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We study the influence of a randomly switching reproduction-predation rate onthe survival behavior of the non-spatial cyclic Lotka-Volterra model, alsoknown as the zero-sum rock-paper-scissors game, used to metaphorically describethe cyclic competition between three species. In large and finite populations,demographic fluctuations (internal noise) drive two species to extinction in afinite time, while the species with the smallest reproduction-predation rate isthe most likely to be the surviving one ("law of the weakest"). Here, we modelenvironmental (external) noise by assuming that the reproduction-predation rateof the "strongest species" (the fastest to reproduce/predate) in a given staticenvironment randomly switches between two values corresponding to more and lessfavorable external conditions. We study the joint effect of environmental anddemographic noise on the species survival probabilities and on the meanextinction time. In particular, we investigate whether the survivalprobabilities follow the law of the weakest and analyze their dependence of theexternal noise intensity and switching rate. Remarkably, when, on average,there is a finite number of switches prior to extinction, the survivalprobability of the predator of the species whose reaction rate switchestypically varies non-monotonically with the external noise intensity (withoptimal survival about a critical noise strength). We also outline therelationship with the case where all reaction rates switch on markedlydifferent time scales.

Survival behavior in the cyclic Lotka-Volterra model with a randomly switching reaction rate