Clines With Partial Panmixia In An Environmental Pocket

Last modified: 2014-04-01

#### Abstract

In geographically structured populations, global panmixia (i.e., random mating) can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an isotropic environmental pocket in $n$ dimensions is investigated. Migration and selection are both weak; the former is homogeneous and isotropic; the latter is directional. If the scaled panmictic rate $\beta \geq 1$, then the allele favored in the pocket is ultimately lost. For $\beta <1$, 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 $a_2$. As expected intuitively, the cline becomes more difficult to maintain, i.e., the critical radius $a_n$ increases for $n=1,2,3,\ldots$, as $\alpha$, $\beta$, or $n$ increases.