Stochastic Spatial Processes

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9971868Cox This research deals with several topics in the theory and application of stochastic spatial processes. Such processes are stochastic models for large systems with many interacting components. Among the phenomena these systems model are competition of species, epidemics, spread of genetic traits, and catalytic reactions. Research on several specific problems will be considered. The first project involves a model for mutually catalytic branching. The goal is to understand the simultaneous evolution of two spatially distributed populations undergoing random motion and reproduction, where individuals from one population reproduce and die only in the presence of individuals from the other population. The second project involves measure-valued scaling limits of interacting particle systems. The goal is to study further the phenomenon, only recently discovered, that certain lattice systems with strong local interactions converge, when scaled appropriately, to super-Brownian motion. In particular, a detailed description of this phenomenon for the voter model in two or more dimensions is a primary goal; both spatial and genealogical structure will be studied. The third project involves questions which arise in the study of mathematical models of hybrid zones. The goal here is to describe the large scale behavior of models with a spatially dependent selection mechanism, especially when the selective advantage parameter is small. This research involves several topics in the theory and application of stochastic spatial processes. The goal of this research is to obtain a better qualitative understanding of some of the complex phenomena exhibited by systems made up of a large number of interacting components, with the interaction having a stochastic or random element. Such systems occur in many applications, one notable area being theoretical biology and ecology, where there is great interest in improving older mathematical models which do not allow for explicit spatial dependence (as do these models). Stochastic spatial models are used to study questions involving predator-prey interactions, species coexistence, and genetic diversity. These models often have dozens of parameters and interaction rules; consequently, they are generally not amenable to rigorous mathematical analysis. The investigator plans research on several specific models which retain the interesting qualitative behavior of more complicated models, but are "simple" enough to allow for the possibility of mathematical analysis. By studying these models, the investigator seeks to give rigorous mathematical arguments which explain the observed behavior of the models. The investigator also plans to study what appears to be an intimate connection between two apparently different types of stochastic spatial models (discrete models with strong interactions and continuous models with weak interactions).