Stochastic measure-valued models for populations expanding in a continuum

Abstract

We model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-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 → +∞, the limiting process, corresponding to the ∞-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 ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. 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 ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.

Latent mutations in the ancestries of alleles under selection

“…Spatially heterogeneous populations in which reproduction rates, death rates, mutation rates and selection strength can depend both on spatial position and local population density present challenges. This is because the population dynamics now take place in high or infinite dimension (Hallatschek and Nelson, 2008;Barton et al, 2010;Durrett and Fan, 2016;Louvet and Véber, 2023;Etheridge et al, 2023). For example, the spatial version of (1), the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation introduced by Shiga (1988), is a stochastic partial differential equation that arises as the scaling limit of various discrete models under weak selection (Müller and Tribe, 1995;Durrett and Fan, 2016;Fan, 2021).…”