A central limit theorem for the spatial Lambda Fleming-Viot process with
selection
(2015)cite arxiv:1512.06163Comment: minor corrections and typos.Abstract
We study the evolution of gene frequencies in a population living in
$R^d$, modelled by the spatial Lambda Fleming-Viot process with
natural selection (Barton, Etheridge and Veber, 2010 and Etheridge, Veber and
Yu, 2014). We suppose that the population is divided into two genetic types,
$a$ and $A$, and consider the proportion of the population which is of type $a$
at each spatial location. If we let both the selection intensity and the
fraction of individuals replaced during reproduction events tend to zero, the
process can be rescaled so as to converge to the solution to a
reaction-diffusion equation (typically the Fisher-KPP equation, as in
Etheridge, Veber and Yu, 2014). We show that the rescaled fluctuations converge
in distribution to the solution to a linear stochastic partial differential
equation. Depending on whether offspring dispersal is only local or if large
scale extinction-recolonization events are allowed to take place, the limiting
equation is either the stochastic heat equation with a linear drift term driven
by space-time white noise or the corresponding fractional heat equation driven
by a coloured noise which is white in time. If individuals are diploid (i.e.
either $AA$, $Aa$ or $aa$) and if natural selection favours heterozygous ($Aa$)
individuals, a stable intermediate gene frequency is maintained in the
population. We give estimates for the asymptotic effect of random fluctuations
around the equilibrium frequency on the local average fitness in the
population. In particular, we find that the size of this effect - known as the
drift load - depends crucially on the dimension $d$ of the space in which the
population evolves, and is reduced relative to the case without spatial
structure.