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

Abstract

We study the evolution of gene frequencies in a population living in R d , modelled by the spatial Λ-Fleming-Viot process with natural selection ([BEV10], [EVY14]). 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, [EVY14]). 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.AMS 2010 subject classifications. Primary: 60G57 60F05 60J25 92D10. Secondary: 60G15.2. Update q as in (6).Note that if we let w = |B(x, r)| −1 B(x,r) q t − (z) dz, then at a neutral reproduction event, P (k = a) = w and at a selective event, P (k = a) = w − F (w).Remark. The existence of a unique Ξ-valued process following these dynamics under condition (4) is proved in [EVY14, Theorem 1.2] in the special case F (w) = w(1 − w) (in the neutral case s = 0, this was done in [BEV10]). In our general case, the condition on w − F (w) allows us to define a dual process and hence prove existence and uniqueness in the same way as in [EVY14].We shall consider two different distributions µ for the radii of events, i) the fixed radius case : µ(dr) = δ R (dr) for some R > 0, ii) the stable radius case : µ(dr) = 1 r≥1 r d+α+1 dr for a fixed α ∈ (0, 2 ∧ d).

Rescaling limits of the spatial Lambda-Fleming-Viot process with selection

“…Notice also that the scaling of s n (relative to u n ) that leads to a nontrivial limit is independent of spatial dimension. In contrast, in [25], the authors consider a different scaling for the parameters and prove a similar convergence result and a central limit theorem, in which the order of magnitude and the limit of the fluctuations around the deterministic EJP 25 (2020), paper 120.…”

“…Variants of the SLFV that incorporate forms of natural selection already appear in a number of studies [6,17,18,19,25], but without a detailed discussion of the construction of the stochastic processes, or whether they are well-defined when the geographic space in which the population evolves is infinite. Our first contribution is to formulate and construct an SLFV with natural selection.…”

Spatial population models

“…Finally we mention that there are very different scaling regimes that one can consider. Forien & Penington [47] consider selection in favour of heterozygosity (the opposite of that considered in Section 6.5) in a regime in which selection is strong enough relative to genetic drift (governed by the impact of events) that the allele frequencies are maintained at an approximately constant intermediate frequency. They then prove a 'Central Limit Theorem' for the fluctuations about that limit and use it to investigate 'drift load', that is the loss of fitness in a population that is subject to selection as a result of genetic drift.…”

Stochastic measure-valued models for populations expanding in a continuum

“…The original version of the SLFV does not account for the presence of a selectively favoured genetic type, but it can be modified in order to incorporate selection : see [22] for different forms of fixed selection mechanisms, and [7,10,31] for ways to introduce fluctuating selection. Our approach will be based on a version of the SLFV with selection introduced in [22] and rigourously constructed in [19]. Most of the work on the SLFV with selection involved investigating scaling limits under different forms of weak selection (see also [16,17]).…”

“…We fill empty areas with type 0 "ghost" individuals, which have a strong selective disadvantage against "real" type 1 individuals. This model is a special case of the SLFV with selection introduced in [19,22] : natural selection acts during all reproduction events, and the fraction of individuals replaced during a reproduction event is constant equal to 1. Letting the selective advantage k of type 1 individuals over type 0 individuals grow to +∞, and without rescaling time nor space, we obtain a new model for expanding populations, the ∞-parent SLFV.This model is reminiscent of the Eden growth model [13], but with an associated dual process of potential ancestors, making it possible to investigate the genetic diversity in a population sample.…”