Evolution at a multiallelic locus under migration and uniform selection
The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The drift and diffusion coefficients may depend on position, but the selection coefficients do not. It is established that if $${\hat{p}}$$ is a uniform equilibrium point under pure selection, then $${\hat{p}}$$ is a migration-selection equilibrium, and that generically the introduction of migration does not change the stability of $${\hat{p}}$$ . It is also proved that if $${\hat{p}}$$ is a uniform, globally asymptotically stable, internal equilibrium point under pure selection, then the gene frequencies converge to $${\hat{p}}$$ when both migration and selection are present. Thus, in this case, after a sufficiently long time, there is no genetic indication of the spatial distribution of the population.