On the effects of a wide opening in the domain of the (stochastic) Allen-Cahn equation and the motion of hybrid zones

Abstract

We are concerned with a special form of the (stochastic) Allen-Cahn equation, which can be seen as a model of hybrid zones in population genetics. Individuals in the population can be of one of three types; $aa$ are fitter than $AA$, and both are fitter than the $aA$ heterozygotes. The hybrid zone is the region separating a subpopulation consisting entirely of $aa$ individuals from one consisting of $AA$ individuals. We investigate the interplay between the motion of the hybrid zone and the shape of the habitat, both with and without genetic drift (corresponding to stochastic and deterministic models respectively). In the deterministic model, we investigate the effect of a wide opening and provide some explicit sufficient conditions under which the spread of the advantageous type is halted, and complementary conditions under which it sweeps through the whole population. As a standing example, we are interested in the outcome of the advantageous population passing through an isthmus. We also identify rather precise conditions under which genetic drift breaks down the structure of the hybrid zone, complementing previous work that identified conditions on the strength of genetic drift under which the structure of the hybrid zone is preserved. Our results demonstrate that, even in cylindrical domains, it can be misleading to caricature allele frequencies by one-dimensional travelling waves, and that the strength of genetic drift plays an important role in determining the fate of a favoured allele.