Spectral Analysis of Population Time Series using Nonlinear Stochastic Models Quantitative Tools and Exeriments ABSTRACT Understanding the interactions between nonlinearities in dynamical models (both physical and biological) and stochastic variation in the parameters of dynamical models is a fundamental challenge in many branches of the natural sciences, including ecology and population biology. Nonlinear effects in population dynamics interact with environmental variability in ways that are, thus far, poorly understood. This project seeks to understand the interactions between, and the relative importance of, internally and externally generated sources of population fluctuation. It also seeks to understand the relationship between the domination of many field population time series by low frequencies and the domination of many environmental fluctuations by low frequencies. The project aims to: 1) Develop new computational mathematical methods for fitting nonlinear stochastic models to ecological time series data in the frequency-domain; 2) Develop methods of inferring the spectral properties of environmental perturbations that affect empirical population time series; 3) Apply the new frequency-domain fitting techniques to existing data from population dynamics experiments on the flour beetle Tribolium that did not impose environmental variability, and to models of these data that are already well tested in the time-domain; 4) Design, conduct and analyze experiments to test the impact of red-shifted, white, and blue-shifted environment al variability on systems exhibiting known population dynamics (by analogy to light, variability dominated by low-frequency fluctuations is called red-shifted, variability dominated by high-frequency fluctuations is called blue-shifted, and variability with all frequencies equally represented is called white); and 5) Apply gained insights to important ecological systems in the field (by contrast to those in the laboratory). The inherent biological processes that determine the dynamics of some animal populations can lead them to fluctuate chaotically. Variations in the external environment (for example, weather or anthropogenic perturbations) can also lead to unpredictable fluctuations of animal populations. The interactions between the environmental fluctuations and the inherent biological processes can produce large, complex and varied effects that presently defy simple explanations. Small changes in the nature of the environmental noise affecting a system can sometimes be magnified and transfigured by the internal system dynamics to produce substantial and diverse qualitative changes in how a population fluctuates. This project seeks to develop mathematical and statistical tools to understand and model the interactions between internally driven fluctuations and externally driven fluctuations of animal populations. The research focuses on the effects of changing the frequencies of the dominant environmental fluctuations. Preliminary modeling predicts, for instance, that the average population size in a laboratory population of Tribolium flour beetles, widely used as an experimental model for animal population dynamics, can be increased or decreased as the dominant environmental-variation frequency changes across a fixed range, depending on the internal dynamics of the system. Understanding the complex interaction between environmental noise and internal dynamics is essential for predicting which effect will occur, and therefore could have implications when larger populations are desirable (endangered species), or undesirable (pests, including insect vectors of disease). New experiments on the flour beetle system will investigate the interactions between environmental variability and internal dynamics. The knowledge gained from these experiments and the novel analytical techniques developed in this project may have practical importance in pest control, conservation, and the control of human disease.
