Stationary states and their stability of the stepping stone model involving mutation and selection
type: Item
instance of: scholarly article
title: Stationary states and their stability of the stepping stone model involving mutation and selection
zbMATH Open document ID: 0642.92008
DOI: 10.1007/BF01845994
author: Tokuzo Shiga
author: Kohei Uchiyama
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 4047485
zbMATH Keywords: stepping stone genetic model
zbMATH Keywords: two alleles
zbMATH Keywords: colonies
zbMATH Keywords: mutation
zbMATH Keywords: selection migration
zbMATH Keywords: random genetic drift
zbMATH Keywords: non-overlapping generations
zbMATH Keywords: discrete- time Markov chain
zbMATH Keywords: infinite-dimensional diffusion approximation
zbMATH Keywords: regularity of finite dimensional marginal distributions of the stationary states
zbMATH Keywords: dual process
zbMATH Keywords: branching random walk with interaction
reviewed by: Carlos A. dos Santos Braumann
MaRDI profile type: Publication
cites work: Oriented percolation in two dimensions
cites work: Q3935985
cites work: Ergodic theorems for weakly interacting infinite systems and the voter model
cites work: Diffusions on an infinite dimensional torus
cites work: Ergodic properties of the equilibrium measure of the stepping stone model in population genetics
cites work: Q5185817
cites work: Convergence of stochastic processes
cites work: Limit diffusions of some stepping-stone models
cites work: Results for the stepping stone model, for migration in population genetics
cites work: An interacting system in population genetics. I
cites work: Infinite dimensional stochastic differential equations and their applications
cites work: Continuous time multi-allelic stepping stone models in population genetics
cites work: The Existence of Probability Measures with Given Marginals
cites work: Criteria for classifying general Markov chains
cites work: Spatial growth of a branching process of particles living in \(R^ n\).
full work available at URL: https://doi.org/10.1007/bf01845994
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4047485