Stationary solutions and their stability for Kimura's diffusion model with intergroup selection
type: Item
instance of: scholarly article
title: Stationary solutions and their stability for Kimura's diffusion model with intergroup selection
zbMATH Open document ID: 0645.92013
author: Yukio Ogura
author: Norio Shimakura
published in: Journal of Mathematics of Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: \textit{M. Kimura}'s [Proc. Natl. Acad. Sci. U.S.A. 80, 6317-6321 (1983; Zbl 0543.92013)] diffusion model for a random mating diploid species in population genetics admitting intergroup selection, with special reference to evolution of an altruistic trait, is studied. The model is described by a second order parabolic differential equation on the interval [0,1] with degenerate coefficients including the mean field. The existence of a unique solution being proved by the second author, J. Math. Kyoto Univ. 25, 775-788 (1985; Zbl 0615.92010) and by \textit{T. Shiga}, ibid. 27, 195-215 (1987), the present authors first enumerate the stationary distributions for the relative gene frequencies of the altruistic allele and then inspect their stability. The model is divided into seven cases according to the relations of the six parameters, representing the effective size of each deme, mutation rates, selection, migration and intergroup competition rates. In the three of them, there are only `trivial' stationary distributions, i.e. the relative gene frequency is equal to zero with probability one or equal to one, and in the other four cases, there are one or infinitely many stationary distributions with density functions. Finally, the monotone dependence of the stationary distribution of the gene frequencies on the parameters is obtained, proving one of the main results of M. Kimura, which gives a criterion for predominance of the altruistic allele or of the other allele by the sign of a simple index defined by the given parameters.
reviewed by: Yukio Ogura
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 4053415
zbMATH Keywords: random mating diploid species
zbMATH Keywords: population genetics
zbMATH Keywords: intergroup selection
zbMATH Keywords: evolution of an altruistic trait
zbMATH Keywords: second order parabolic differential equation
zbMATH Keywords: mean field
zbMATH Keywords: altruistic allele
zbMATH Keywords: stability
zbMATH Keywords: mutation rates
zbMATH Keywords: migration
zbMATH Keywords: competition rates
zbMATH Keywords: relative gene frequency
zbMATH Keywords: stationary distribution of the gene frequencies
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520719
OpenAlex ID: W4250277751
DOI: 10.1215/KJM/1250520719
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description: scientific article; zbMATH DE number 4053415