Stationary solutions and their stability for Kimura's diffusion model with intergroup selection
\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.
second order parabolic differential equation
stationary distribution of the gene frequencies
scientific article; zbMATH DE number 4053415
scientific article