In many cases, the population can reach absorbing states such as genetic fixation or population extinction, which are inevitable in the presence of fluctuations.

Estimating the mean time to reach these states is of great importance in evolution and ecology.

Here we look at a canonical setting, of a natural population residing in two locations (patches) coupled by migration, and where the environmental conditions fluctuate. Chances of extinction in the simpler problem of a single patch have been derived long ago and are a fundamental result in the field. For multiple coupled patches, the problem has thus far remained open.

We find that, unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path (the instanton). We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time.