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J Math Biol. 1977 Dec 27;5(1):61-73.

Migration and mutation in stochastic models of gene frequency change. I. The island model.

Abstract

Migration has in the past been introduced deterministically into stochastic gene frequency models. Migration at rate m then reduces the between-population variability by a factor of (1 - m)2 each generation. We show that with stochastic migration, whether of fixed or variable numbers of individuals, a positive term delta m is added to the variance. As a result of the delta m term, the equilibrium value of the between-population variability is increased compared to the corresponding value for deterministic migration by a factor of approximately (1 - m)-2 for small m. An equivalent result is derived for mutation, using the infinite allele model for a single population. We show in addition that these results may be derived much more simply by use of identity-by-descent probability methods, but only if a modified definition of the probability of identity-by-descent is used, involving the sampling with instead of without replacement of pairs of genes from the population.

PMID:
753908
[Indexed for MEDLINE]
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