## Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Clines With Partial Panmixia In An Environmental Pocket
Linlin Su

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.