This research involves several topics in the theory and application of interacting particle systems. These processes are stochastic models for large systems of interacting components. Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, and more. The investigator will pursue research on several specific problems. The first project involves the mutually catalytic branching process, which models the evolution of two populations whose growth rates depend on one another. The second project concerns the relationship between low density interacting particle systems and measure-valued diffusions, especially the convergence of rescaled versions of the former to super-Brownian motion. The third project treats some questions from mathematical genetics, especially limiting results for the stepping stone model on the two dimensional integer lattice. In the fourth project a class of generalized branching random walks is considered. The key to resolving the main question of local extinction versus stability for this class appears to be the analysis of a particular case with a certain minimal branching rate which takes a form not previously considered. This research involves several topics in the theory and application of interacting particle systems or stochastic spatial processes. The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particle systems model well. These are large systems made up of many interacting components, usually with a stochastic or random element. For example, a simple model for the evolution of a genetic trait through a spatially distributed population fits into this framework. It incorporates the movement of individuals across spatially distributed colonies, and mutation at a small rate. The investigator hopes to show that this type of model can give more accurate results than the traditional, simpler non-spatial models widely used. Part of the proposed research concerns very specific models and questions such as this one, and part of the proposed research will aim at developing general mathematical techniques for handling models of this type.