Rescaling limits of the spatial Lambda-Fleming-Viot process with selection

Abstract

We consider the spatial $\Lambda $-Fleming-Viot process model for frequencies of genetic types in a population living in $\mathbb {R}^{d}$, with two types of individuals ($0$ and $1$) and natural selection favouring individuals of type $1$. We first prove that the model is well-defined and provide a measure-valued dual process encoding the locations of the “potential ancestors” of a sample taken from such a population, in the same spirit as the dual process for the SLFV without natural selection [7]. We then consider two cases, one in which the dynamics of the process are driven by purely “local” events (that is, reproduction events of bounded radii) and one incorporating large-scale extinction-recolonisation events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial $\Lambda $-Fleming-Viot processes indexed by $n$, and we assume that the fraction of individuals replaced during a reproduction event and the relative frequency of events during which natural selection acts tend to $0$ as $n$ tends to infinity. We choose the decay of these parameters in such a way that when reproduction is only local, the measure-valued process describing the local frequencies of the less favoured type converges in distribution to a (measure-valued) solution to the stochastic Fisher-KPP equation in one dimension, and to a (measure-valued) solution to the deterministic Fisher-KPP equation in more than one dimension. When large-scale extinction-recolonisation events occur, the sequence of processes converges instead to the solution to the analogous equation in which the Laplacian is replaced by a fractional Laplacian (again, noise can be retained in the limit only in one spatial dimension). We also consider the process of “potential ancestors” of a sample of individuals taken from these populations, which we see as (the empirical distribution of) a system of branching and coalescing symmetric jump processes. We show their convergence in distribution towards a system of Brownian or stable motions which branch at some finite rate. In one dimension, in the limit, pairs of particles also coalesce at a rate proportional to their collision local time. In contrast to previous proofs of scaling limits for the spatial $\Lambda $-Fleming-Viot process, here the convergence of the more complex forwards in time processes is used to prove the convergence of the dual process of potential ancestries.