Electronic Journal of Probability

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

Raphaël Forien and Sarah Penington

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We study the evolution of gene frequencies in a population living in $\mathbb{R} ^d$, modelled by the spatial $\Lambda $-Fleming-Viot process with natural selection. 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). 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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 5, 68 pp.

Received: 9 March 2016
Accepted: 16 December 2016
First available in Project Euclid: 17 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60F05: Central limit and other weak theorems 60J25: Continuous-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60G15: Gaussian processes

generalised Fleming-Viot process population genetics limit theorems Fisher-KPP equation stochastic heat equation

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Forien, Raphaël; Penington, Sarah. A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection. Electron. J. Probab. 22 (2017), paper no. 5, 68 pp. doi:10.1214/16-EJP20. https://projecteuclid.org/euclid.ejp/1484622022

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