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 an. For a step-environment without dominance, simple, explicit formulas are derived for a1 and a3; an equation with a unique solution and simple, explicit approximations are deduced for a2. 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 an increases for n=1,2,3,… as α, β, or n increases.