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Evolution of an altruistic trait through group selection as studied by the diffusion equation method. (English) Zbl 0611.92017
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 evolution of an altruistic trait.
Let $$D_ K=c/(v+v'+m)-4N_ es'$$, 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, $$N_ e$$ 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 $$c(x-\bar x)$$ relative to the average deme. Then, group selection overrides individual selection if $$D_ K>0$$, while individual selection prevails if $$D_ K<0$$.

##### MSC:
 92D15 Problems related to evolution 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)