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The survival of natural populations may be greatly affected by environmentalconditions that vary in space and time. We look at a population residing in twolocations (patches) coupled by migration, in which the local conditionsfluctuate in time. We report on two findings. First, we find that unlike rareevents in many other systems, here the histories leading to a rare extinctionevent 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 theDonsker-Varadhan formalism which treats rare events of atypical averages over along time. It provides a detailed description of the statistics of historiesleading to the rare event. The framework applies to rare events in a broadclass of systems driven by non-Gaussian noise. Secondly, applying thisframework to the population-dynamics model, we find a novel phase transition inits extinction behavior. Strikingly, a patch which is a sink (where individualsdie more than are born), can nonetheless reduce the probability of extinction,even if it normally lowers the population's size and growth rate.

Extinctions of coupled populations, and rare-event dynamics under non-Gaussian noise