Stationary states and their stability of the stepping stone model involving mutation and selection
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Probability Theory and Related Fields 73 (1): 87-117 (1986)

The stepping stone model is a generalization of the Wright-Fisher model in population genetics, by taking account of geographical structure, 7, -8, ,12. For this we are particularly interested in the influence of geographical factors on stationary states in the genetical evolution. The model is originally for- mulated as a discrete time Markov process with values in an infinite product space which describes an evolution of a population consisting of a number of colonies, having non-overlapping generations. However we here treat a dif- fusion model which, obtained by taking a diffusion approximation, is an infinite dimensional diffusion process. In the absence of mutation and selective force one of the authors obtained in ,13 a complete characterization of stationary states in terms of migration rates as geographical factors. In the present paper we consider the stepping stone model involving mutation and selection, and investigate stationary states and their stability paying our attention to the mutual influences of mutation, selection and migration. Furthermore we also discuss regularity of finite di- mensional marginal distributions of the stationary states. For the discrete time model it is to be noted that analogues to some of our results are recently obtained by Itatsu in ,6, but our method can essentially cover the discrete time case. Before stating the present problem and our results we here give a de- scription of the discrete time model and briefly survey the diffusion approxima- tion of it. Let S be a countable set. Each element i of S corresponds to a sub- population, which is called a colony. Assuming that there are two alleles A 1 and A 2 in each colony, we denote by 0 < x i < l the gene frequency of the A 1- allele in the colony i. We assume that a genetical evolution is caused by migration among colonies, and mutation, selection and random sampling drift within each colony.
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