9306271 Cushing The investigators conduct an interdisciplinary research program to test nonlinear population theory: they construct and analyze mathematical models, design and implement biological experiments, develop and apply statistical techniques for the analysis of data. For the biological experiments an organism is used that is easy to culture, has a short generation time (i.e. yields long time series data) and allows an accurate census of animal numbers: flour beetles of the genus Tribolium. In the first part of the study the objectives are model identification and parameter estimation. In the second part, the concern is to document transitions in the qualitative behavior of the demographic dynamics. Rates of reproduction and adult mortality are manipulated in order to cross boundaries in parameter space from stable equilibria, to periodic cycles, to chaos. In phase three the objectives are to test hypotheses concerning the existence of these unusual demographic dynamics and develop methods for identifying these phenomena in experimental data. A major contribution of this project is an unequivocal example of experimentally manipulated transitions between qualitatively different dynamical behaviors of a biological population as predicted by a mathematical model. In the last ten years or so, the recognition that simple equations can generate complex dynamics has led to an outpouring of fascinating theoretical possibilities for the explanation of population time series data. Understanding the observed fluctuations in animal population numbers is a central question in population biology; it has far-reaching applications in areas ranging from food production and pest control, to the management of renewable resources, to the conservation of species diversity. The hypothesis that fluctuations are the result of nonlinear dynamic forces has proved to be elusive to test due to the difficulties of gathering adequate ecological data, of experimentall y manipulating ecological systems, and of evaluating complex mathematical models with ecological data. In this research project the investigators' approach to testing nonlinear population theory is to connect mathematical models rigorously with experimental biological data by means of newly developed statistical methods for nonlinear time series. The project is unique in its interdisciplinary approach because it involves both theory and experimentation and utilizes the talents of the biologist, statistician, and mathematician. It is unusual in the field of population biology to have an interdisciplinary effort in which investigators from all of these disciplines are involved in all aspects of the project, from experimental design and implementation, through theoretical modeling and analysis, to statistical testing and verification. The ultimate goal is to demonstrate the usefulness and importance of nonlinear mathematics in gaining a rigorous understanding of the dynamics of animal populations and in particular of fluctuations in population numbers, be these fluctations regular or "chaotic."
