Stochastic Spatial Models

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9628290 Cox ABSTRACT This proposal involves research on several topics in the theory and application of interacting particle systems. Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, as well as many others. The investigator will pursue research on several specific problems. The first project involves the parabolic Anderson model as an interacting diffusion, and associated critical phenomena. The second project concerns the behavior of one-dimensional interfaces regions. This continues work on a model for hybrid zones or "narrow" coexistence zones of different species. The third project attempts to model species abundance distributions with an interacting particle system model. This builds on related work modeling the relationship between species number and area. The fourth project involves possible extensions of a recently discovered comparison technique for interacting systems. Since exact calculations are usually difficult or impossible to make, comparison methods are important tools for analyzing new systems. This proposal involves research on several topics in the theory and application of interacting particle systems. . Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, as well as many others. The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particle systems model well. For example, a simple model for the evolution and extinction of species, to study species abundance questions, fits into this framework. It is based on the birth, death, movement, and reproduction, with a small mutation rate, of individuals. The investigator hopes to show that this type of model can explain the data patterns observed in the field. Part of the proposed research concerns very specific models and questions such as this one, and part of the proposed researc h will aim at developing general mathematical techniques for handling models of this type.