Clines with partial panmixia in an environmental pocket

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines (i.e., asymptotically stable equilibria) maintained by migration and selection in an isotropic environmental pocket in n dimensions is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and isotropic; the latter is directional. If the scaled panmictic rate β ≥ 1 , then the allele favored in the pocket is ultimately lost. For β < 1 , a cline is maintained if and only if the scaled radius a of the pocket exceeds a critical value a n . For a step-environment without dominance, simple, explicit formulas are derived for a 1 and a 3 ; an equation with a unique solution and simple, explicit approximations are deduced for a 2 . The ratio of the selection coefficients outside and inside the pocket is − α . As expected intuitively, the cline becomes more difficult to maintain; i.e., the critical radius a n increases for n = 1 , 2 , 3 , … as α , β , or n increases.