The principal focus of the project will be on stochastic models of population dynamics and related interacting particle systems. These include models of spatial growth and competition, and models of epidemics and endemics in geographically structured populations. A substantial part of the research will be directed to the behavior of certain interacting systems on noneuclidean and other nonstandard graphs. Particular problems to be investigated will include coexistence of competing species, growth and spread of populations, weak-to-strong survival transitions, and spatial propagation of epidemics and endemics. These have important connections with other developing areas of probability, especially percolation, first-passage percolation, and random graph theory. Secondary foci of the project will be on (1) analytic techniques for studying systems of generating functions, both finite and infinite, especially those that arise in random walk and related combinatorial enumeration problems on tree-like structures; and (2) signal extraction and inference procedures for time series generated by chaotic dynamical systems. Stochastic interacting particle systems are widely used as models in various areas of science, especially in statistical physics and in population biology, but also in economics. These systems consist of "particles" (which in different contexts may represent molecules, biological organisms, economic agents, etc.) that interact in various ways, but generally in a non-deterministic fashion. Mathematical studies of such systems seek to understand how large-scale collective phenomena (such as magnetism, propagation of epidemics in large populations, tumor growth, etc.) arise from and are determined by the details of the individual particle interactions. An important goal of this project is to contribute to our understanding of how the geometry describing the interactions between particles affects the nature of macroscopic behavior, and in particular, to understand how the behavior of certain model particle systems may vary in unusual geometries that may describe social and biological connections in certain animal and human populations.
