Stochastic measure-valued models for populations expanding in a continuum

TL;DR

This paper introduces and rigorously constructs stochastic measure-valued models for populations expanding in a continuum, analyzing the limit of models with increasing parental complexity and linking population genetics with stochastic growth theories.

Contribution

It provides a new rigorous construction of the $ abla$-parent SLFV as a limit of k-parent models, connecting population genetics and percolation-based growth models.

Findings

The $ abla$-parent SLFV is the limit of k-parent SLFVs as k approaches infinity.

Alternative coupling construction for SLFVs with different selection strengths.

Three characterizations of the $ abla$-parent SLFV linking genetics and growth models.

Abstract

We model spatially expanding populations by means of two spatial $\Lambda$-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the $\infty$-parent SLFV. In order to do so, we fill empty areas with type 0 ''ghost'' individuals with a strong selective disadvantage against ''real'' type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k $\rightarrow$ +$\infty$, the limiting process, corresponding to the $\infty$-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the $\infty$-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k $\rightarrow$ +$\infty$. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the $\infty$-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.