Abstract
We consider the spatial Lambda-Fleming-Viot process model 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 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. We then consider two cases, one in which the dynamics of the process are driven by events of bounded radii and one incorporating large-scale 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. We also consider the process of `potential ancestors' of a sample of individuals taken from these populations, which we see as 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.
Highlights
We consider the spatial Λ-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
The limits obtained here assume that the local population densities are high, complementing results of [18, 19] which address the interaction of natural selection and genetic drift when local population densities are small
This intuitive idea of how the SLFV with fecundity selection should evolve suggests a natural choice of operator L on functions of the form (1.5), see (1.9), and we shall show in Theorem 1.2 that for any probability measure P on Mλ describing the law of the initial condition, the martingale problem for (L, P ) has a unique solution on the space of all measurable Mλ-valued paths
Summary
The main innovation in the SLFV is that reproduction in the population is based on a Poisson point process of events, rather than on individuals. In order to gain a feeling for the process, let us first give a non-rigorous description based on the two independent Poisson point processes of “neutral” and “selective” events mentioned above This intuitive idea of how the SLFV with fecundity selection should evolve suggests a natural choice of operator L on functions of the form (1.5), see (1.9), and we shall show in Theorem 1.2 that for any probability measure P on Mλ describing the law of the initial condition, the martingale problem for (L, P ) has a unique solution on the space of all measurable Mλ-valued paths. The second construction, which requires Condition (1.7), allows for the sort of selection considered here, again, the actual proof of existence is only provided in the neutral case
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