Duality and Fixation in $\Xi$-Wright-Fisher processes with frequency-dependent selection

TL;DR

This paper introduces a generalized population model with frequency-dependent selection and skewed offspring distribution, analyzing its scaling limits and ancestral processes through duality, and characterizing extinction probabilities.

Contribution

It develops a new discrete-time model incorporating potential parental selection and derives its continuous limits using duality methods.

Findings

Weak convergence to a $ ext{Xi}$-Fleming-Viot process with frequency-dependent selection

Characterization of ancestral processes as branching-coalescing processes with multiple collisions

Extinction probabilities linked to ancestral process ergodicity

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 $\textit{potential}$ 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 sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing 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.