Stochastic Models in Population Genetics and Ecology
These research projects fall within the areas of mathematical population genetics and interacting particle systems. The unifying thrust is the study of mathematical models of biological systems at the population level. The interplay between various genetic influences such as mutation, selection, and population structure can sometimes confound one's ability to draw valid inference. The first goal is to develop methods of estimation and inference for complex molecular genetic data arising from populations with various kinds of structure. The investigators explore sampling distributions and likelihood methods in the presence of these confounding effects. The second goal involves the study of genealogical processes (i.e., coalescent theory) for populations which are subject to various selective forces, geographical segregation, or other hierarchical structuring. Both sampling theory and coalescent methods provide a framework for computational methods such as Markov Chain Monte Carlo and Importance Sampling. The third goal is to study spatial interactions which arise in ecology and epidemiology using interacting particle systems. The problems addressed include several new nonlinear voter models with a cluster-breaking feature and new work on the two-stage contact process, with applications to metapopulation models. Previous models of the ecology and genetics of natural populations have been based on many simplifying assumptions. It is now important to produce more realistic models to address ecological and evolutionary problems that impact society. In addition to the inherent randomness in real biological systems, spatial structure is often of critical importance. Examples of these problems include the evolution and spread of infectious diseases, invasions of exotic species, and survival of species in fragmented habitats. Furthermore, the technical revolution in gene sequencing is producing huge data sets that can only be deciphered in a sophisticated mathematical framework. The new field of bioinformatics relies heavily on an evolutionary perspective. This interdisciplinary work is part of an ongoing effort to develop and employ new mathematical methods for the purpose of addressing these complex biological problems at the population level.