The spatial Lambda-Fleming-Viot process with fluctuating selection

Abstract

We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population.We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space.We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from Hallatschek and Nelson (2008) and Durrett and Fan (2016), in numerical experiments with a closely related model based on the classical Moran model.

Genealogies in bistable waves

“…[DK99]). The idea of using these markers, or 'tracers', in the context of expanding biological populations goes back at least to Hallatschek and Nelson [HN08], and has subsequently been used, for example, by Durrett and Fan [DF16], Birzu et al [BHK18] and Biswas et al [BEK18]. The idea is that at some time t 0 , a subset of the type A individuals are labelled as 'tracers'.…”

Spatial Population Models

“…The forms of selection that we have considered all assume a constant environment. In Biswas, Etheridge & Klimek [12], we consider a population in which there are two genetic types, and which one is favoured depends on both space and time. In a regime in which selection fluctuates rapidly relative to the timescale of evolution, a scaling limit is obtained which is stochastic in all dimensions.…”

Multidimensional $Λ$-Wright-Fisher processes with general frequency-dependent selection

“…Multitype models are widespread in the literature, both in population genetics (see for example [7,21,45,46]) and in eco-evolutionary models (for instance [15,17,14,9,10]). There are also interesting instances of colouring (2-types) models in which individuals observe several potential parents and are coloured according to some rule depending on the observed sample [8,18,28] and questions regarding the existence and (lack of) uniqueness of a colouring rule that leads to a prescribed stochastic differential equation are addressed in [18]. The literature that accounts for mutations is bast [23,22,3].…”