  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.
