The spatial Lambda-Fleming-Viot process with fluctuating selection

TL;DR

This paper models populations with fluctuating environmental selection using Lambda-Fleming-Viot processes, deriving stochastic differential equations and SPDEs to describe genetic variation dynamics in spatial and non-spatial settings.

Contribution

It introduces a novel scaling limit for Lambda-Fleming-Viot processes in fluctuating environments, extending to spatial models and deriving associated stochastic PDEs.

Findings

Scaling limit as a stochastic differential equation in non-spatial case

Derivation of a stochastic PDE for spatial populations in higher dimensions

Environmental fluctuations induce stochasticity in the spatial model, even when genetic drift diminishes.

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.