Population switching under a time-varying environment

We study the switching dynamics of a stochastic population subjected to a deterministically time-varying environment. Our approach is demonstrated on a problem of population establishment, which is important in ecology. At the deterministic level, the model we study gives rise to a critical population size beyond which the system experiences establishment. Notably the latter has been shown to be strongly influenced by the interplay between demographic and environmental variations. Here we consider two prototypical examples of a time-varying environment: a temporary change in the environment, and a periodically varying environment. By employing a semiclassical approximation we compute, within exponential accuracy, the change in the establishment probability and mean establishment time of the population, due to the environmental variability. Our analytical results are verified by using a modified Gillespie algorithm which accounts for explicitly time-dependent reaction rates. Importantly, our theoretical approach can also be useful in studying switching dynamics in gene regulatory networks under external variations.