Abstract

. Introduction The purpose o f th e present paper is to investigate a n interacting system de- scribing an evolution of gene frequencies in population genetics. In particular, we are interested in the effect of geographical structure in the process of evolution. We shall first explain our interacting system. Let S be a countable set. Each element i of S is called " a c o lo n y ". W e assume that there are two alleles A and B at each colony and denote by x (1—x ) the gene frequency of the A-allele (resp. the B-allele) for the colony i e S . A time evolution of gene frequencies is caused by the following three factors; (1) mutation in each colony, (2) migration among colonies, and ( 3 ) random sampling drift. It is described by means of the following stochastic differential equation,

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