The effect of domain geometry and long-range dispersal on the motion of hybrid zones

Thesis

We consider two different population models that relate to hybrid zones. In both cases individuals in the population can be of one of three types: aa which may be fitter than AA, and both 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.

First, 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.

The second set of results regard the study of how hybrid zones are affected by long-range dispersal of the individuals in the population, assuming both homozygotes are equally fit. We ignore genetic drift and use a deterministic model, the fractional Allen-Cahn equation to study the population. We show not only that under a suitable rescaling the motion of hybrid zones converges to motion under mean curvature flow, but also that there is a family of possible rescalings that gives this result. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators, and use this coupling to quantify how the large jumps of the stable motion affect our main result.

Authors: Letter Restuccia, IP

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: probability; hybrid zones; population genetics; SLFV