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.