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

Abstract

We consider the spatial Λ-Fleming-Viot process model for frequencies of genetic types in a population living in R d , with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We first prove that the model is well-defined and provide a measure-valued dual process encoding the locations of the "potential ancestors" of a sample taken from such a population, in the same spirit as the dual process for the SLFV without natural selection [7]. We then consider two cases, one in which the dynamics of the process are driven by purely "local" events (that is, reproduction events of bounded radii) and one incorporating large-scale extinctionrecolonisation events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial Λ-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 spatial Lambda-Fleming-Viot process with fluctuating selection

“…. , χ j 0 and, just as in ETHERIDGE, VÉBER and YU (2020, [15]) Section 1.2, this duality must be defined 'weakly', that is by integrating against a suitable test function ψ(χ 1 0 , . .…”

“…We suppose that the population, which is distributed across R d , is subdivided into two genetic types {a, A}. As explained in detail in ETHERIDGE, VÉBER and YU (2020, [15]), which in turn borrows results from VÉBER and WAKOLBINGER (2015, [49]), formally, at each time the state of the population is described by a measure M t on R d × K, where K = {a, A}, whose first marginal is Lebesgue measure on R d . At any fixed time there is a density w(t,…”

“…Existence of the SLFVFS is guaranteed by the methods of ETHERIDGE, VÉBER and YU (2020, [15]). Indeed, we could have taken different measures µ and ν r according to whether events are selective or neutral.…”

“…Also mirroring that setting, the resulting 'moment duality' is sufficient to guarantee uniqueness of the SLFVFS. Since we do not use the duality in what follows, we refer the reader to ETHERIDGE, VÉBER and YU (2020, [15]) for details.…”

“…We are interested in the effects of fluctuating selection over large spatial and temporal scales and so we shall consider a rescaling of our model. ETHERIDGE, VÉBER and YU (2020, [15]) consider the corresponding process in which selection does not fluctuate with time, but instead always favours type A (say). In that setting it is shown that if impact scales as u n =ū/n 1/3 and selection scales as s n = s/n 2/3 , then as n → ∞, w(nt, n 1/3 x) (or rather a local average of this quantity) converges in d ≥ 2 to the solution to the deterministic Fisher-KPP equation, and to the solution of the corresponding stochastic p.d.e.…”