An integro-PDE model from population genetics

We investigate an integro-partial differential equation that models the evolution of the frequencies for two alleles at a single locus under the joint action of migration, selection, and partial panmixia (i.e., global random mating). We extend previous analyses [T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol. 81 (2012) 45–68] on the maintenance of both alleles from conservative to arbitrary migration and prove the uniqueness and global asymptotic stability of the nontrivial equilibrium. For conservative migration, we show that increasing the rate of panmixia makes it harder to maintain the allele with the smaller average fitness in the population. In terms of the selection function, we estimate the dependence on the panmictic rate of the minimal value of the selection intensity for the persistence of the allele with the smaller average fitness. We also show that, at least in an average sense, increasing panmixia flattens the cline.