Misc,
Rescaling limits of the spatial Lambda-Fleming-Viot process with selection
(2014)cite arxiv:1406.5884.
Abstract
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 with selection is well-defined. Next, we consider two cases, one
in which the dynamics of the process are driven by purely local events and one
incorporating large-scale extinction-recolonisation events. 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 the frequency of the less favoured type converges in distribution to
the solution to the Fisher KPP-equation (with noise in one dimension) when
reproduction is only local, or to the solution to an analogous equation in
which the Laplacian is replaced by a fractional Laplacian when large-scale
extinction-recolonisation events occur. We also define the process of potential
ancestors of a sample of individuals taken from these populations, and 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 local time together.
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 potential
ancestries.
