Stochastic Spatial Processes

点击翻译按钮获取中文摘要

(1) 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 from theoretical ecology and biology 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. The primary model of interest is a model for competition, a stochastic spatial Lotka-Volterra model. The objectives are to determine the parameter regions which correspond to survival of one species and/or 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. The investigator will supplement this approach using techniques from hydrodynamics to try to show that the previously obtained survival and coexistence conditions are sharp. A second competition model to be studied is the multitype contact process. The main questions here are also the issues of survival and coexistence of competing types. Another model to be studied is one of gene flow from modified crops into a natural or wild population. The goal here is to understand the length of the time it takes for the modified type to invade the wild population. Mathematically, this becomes a question of the behavior of coalescing random walks systems in two dimensions.(2) This proposal involves research in the theory of interacting particle systems and stochastic spatial processes. These 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 large scale behavior of these systems depends on the small scale, individual interaction rules. 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 techniques established for some specific models to handle more general ones, thus justifying their use in applications.