Zero-One Survival Behavior of Cyclically Competing Species

The coexistence of competing species is, due to unavoidable fluctuations, always transient. In this Letter, we investigate the ultimate survival probabilities characterizing different species in cyclic competition. We show that they often obey a surprisingly simple, though nontrivial behavior. Within a model where coexistence is neutrally stable, we demonstrate a robust zero-one law: When the interactions between the three species are (generically) asymmetric, the weakest species survives at a probability that tends to one for large population sizes, while the other two are guaranteed to extinction. We rationalize our findings from stochastic simulations by an analytic approach.