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, [23]) and DURRETT and FAN (2016, [13]), in numerical experiments with a closely related model based on the classical Moran model.
Highlights
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
We shall suppose that our population occurs in just two types, {a, A} and that the environment fluctuates between two states, in the first of which a, and in the second of which A, is favoured
We suppose that selection is sufficiently strong that if the environment did not fluctuate, the favoured type would rapidly fix in the population, but that there is a ‘balance’ between the two environments so that both types can be maintained at non-trivial frequencies for long periods of time
Summary
We outline the biological context for this work. not a prerequisite for understanding the mathematics of subsequent sections, it explains our motivation for tackling this particular scaling limit. He assumes that selection is strong enough that the duration of the sweeps causing the hitchhiking events that affect a given locus is small compared to the time between them so that we can ignore the possibility that a locus will be subject to two simultaneous hitchhiking events He establishes that the first two moments of the change in allele frequency at the neutral locus over the course of a hitchhiking event take exactly the same form as if they had been produced by genetic drift over a single generation of reproduction. In the absence of spatial structure, allele frequency dynamics under fluctuating selection are identical to those under within-generation fecundity variance polymorphism In this setting, TAYLOR (2013, [47]) shows that the effects on the genealogy at a linked neutral locus will differ. Our work here is a step towards a tractable framework in which to consider the combined effects of spatial and temporal fluctuations
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