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The time to fixation of a strongly beneficial mutant in a structured population - a graphical stochastic analysis

We discuss a system that describes the evolution of the vector of relative frequencies of a beneficial allele in d colonies, starting in (0,...,0) and ending in (1,...,1). Its diffusion part consists of Wright-Fisher noises that model the random reproduction, its drift part is a linear interaction term coming from the gene flow between the colonies, together with a logistic growth term due to the selective advantage of the allele, and a term which makes the entrance from (0,...,0) possible. It turns out that there are d extremal ones among the solutions of the system, each of them corresponding to one colony in which the beneficial mutant originally appears. We explain how these solutions can be represented in terms of a random graph that describes individual genealogies (the so called ancestral selection graph), and how this graphical representation helps to analyze the asymptotic distribution of the fixation time in the limit of a large selection coefficient. All this will be embedded in a short review of some classics in mathematical population genetics.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46