Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model

Cyclic dominance of species has been identified as a potential
mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A.
Riley, M. W. Feldman and B. J. M. Bohannan Nature 418, 171 (2002)]
and B. Kirkup and M. A. Riley Nature 428, 412 (2004)]. Through
analytical methods supported by numerical simulations, we address
this issue by studying the properties of a paradigmatic non-spatial
three-species stochastic system, namely, the "rock-paper-scissors"
or cyclic Lotka-Volterra model. While the deterministic approach
(rate equations) predicts the coexistence of the species resulting
in regular (yet neutrally stable) oscillations of the population
densities, we demonstrate that fluctuations arising in the system
with a finite number of agents drastically alter this picture and
are responsible for extinction: After long enough time, two of the
three species die out. As main findings we provide analytic
estimates and numerical computation of the extinction probability
at a given time. We also discuss the implications of our results
for a broad class of competing population systems.