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 theory—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 approach—one 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 dynamics— a cascade called a "route-to-chaos." Our approach focuses on the bifurcations and a route-to-chaos predicted by a mathematical model 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 of assorted types, stability and instability, transient dynamics, parameter estimation, model validation, and stochastic -ity. We will find that the mix of nonlinearity and stochasticity produces a level of complexity, a full understanding of which 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 of models using data, and the use of models to design and implement experiments that test model predictions. The importance of these 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 predictions — 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 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, DerekSperry, DavidWood (University of Arizona); 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 Nunez, 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.

Department of Mathematics, Program in Applied Mathematics

University of Arizona

R. F. Costantino

Department of Ecology and Evolutionary Biology

University of Arizona

Brian Dennis

Department of Fish 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

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