(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.
. These processes are well suited as mathematical models of many types of real world phenomena, and appear, for example, in theoretical ecology, genetics, epidemiology and evolutionary game theory. The distinguishing feature of these models is that they incorporate random interactions which may vary from one location to another. This is particularly important in situations involving competition, where a main question is whether or not coexistence of different types is possible. For this question, predictions made by more traditional mathematical models, typically deterministic and nonspatial, are often incorrect. Predictions made by appropriate stochastic spatial processes are usually more accurate, but the models are significantly more difficult to analyze rigorously. In this project a new mathematical approach to the study of stochastic spatial processes was developed, along with new mathematical results in the theory of random walks and measure-valued processes. In particular, a new criterion for coexistence was found and shown to be applicable to a wide range of models. It was applied to several important specific models, proving rigorously that coexistence was possible in some cases but not in others. In addition, new methods and results were obtained for the long term behaviour of some related stochastic spatial processes, including models for the spread of disease and of a genetic trait in a spatially distributed population.
