Martina Favero: Stochastic processes in population genetics under strong selection

Abstract

Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of genetic frequencies and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the sampling probability, a crucial quantity in inference. After a general introduction, we consider the asymptotic regime where the selective advantage of one genetic type grows to infinity, while the other parameters remain fixed. In this regime, we show that the Wright-Fisher diffusion can be approximated either by a Gaussian process or by independent continuous-state branching processes with immigration. While the first process becomes degenerate at stationarity, the latter does not and provides a simple, analytic approximation for the leading term of the sampling probability. We characterise all the remaining terms using another approach based on a recursion formula. Finally, we study the asymptotic behaviour of the ancestral graph and establish an asymptotic duality relationship between this and the diffusion. (Joint work with Paul Jenkins)