Problems in Probability Motivated by Questions in Ecology

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9877066This investigator will study a number of questions that arise from ecology. An important direction of the research will be to prove general results in support of the picture advocated by Durrett and Levin (1994): the behavior of stochastic spatial models can be predicted from that of the mean field ODE obtained by pretending all sites are always independent. He will also continue work on hybrid zones, progressing now to the derivation of PDE limits for these systems as selection tends to 0, and the study of spatially inhomogeneous models to understand the structure of mosaic hybrid zones that have been seen in field work on the cricket species Gryllus firmus and G. pennsylvanicus. Other projects include understanding limits to diversity by analyzing Tilman's competition model on a finite grid, studying the asymptotic behavior of an evolutionary arms race, and demonstrating the existence of chaotic oscillations in a simple discrete time model of gypsy moth populations.The basic question that drives this research is: when does the spatial distribution of individuals change the behavior of a biological system? The comparison is made to a homogeneously mixing system in which each individual interacts equally with all the others. In the past five years the investigator and his collaborators have developed some rules for predicting the answer to this question and have applied them to a variety of biological systems. In the next three years they will apply this theory to some new examples. One of the most important questions they will study is: What are the limits to diversity? They will address this question mathematically by investigating the number of species that can coexist in a stochastic competition model. Other questions will concern the fluctuations of the size of gypsy moth populations and the structure of hybrid zones that result from spatially varying selection.