Clines with partial panmixia across a geographical barrier

In geographically structured populations, partial global panmixia can be regarded as the limiting case of long-distance migration. In the presence of a geographical barrier, an exact, discrete model for the evolution of the gene frequencies at a multiallelic locus under viability selection, local adult migration, and partial panmixia is formulated. For slow evolution, from this model a spatially unidimensional continuous approximation (a system of integro-partial differential equations with discontinuities at the barrier) is derived. For (i) the step-environment, (ii) homogeneous, isotropic migration on the entire line, and (iii) two alleles without dominance, an explicit solution for the unique polymorphic equilibrium is found. In most natural limiting cases, asymptotic expressions are obtained for the gene frequencies on either side of the barrier.