Multiple extinction routes in stochastic population models

Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multipopulation systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time, but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first, causing a rapid extinction of the predators. Assuming large subpopulation sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the subpopulations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state, and the empty state.