  The survival of natural populations may be greatly affected by environmental
conditions that vary in space and time. We look at a population residing in two
locations (patches) coupled by migration, in which the local conditions
fluctuate in time. We report on two findings. First, 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. 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. It provides a detailed description of the statistics of histories
leading to the rare event. The framework applies to rare events in a broad
class of systems driven by non-Gaussian noise. Secondly, applying this
framework to the population-dynamics model, we find a novel phase transition in
its extinction behavior. Strikingly, a patch which is a sink (where individuals
die more than are born), can nonetheless reduce the probability of extinction,
even if it normally lowers the population's size and growth rate.
