Stationary states and their stability of the stepping stone model involving mutation and selection
Consider a stepping stone genetic model with two alleles \(A_ 1\) and \(A_ 2\) and assume the population is divided into subpopulations or colonies (each one with N individuals) located at different sites in a d- dimensional space. Let the set of sites be \(S={\mathbb{Z}}^ d\) and let \(x_ i\) be the frequency of allele \(A_ 1\) in site \(i\in S\). Allow mutation, selection migration and random genetic drift and assume non-overlapping generations. The model is a discrete-time Markov chain with values in \(\{0,1/N,2/N,...,1\}^ S\). Assuming mutation and migration rates and selection coefficients to be of order 1/N, an infinite-dimensional diffusion approximation is obtained. Assume mutation rates and selection coefficients to be site independent and migration rates between sites i and j to depend only on j-i (componentwise subtraction) and satisfy an irreducibility assumption. Previous papers of the first author [see J. Math. Kyoto Univ. 22, 1-40 (1982; Zbl 0539.92012)] have obtained a complete characterization of stationary states and their stability in the absense of selection and mutation. Such results are extended now to the general case. Furthermore, the authors discuss regularity of finite dimensional marginal distributions of the stationary states. The basic tool used in the proofs is the dual process, which has been successfully used in the theory of interacting particle systems; this dual process is what the authors call a branching random walk with interaction.
infinite-dimensional diffusion approximation
regularity of finite dimensional marginal distributions of the stationary states
scientific article; zbMATH DE number 4047485
scientific article