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MOTOO KIMURA, Evolution of an Altruistic Trait through Group Selection as Studied by the Diffusion Equation Method, Mathematical Medicine and Biology: A Journal of the IMA, Volume 1, Issue 1, 1984, Pages 1–15, https://doi.org/10.1093/imammb/1.1.1
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Abstract
A diffusion model is formulated which incorporates the process of group selection (i.e. interdeme competition) in addition to mutation, migration, individual selection, and random genetic drift. A condition is obtained for group selection to prevail over individual selection in the evolutionv of an altruistic trait. Let DK = c /( v + v ′ + m ) − 4 N e s ′, where v ′ and v are mutation rates to and from the altruistic allele (A′), m is the migration rate (assuming Wright's island model), s ′ is the selective disadvantage of A′ with respect to individual selection, Ne is the effective population size of each deme (group) and c is a positive constant such that a deme having A′ with frequency x has the advantage c ( x − x ¯) relative to the average deme. Then, group selection overrides individual selection if DK >0, while individual selection prevails if DK <0.