Abstract
We consider the spatial LambdaFlemingViot 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 welldefined and provide a measurevalued 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 largescale extinction/recolonisation events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial LambdaFlemingViot 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 measurevalued process describing the local frequencies of the less favoured type converges in distribution to a (measurevalued) solution to the stochastic FisherKPP equation in one dimension, and to a (measure valued) solution to the deterministic FisherKPP equation in more than one dimension. When largescale extinctionrecolonisation 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 LambdaFlemingViot 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.
Original language  English 

Journal  Electronic Journal of Probability 
Publication status  Accepted/In press  6 Sep 2020 
Keywords
 Generalised FlemingViot process
 natural selection
 limit theorems
 duality
 symmetric stable processes
 population genetics
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Profiles

Dr Feng Yu
 Statistical Science
 Probability, Analysis and Dynamics
 School of Mathematics  Senior Lecturer in Statistics
 Probability
Person: Academic , Member