Lambda-lookdown model with selection

TL;DR

This paper introduces a stochastic model for population genetics with two types, incorporating selection and duality with the Lambda-coalescent, and analyzes fixation and asymptotic behavior in large populations.

Contribution

It formulates the infinite population Lambda-lookdown model with selection and establishes convergence to a stochastic differential equation, linking fixation to coalescent properties.

Findings

Type proportions converge to SDE solutions as population size grows.

Fixation occurs in finite time if the coalescent comes down from infinity.

Provides asymptotic results for the Bolthausen-Sznitman coalescent.

Abstract

The goal of this paper is to study the lookdown model with selection in the case of a population containing two types of individuals, with a reproduction model which is dual to the $\Lambda$-coalescent. In particular we formulate the infinite population "$\Lambda$-lookdown model with selection". When the measure $\Lambda$ gives no mass to 0, we show that the proportion of one of the two types converges, as the population size $N$ tends to infinity, towards the solution of a stochastic differential equation driven by a Poisson point process. We show that one of the two types fixates in finite time if and only if the $\Lambda$-coalescent comes down from infinity. We give precise asymptotic results in the case of the Bolthausen-Sznitman coalescent. We also consider the general case of a combination of the Kingman and the $\Lambda$-lookdown model.