Branching random walks in random enviroment and their applications to population genetics

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We propose to study various branching random walk systems with random branching rates. The state space that we focus on is the N-dimensional hypercube, serving as a model for long gene sequences with mutations occurring by flip of a gene. We throughout consider the limit of large N, coupled with characteristic parameters such as time and sample size.The random branching rates form a fitness landscape that governs the selection. We first focus on uncorrelated landscapes and start with a population of one randomly picked gene sequence. We want to study the concentration properties of the system, the evolution of the mean fitness of the population and the aging properties. We will attack these questions both with probabilistic and analytical, i.e., spectral theoretical, methods. In another workload, we will consider Gaussian correlated fitness landscapes, namely, Sherrington-Kirkpatrick models of spin glasses. The key to the study of these models is Gaussian comparison techniques, and we shall adopt these tools for studying branching systems.Finally, we will consider theoretical models of experiments on evolution which involves a strategy of restarting the system with a part of the population at the end of some time periods. We plan to develop a thorough understanding of appropriate choices of the scales for the time lags in order to be able to deduce useful conclusions from the experiment.