparently periodic-pattern in the abundance of Norwegian lemmings. With some of the last of his money, Elton bought the book, brought it with him back to Oxford, and had it translated into English. In 1924, Elton published the pioneering article "Periodic Fluctuations in the Number of Animals: Their Cause and Effects" (Elton 1924), based largely on Collett's data (Stenseth and Ims 1993a; Crowcroft 1991).

About the same time, Elton read The Conservation of the Wild Life of Canada by Gordon Hewitt, which contained graphs of the annual fur returns of the Hudson's Bay Company showing remarkably regular oscillations in the numbers of lynx and snowshoe hare pelts (Crowcroft 1991:4). Elton was appointed biological consultant to the Hudson's Bay Company in 1925, and examined the company's records to trace the dynamics of Canada lynx populations back to 1736. The results of this research were eventually published in 1942 (Elton and Nicholson 1942). A second line of attack consisted of empirically studying fluctuations in the numbers of British voles, using Oxford as a base (Crowcroft 1991:6). While Elton and his group were engaged in these empirical studies, momentous changes were occurring in the field of theoretical ecology.

1.1.2 Modeling Nature

By a curious coincidence, the mathematical study of population oscillations started practically at the same time as Elton was puzzling over lemming cycles (Lotka 1925; Volterra 1926). The two traditions, the empirical and the mathematical, although having started almost simultaneously, developed largely separately. Only three-quarters of a century later we are starting to see a true synthesis.

Theory is important because there is a tendency for common phenomena to be overlooked or misinterpreted in the absence of a well-known body of theory (Abrams 1998:211). One ecological illustration of this tendency is the meager experimental evidence for apparent competition that Holt (1977) could marshal in the article where he proposed the concept, compared with the large body of evidence reviewed by Holt and Lawton (1993) seventeen years later (Abrams 1998). So it was at the beginning of the study of population cycles. In his first paper on population cycles, Elton wrote: "It will be shown in the body of this paper that the periodic fluctuations in the numbers of certain animals there dealt with, must be due to climatic variations" (Elton 1924:119). When Volterra's 1926 article appeared in Nature, Julian Huxley, Elton's former tutor at Oxford, brought it to him, and Elton immediately realized its importance. The generation of population cycles through endogenous causes was new and unexpected (Kingsland 1995:127).

1.1.3 The Balance of Nature

Whereas the study of population oscillations originated with the empirical work of Elton and the theoretical work of Lotka and Volterra, time-series analysis of population fluctuations can be traced to the famous debate about population regulation, which crystallized at the 1957 meeting in Cold Spring Harbor. One of the protagonists in the debate was A. J. Nicholson, who developed the theory of population regulation by density-dependent mechanisms (Nicholson 1933, 1954). Nicholson's views were supported by Elton, who wrote, "it is becoming increasingly understood by population ecologists that the control of populations, i.e., ultimate upper and lower limits set to increase, is brought about by density-dependent factors" (Elton 1949:19). Andrewartha and Birch (1954:649) disagreed: density-dependent factors "are not a general theory because . . . they do not describe any substantial body of empirical facts." The debate reached a peak at the Cold Spring Harbor Symposium (Andrewartha 1957; Nicholson 1957). It has continued ever since, reaching another peak of intensity during the 1980s (a review in Turchin 1995b), although currently some consensus is apparently beginning to emerge (section 5.4).

An interesting thing happened while the regulation debate was raging. First, empirical ecologists began collecting long-term data on population fluctuations of a wide variety of organisms. It is curious that a lot of long-term data sets were started during the 1940s and 1950s (i.e., just when the debate was at one of its peaks!). Next, quantitative ecologists started analyzing these time-series data (Moran 1953; Bulmer 1974; Berryman 1978; Royama 1981; Potts et al. 1984; Turchin 1990) using, in the beginning, such linear approaches as the Box-Jenkins time-series analysis. Then, ecologists (most notably, Robert May) participated in the nonlinear dynamics revolution (Gleick 1988). When physicists invented the new technique of attractor reconstruction in time-delayed coordinates (Takens 1981; Packard et al. 1980), some ecologists began applying it to ecological time series (Schaffer 1985). Classical time-series analyses and nonlinear dynamics approaches were eventually merged in a synthetic approach to the analysis of ecological data (these approaches will be discussed in part II), and applied to issues ranging beyond mere density dependence. Presently, we are seeing how these nonlinear time-series methods are being merged with the theoretical tradition (see chapter 8), and there are also promising beginnings of the synthesis between the population-regulation analyses and experimental approaches (Cappuccino and Harrison 1996).

1.2 GENERAL PHILOSOPHY OF THE APPROACH

Most ecologists do their science without giving much thought to the broad philosophical issues underlying what they do. Among those ecologists who do worry about philosophical foundations, the most vocal, and not afraid of making strong recommendations, are the Popperians (e.g., Chitty 1996; Murray 2000; Lambin et al. 2002). Other ecologists take the view that there are many ways of doing ecology, and one should not be too dogmatic about it (e.g., Fagerstrom 1987; Pickett et al. 1994). I believe that such philosophical discussions are important, because they affect how we do ecology. Furthermore, one of the broad themes of this book is methodological (see the preface): what are the best approaches to solving the puzzle of population cycles? Thus, I need to describe the philosophical basis of the general approach that I advocate.

While Popper's idea that all theories have to be testable in order even to be called scientific seems quite reasonable to me, I find the rest of his philosophy of science, at least as expounded by his ecological disciples, not to be a very useful way of doing science. I am particularly bothered by the emphasis of Popperians on falsificationism as the way of doing science. First, the view that data are "hard facts" is untenable for methodological and psychological reasons (see Fagerstrom 1987 for a very clear discussion of this point). Thus, it is not true that in any contest between theory and data, it is theory that should necessarily lose. Second, I don't think that ecologists are in the business of rejecting theories. "Ecologists, like many others, do not reject theories for the futile reason that they are wrong; theories are retained until better ones emerge" (Fagerstrom 1987). A very good idea of how futile a rejectionist program can be is conveyed by the book of Dennis Chitty (1996), Do Lemmings Commit Suicide? There Chitty relates how a consistent application of the rejectionist approach led him to reject all hypotheses that could be tested, leaving him with the explanation that nobody could figure out how to test.

I think that we (ecologists) are, instead, in the business of deciding which of the available alternative theories is the best, or "least wrong" (I shall make this idea more precise later in this section). One thing that any scientist has to come to terms with is that all our theories are, in the final account, wrong (the alternative of not being wrong is to become untestable, that is, nonscientific). The more explicitly we formulate our theories (which, at least in the context of population dynamics, means translating them into mathematical statements) the more wrong they become, simply because our simple theories can never capture all the complexity and detail of nature. So falsifying theories is trivial: just collect detailed data about any aspect of the theory, and you are certain to show that the theory is wrong. If you have not, it simply means that you either collected too few data points or did not measure them carefully enough.

If all our theories are a priori wrong, what can we do? Well, science is still the search for truth, but any scientific truth that we find is both approximate and tentative. Approximate, because of the reasons discussed in the paragraph above; tentative, because we have no guarantee that somebody smarter or possessing better data and analytical tools will not come up with a better "truth" sometime in the future. Therefore, we should not be in the business of rejecting theories, as ecological Popperians would have us do, but in the business of contrasting two or more theories with each other, using the data as an arbiter. The corollary of this approach is that our best theory may not explain or predict data very well, but we should still use it until we have something better. Even the theory that explains only 10% variation in the data is useful, because it sets a standard to be bettered.

In the rest of the section, I make this idea precise for the specific context of population dynamics. The basic notions are three: (1) define very carefully what you are trying to explain; (2) translate your verbal theories into explicit mathematical models (note the plural here); and (3) use formal statistical methods to quantify the relative ability of the rival models to predict data. Data may already be available, or they may be specifically collected to distinguish between predictions of the rival hypotheses (the latter constitutes an experiment).

1.2.1 Defining the Phenomenon to Be Explained

The broad question that I address in this book is, why do population numbers change with time? Or, to put it more succinctly, "why do populations behave as they do?" (Royama 1992:1). In any particular case study, this broad question can be broken into more specific issues. First, are dynamics of the studied population characterized by a stationary distribution of densities? (This is the issue of population regulation.) If yes, there is some characteristic mean level around which the population fluctuates, and fluctuations are characterized by a certain (finite) variance. What ecological mechanisms are responsible for setting this mean level? (This is the focus of population statics.) What mechanisms set the amplitude of fluctuations? Finally, are there detectable statistical periodicities, and what is the order and trajectory stability characterizing dynamics?

At the most general level, the phenomenon to be explained is quantified by a temporal record of population fluctuations, or time-series data. Time-series data are often available even before the beginning of the formal inquiry into dynamics of a particular population (although we often have to do with an index of population rather than an absolute measure of population density; examples include fur returns, bag records, and pheromone trap catches). If time-series data are not available, a systematic program for their collection should be initiated immediately. (One should not worry too much about limited usefulness of short time series; after all, it may take many decades to approach the solution, by which time time-series data will be long enough to be useful!)

I will call the density measurements of the "focal species" (the one whose dynamics we are trying to understand), {N_t }, the primary data.¹ We may have time-series data on other aspects of system dynamics available (e.g., temporal changes in mean body mass, fluctuations in the availability of food, and densities of predators or parasitoids). Such ancillary data may be extremely useful, but are secondary in the sense that we do not require that our explanation of the focal species dynamics would account for all of them. For example, if we are studying a forest defoliator, then a model based on plant quality does not need to explain why parasitism rates vary (perhaps parasitoids are simply responding to the oscillations of their food supply, without a detectable feedback effect on defoliator densities). Vice versa, a parasitism-based explanation does not need to account for changes in plant quality. Of course, the model based on a particular factor has to be consistent with time-series data for this factor.

A focus on the primary data permits us to use the same metric when comparing hypotheses based on very different factors. One particular metric that I will use extensively is the coefficient of prediction, R _pred² (the proportion of variance in log-transformed density explained by the hypothesis). However, this is not the only metric that can be employed to quantitatively compare the performance of different hypotheses.

Continues...

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Table of Contents

Preface xi
Mathematical Symbols xv
Part I: THEORY
1. Introduction 3
1.1 At the Sources 3
1.1.1 The Puzzle of Population Cycles 3
1.1.2 Modeling Nature 4
1.1.3 The Balance of Nature 5
1.2 General Philosophy of the Approach 6
1.2.1 Defining the Phenomenon to Be Explained 8
1.2.2 Formalizing Hypotheses as Mathematical Models 11
1.2.3 Contrasting Models with Data 14

2. Population Dynamics from First Principles 17
2.1 Introduction 17
2.2 Exponential Growth 19
2.2.1 Derivation of the Exponential Model 20
2.2.2 Comparison with the Law of Inertia 22
2.2.3 "Laws": Postulates, Theorems, Empirical Generalizations? 25
2.3 Self-Limitation 26
2.3.1 Upper and Lower Density Bounds 26
2.3.2 Formalizing the Notion of Self-Limitation 27
2.3.3 The Logistic Model 29
2.4 Consumer-Resource Oscillations 30
2.4.1 Three More Postulates 31
2.4.2 The Lotka-Volterra Predation Model 33
2.5 Process Order 36
2.6 Synthesis 44

3. Single-Species Populations 47
3.1 Models without Population Structure 47
3.1.1 Continuous-Time Models 48
3.1.2 Discrete-Time Models 52
3.1.3 Delayed Differential Models 56
3.2 Exogenous Drivers 58
3.2.1 Stochastic Variation 60
3.2.2 Deterministic Exogenous Factors 61
3.3 Age-and Stage-Structured Models 64
3.3.1 Mathematical Frameworks 65
3.3.2 An Example: Flour Beetle Dynamics 68
3.4 Second-Order Models 70
3.4.1 Maternal Effect Hypothesis 70
3.4.2 Kin Favoritism Model 72
3.5 Synthesis 76

4. Trophic Interactions 78
4.1 Responses of Predators to Fluctuations in Prey Density 79
4.1.1 Functional Response 79
4.1.2 Aggregative Response 88
4.1.3 Numerical Response 90
4.2 Continuous-Time Models 93
4.2.1 Generalized Lotka-Volterra Models 94
4.2.2 Models Not Conforming to the LV Framework 99
4.2.3 Anatomy of a Predator-Prey Cycle 102
4.2.4 Generalist Predators 104
4.3 Discrete-Time Models: Parasitoids 108
4.3.1 Functional and Numerical Responses 109
4.3.2 Dynamical Models 111
4.4 Grazing Systems 112
4.4.1 Grazer's Functional Response 113
4.4.2 Dynamics of Vegetation Regrowth 117
4.4.3 Dynamics of Grazer-Vegetation Interactions 120
4.4.4 Plant Quality 123
4.5 Pathogens and Parasites 127
4.5.1 Transmission Rate 127
4.5.2 Microparasitism Models 128
4.5.3 Macroparasitism Models 131
4.6 Tritrophic Models 133
4.7 Synthesis 136

5. Connecting Mathematical Theory to Empirical Dynamics 137
5.1 Introduction 137
5.2 Qualitative Types of Deterministic Dynamics 139
5.2.1 Attractors 139
5.2.2 Sensitive Dependence on Initial Conditions 140
5.3 Population Dynamics in the Presence of Noise 146
5.3.1 Simple Population Dynamics 146
5.3.2 Stable Periodic Oscillations 147
5.3.3 Chaotic Oscillations 148
5.3.4 Quasi-Chaotic Oscillations 151
5.3.5 Regular Exogenous Forcing 153
5.3.6 Synthesis 153
5.4 Population Regulation 154
5.4.1 Definition of Density Dependence 155
5.4.2 Regulation: Evolution of the Concept 156
5.4.3 The Stationarity Definition of Regulation 156
5.4.4 Beyond Stationarity: Stochastic Boundedness 157
5.4.5 Synthesis 158

Part II: DATA
6. Empirical Approaches: An Overview 163
6.1 Introduction 163
6.2 Analysis of Population Fluctuations 164
6.2.1 The Structure of Density Dependence 164
6.2.2 Probes: Quantitative Measures of Time-Series Patterns 165
6.2.3 Phenomenological versus Mechanistic Approaches 167
6.3 Experimental Approaches 168
7. Phenomenological Time-Series Analysis 173
7.1 Basics 173
7.1.1 Variance Decomposition 173
7.1.2 Data Manipulations Prior to Analysis 175
7.1.3 Diagnostic Tools 178
7.2 Fitting Models to Data 183
7.2.1 General Framework 183
7.2.2 Choosing the Base Lag 186
7.2.3 Functional Forms 188
7.2.4 Model Selection by Cross-Validation 191
7.3 Synthesis 195

8. Fitting Mechanistic Models 197
8.1 Model Selection 198
8.2 Analysis of Ancillary Data 200
8.3 One-Step-Ahead Prediction 201
8.4 Trajectory Matching 203
8.5 Fitting by Nonlinear Forecasting 205

Part III: CASESTUDIES
9. Larch Budmoth 213
9.1 Introduction 213
9.2 Analysis of Time-Series Data 217
9.3 Hypotheses and Models 220
9.3.1 Plant Quality 220
9.3.2 Parasitism 229
9.3.3 Putting It All Together: A Parasitism-Plant Quality Model 235
9.4 Synthesis 237

10. Southern Pine Beetle 239
10.1 Introduction 239
10.2 Analysis of Time-Series Data 240
10.3 Hypotheses and Models 243
10.3.1 General Review of Hypotheses 243
10.3.2 Interaction with Hosts 247
10.3.3 Interaction with Parasitoids 253
10.3.4 The Predation Hypothesis 255
10.4 An Experimental Test of the Predation Hypothesis 259
10.4.1 Rationale 259
10.4.2 Results 264
10.5 Synthesis 271

11. Red Grouse 272
11.1 Numerical Patterns 273
11.2 Hypotheses and Models 281
11.2.1 Overview 281
11.2.2 Parasite-Grouse Hypothesis 282
11.2.3 Kin Favoritism Hypothesis 285
11.3 Experiments 289
11.3.1 Density Manipulation 289
11.3.2 Parasite Manipulation 291
11.4 Synthesis 294

12. Voles and Other Rodents 296
12.1 Introduction 296
12.2 Analysis of Time-Series Data 297
12.2.1 Methodological Issues 297
12.2.2 Numerical Patterns 301
12.3 Hypotheses and Models 310
12.3.1 Maternal Effect Hypothesis 311
12.3.2 Interaction with Food 316
12.3.3 Predation 317
12.4 Fitting the Predation Model by NLF 321
12.5 Lemmings 325
12.5.1 Numerical Patterns 326
12.5.2 Testing Alternative Trophic Hypotheses 328
12.5.3 Lemming-Vegetation Dynamics at Barrow 331
12.6 Synthesis 335
12.6.1 Summary of Findings 335
12.6.2 Towar a General Trophic Theory of Rodent Dynamics 339

13. Snowshoe Hare 344
13.1 Introduction 344
13.2 Numerical Patterns 345
13.3 Models 349
13.4 Experiments 356
13.5 Synthesis 362

14. Ungulates 365
14.1 Introduction 365
14.2 Interaction with Food 368
14.3 Interaction with Predators 371
14.4 Numerical Dynamics 376
14.5 Synthesis 381

15. General Conclusions 383
15.1 What Mechanisms Drive Oscillations in Nature? 383
15.2 Structure of Density Dependence 386
15.3 What about Chaos? 390
15.4 Population Ecology: A Mature Science 392

Glossary 397
References 405
Index 437

What People are Saying About This

From the Publisher

"This book gives an excellent introduction to our maturing understanding of how ideas about nonlinear dynamical systems can shed light on the complex behavior of real populations. Turchin first presents the theoretical concepts, next the data, and finally brings the two together in thoughtfully presented case studies. He splendidly achieves his aim of showing how ecological problems can be illuminated by mathematical models which are—in Einstein's words—'as simple as possible, but not more so.' "—Robert M. May, President, Royal Society

"This book is a true landmark in the study of population ecology. Never before has a lucid synthesis of the basic principles and models of population dynamics been so effectively merged with insightful analyses of some of the best data for populations of insects, birds and mammals. Peter Turchin writes with admirable clarity and authority, always retaining a focus on issues that really matter for population ecology."—Ilkka Hanski, University of Helsinki

"Turchin has done an outstanding job and produced a significant milestone in our understanding of population dynamics. I have long been seeking a book like this. It is readable and accessible while going into a good depth that so many undergraduate texts fail. All people working in the field will use it, and professors will use it in courses on population dynamics. The presentation is superb—the text gallops along."—Peter Hudson, University of Stirling

Editorial Reviews

"Turchin has to be congratulated for the conceptual clarity of the book. . . . I especially recommend the book's first two parts to anyone interested in how to model and analyze population fluctuations. . . . Turchin offers researchers and students alike interesting material and a great deal to think about."—-Esa Ranta, Science

"This book contributes profoundly to the literature. . . . [It] may have a huge impact on the field."—-Nils Chr. Stenseth, Nature

"A superbly written text offering many fresh insights both pragmatic and profound. . . . Throughout the book, Turchin manages to present complex material in an informal style with clarity and eloquence."—-Douglas H. Deutschman, Ecology

From the Publisher