Diffusion model for genotype-dependent migration

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A diffusion model for the joint action of genotype-dependent migration and genotype-independent population regulation is derived and investigated. The diploid, monoecious population occupies a finite chain of equally spaced colonies; generations are discrete and nonoverlapping; mating is random in each colony. A boundary-value problem is deduced for the gene frequencies at a multiallelic locus in the absence of mutation, selection, and random drift. This problem is studied for two alleles and constant and uniform population density and drift and diffusion coefficients. All equilibria are shown to be monotone, and explicit conditions for a protected polymorphism are established. Two examples of asymptotically stable clines are presented. It is demonstrated that genotype dependence of the mean displacements is necessary to produce or maintain spatial differentiation of gene frequencies.

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