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

Alison Etheridge, Amandine Veber, Feng Yu

Published 2014-06-23, updated 2018-07-11Version 2

We consider the spatial Lambda-Fleming-Viot process model [BEV10] for frequencies of genetic types in a population living in R^d, with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We consider two cases, one in which the dynamics of the process are driven by purely `local' (fixed radius) events 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 dimensions. 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 define the process of `potential ancestors' of a sample of individuals taken from these populations, which takes the form 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 the local time at zero of their separation.