What We Learned 173 Bibliography 183 Appendix 195

A The Desharnais Experiment 195 B The Bifurcation Experiment 196 C The Chaos Experiment 213

Index 223

Foreword: An Inordinate 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 it—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 [131a].

Low-dimensional chaos has a characteristic signature that can be detected in univariate time series via phase-portrait reconstruction [174]

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 ecology was 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 castaneum. Key to their success has been the ability 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 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

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