The impact of selection in the \Lambda-Wright-Fisher model

TL;DR

This paper investigates how selection influences the long-term behavior of allele frequencies in the b5-Wright-Fisher model, revealing conditions under which disadvantaged alleles can be eliminated, contrasting with classical models.

Contribution

It provides new asymptotic results for the b5-Wright-Fisher process with selection, especially under specific conditions on the measure b5 and the selection coefficient.

Findings

Disadvantaged alleles can vanish asymptotically under certain selection pressures.

The behavior differs from classical Wright-Fisher diffusion where such extinction does not occur.

Martingale methods are used to analyze the dual process and prove the results.

Abstract

The purpose of this article is to study some asymptotic properties of the \Lambda-Wright-Fisher process with selection. This process represents the frequency of a disadvantaged allele. The resampling mechanism is governed by a finite measure \Lambda on [0,1] and the selection by a parameter \alpha. When the measure \Lambda verifies \int_0^1-\log(1-x)x^{-2} \Lambda(dx)<\infty, some particular behaviours in the frequency of the allele can occur. The selection coefficient \alpha may be large enough to compensate the random genetic drift. In other words, for certain selection pressure, the disadvantaged allele will vanish asymptotically. This phenomenon cannot occur in the classical Wright-Fisher diffusion. We study the dual process of the \Lambda-Wright-Fisher process with selection and prove this result through martingale arguments.