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.
%0 Generic
%1 etheridge2014rescaling
%A Etheridge, Alison
%A Veber, Amandine
%A Yu, Feng
%D 2014
%K Fisher-KPP SLFV branching_Brownian_motion scaling_limits selection spatial_coalescent
%T Rescaling limits of the spatial Lambda-Fleming-Viot process with
selection
%U http://arxiv.org/abs/1406.5884
%X 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.
@misc{etheridge2014rescaling,
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.},
added-at = {2016-03-25T19:27:49.000+0100},
author = {Etheridge, Alison and Veber, Amandine and Yu, Feng},
biburl = {https://www.bibsonomy.org/bibtex/27308bb150a3a9c3fa9cdfc879e733c0e/peter.ralph},
interhash = {014949bea794ad5ec788a7a26ca1a032},
intrahash = {7308bb150a3a9c3fa9cdfc879e733c0e},
keywords = {Fisher-KPP SLFV branching_Brownian_motion scaling_limits selection spatial_coalescent},
note = {cite arxiv:1406.5884},
timestamp = {2016-03-25T19:27:49.000+0100},
title = {Rescaling limits of the spatial {Lambda}-{Fleming}-{Viot} process with
selection},
url = {http://arxiv.org/abs/1406.5884},
year = 2014
}