On the boundary classification of $\Lambda$-Wright-Fisher processes with frequency-dependent selection

TL;DR

This paper extends $ ext{Lambda}$-Wright-Fisher processes with frequency-dependent selection beyond their boundary, establishing dualities with fragmentation-coalescence processes and analyzing boundary behaviors and new phenomena in allele dynamics.

Contribution

It introduces extensions of $ ext{Lambda}$-WF processes with selection beyond the boundary, revealing duality relationships and boundary behaviors under various conditions.

Findings

Boundary $1$ can be an exit or entrance boundary depending on selection strength.

Conditions for excursions out from boundary $1$ before absorption at $0$ are established.

New phenomena where deleterious alleles spread instantaneously before vanishing are identified.

Abstract

We construct extensions of the pure-jump $\Lambda$-Wright-Fisher processes with frequency-dependent selection ($\Lambda$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coalescence processes (EFC). One-to-one correspondences between the nature of the boundary $1$ of the $\Lambda$-WF process with selection and the boundary $\infty$ of the block counting process are established. New properties for the $\Lambda$-WF processes with selection and the block counting processes of the simple EFC processes are deduced from these correspondences. Some conditions are provided for the selection to be either weak enough for boundary $1$ to be an exit boundary or strong enough for $1$ to be an entrance boundary. When the measure $\Lambda$ and the selection mechanism satisfy some regular variation properties, conditions are found in order that the extended $\Lambda$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In the latter process, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele which can spread into the population in a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.