  Isolated populations ultimately go extinct because of the intrinsic noise of
elementary processes. In multi-population 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
sub-population sizes in the coexistence state, we compare the probabilities of
each of the two extinction routes and predict the most likely path of the
sub-populations 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.
