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

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Assume a diploid species consisting of an infinite number of competing demes, each having Ne 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 ϕ = ϕ(x;t) be the distribution function of x among demes in the species at time t. Then, we have ∂ϕ/∂t = L(ϕ) + c(x — ¯x)ϕ, where L is the Kolmogorov forward differential operator commonly used in population genetics [i.e., L = (1/2) (∂2/∂x2)Vδx — (∂/∂x)Mδx], and Mδx and Vδx 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 — 4Nes′ 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.

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