The spatial Lambda-Fleming-Viot process with fluctuating selection

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 LambdaFleming-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.