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

We consider the spatial Lambda-Fleming-Viot process model for frequencies of genetic types in a population living in Rd, 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. 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.