Problems in Probability Arising from Ecology and Genetics

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0202935Durrett Durrett proposes to study a variety of problems in probability theory that arise from questions in ecology and genetics. In the area of stochastic spatial models, he will study the asymptotic behavior of multitype voter models where voters have several opinions and are more likely to imitate neighbors with similar opinions. With Ted Cox he will work to extend their conclusions for the stepping stone model to samples of size k 2, and will study equilibrium properties of a hybrid zone which results when each of two alleles is favored in one half of the space. With Ed Perkins he will study measure valued limits of particle systems to produce limit processes with interactions between different types of particles. Not all of the systems Durrett studies will be spatial however: he will work with Jason Schweinsberg to develop approximations of genetic hitchhiking that involve the coalescent with multiple collisions, will study the phase transition in a quasi-species model which models properties of viruses with high mutation rates, and will continue his work with Vlada Limic on Kauffmann's NK model, which is a prototypical example of a fitness landscape with a large number of local maxima. Durrett's research is motivated by a variety of applications. Durrett will work with Ted Cox to extend their work on spatial genetics models to investigate explanations for the surprisingly large amount of linkage disequilibrium (genetic correlation) in the human genome. Understanding the possible sources of patterns in DNA sequences is important for using association mapping to locate genes and to identify footprints of positive selection. In relation to the latter question, Durrett will work with Jason Schweinsberg to develop an approximation to the genetic hitchhiking that occurs with the fixation of favorable mutation, which will allow for the derivation of analytical results for quantities that can currently be understood only by simulation. A major theme of Durrett's research is the influence of a spatial distribution of competitors on equilibrium properties of a system. With Simon Levin, he will study a collection of stochastic spatial models for the evolution of social norms. With Ted Cox, he will investigate hybrid zones where closely related but genetically differentiated populations coexist in close proximity. With Ed Perkins he will investigate scaling limits of stochastic spatial models, in order to construct new interesting examples of interacting measure-valued diffusions and to obtain approximations for particle systems with large range.