This proposal will address questions in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting ``components'' (cells, individuals, particles, plants, etc.). Some examples of the phenomena these systems model are: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Two competition models will be studied, a spatial stochastic Lotka-Volterra model and a multitype contact process. For the Lotka-Volterra model, the objectives are to determine the parameter regions which correspond to survival of one species and coexistence of both species. The project is part of the investigator's ongoing efforts to understand and exploit the scaling relationship between interacting particle systems and measure-valued diffusions. For the multitype contact process, the questions of interest concern survival versus extinction on regular trees. The investigator expects to find new phase dependent survival and extinction phenomena that do not exist in the lattice case. A second general area of this proposal addresses problems motivated by population genetics. The evolution of genetic traits in geographically structured populations is often modeled with Kimura's stepping stone model, where questions about measures of kinship are reformulated in terms of questions about the behavior of coalescing random walk systems. The investigator's previous research includes work on limit theorems for these systems on large, two-dimensional lattice sets with torus or wrap-around boundary conditions. The goal here is to show robustness of these theorems over a wide range of boundary conditions, justifying their use in population genetics.
This proposal involves research in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting components (cells, individuals, particles, plants, etc.). The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particles systems model well, such as models of: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Several specific models will be studied, including a stochastic spatial version of a well known model for competition between species. In addition to work on specific models, the investigator will try to extend the validity of some approximation theorems established for some specific models to handle more general ones, thus justifying their use in applications.
