Model reduction methods for population dynamics with fast-switching environments: reduced master equations, stochastic differential equations, and applications

We study stochastic population dynamics coupled to fast external environments, and combine expansions in the inverse switching rate of the environment, and a Kramers{Moyal expansion in the inverse size of the population. This leads to a series of approximation schemes, capturing both intrinsic and environmental noise. These methods provide a means of efficient simulation and we show how they can be used to obtain analytical results for the fluctuations of population dynamics in switching environments. We place the approximations in relation to existing work on piecewise-deterministic and piecewise-diffusive Markov processes. Finally, we demonstrate the accuracy and efficiency of these model-reduction methods in different research fields, including systems in biology and a model of crack propagation.