Asymptotic Properties for Kimura’s Diffusion Model with Altruistic Allele

We study the diffusion model for a random mating diploid species in population genetics admitting intergroup selection, which was recently proposed by M. Kimura. We first enumerate the stationary distributions for the relative gene frequencies of the altruistic allele and inspect their stability. According to the regions of the parameters, we divide the model into seven cases. 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. But in the other four cases, we can find one or infinitely many stationary distributions with density functions. Under a mild assumption, we also show that, for any initial distribution, the distribution of the gene frequencies goes to one of the stationary distributions as time goes to infinity. We next study how the data affect the distribution of the gene frequencies. Actually, the moment sequence of the distribution depends on the data monotonically. As a result, we verify one of the main results of M. Kimura, which gives a criterion for predominance of the altruistic allele or the other allele by a simple index D = c/m − 4Ns.