  We study individual-based dynamics in finite populations, subject to randomly
switching environmental conditions. These are inspired by models in which genes
transition between on and off states, regulating underlying protein dynamics.
Similarly switches between environmental states are relevant in bacterial
populations and in models of epidemic spread. Existing piecewise-deterministic
Markov process (PDMP) approaches focus on the deterministic limit of the
population dynamics while retaining the randomness of the switching. Here we go
beyond this approximation and explicitly include effects of intrinsic
stochasticity at the level of the linear-noise approximation. Specifically we
derive the stationary distributions of a number of model systems, in good
agreement with simulations. This improves existing approaches which are limited
to the regimes of fast and slow switching.
