Extinction rates of established spatial populations

This work deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a refuge as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In the first class of models the zero population size is a repelling fixed point of the on-site deterministic dynamics. In the second class it is an attracting fixed point, corresponding to what is known in ecology as Allee effect. Assuming a large on-site population size, we develop WKB approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations are closely related to equations obtained, in a different way, by Elgart and Kamenev (2004). We classify possible regimes of population extinction with and without Allee effect and for different types of refuge and solve several examples analytically and numerically. For a very strong Allee effect the extinction problem can be mapped into the over-damped limit of theory of homogeneous nucleation due to Langer (1969). In this regime we predict an optimal refuge size that maximizes the mean time to extinction.

This is a joint work with Pavel Sasorov