Duality and fixation in Xi-Wright-Fisher processes with frequency-dependent selection

Abstract

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of potential parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the populations ancestral process. The scaling limits are, respectively, a two-types Xi-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branchingcoalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process ergodic properties.