Diffusion model of intergroup selection, with special reference to evolution of an altruistic character.

Assume a diploid species consisting of an infinite number of competing demes, each having N(e) reproducing members and in which mating is at random. Then consider a locus at which a pair of alleles A and A' are segregating, where A' is the "altruistic allele," which has selective disadvantage s' relative to A with respect to individual selection, but which is beneficial for a deme in competition with other demes; namely, a deme having A' with frequency x has the advantage c(x - x) relative to the average deme, where c is a positive constant and x is the average of x over the species. Let varphi = varphi(x;t) be the distribution function of x among demes in the species at time t. Then, we have partial differentialvarphi/ partial differentialt = L(varphi) + c(x - x)varphi, where L is the Kolmogorov forward differential operator commonly used in population genetics [i.e., L = (1/2) ( partial differential(2)/ partial differentialx(2))V(deltax) - ( partial differential/ partial differentialx)M(deltax)], and M(deltax) and V(deltax) stand for the mean and variance of the change in x per generation within demes. As to migration, assume Wright's island model and denote by m the migration rate per deme per generation. By investigating the steady state, in which mutation, migration, random drift, and intra- and interdeme selection balance each other, it is shown that the index D = c/m - 4N(e)s' serves as a good indicator for predicting which of the two forces (i.e., group selection or individual selection) prevails; if D > 0, the altruistic allele predominates, but if D < 0, it becomes rare and cannot be established in the species.