  We study the influence of a randomly switching reproduction-predation rate on
the survival behavior of the non-spatial cyclic Lotka-Volterra model, also
known as the zero-sum rock-paper-scissors game, used to metaphorically describe
the cyclic competition between three species. In large and finite populations,
demographic fluctuations (internal noise) drive two species to extinction in a
finite time, while the species with the smallest reproduction-predation rate is
the most likely to be the surviving one ("law of the weakest"). Here, we model
environmental (external) noise by assuming that the reproduction-predation rate
of the "strongest species" (the fastest to reproduce/predate) in a given static
environment randomly switches between two values corresponding to more and less
favorable external conditions. We study the joint effect of environmental and
demographic noise on the species survival probabilities and on the mean
extinction time. In particular, we investigate whether the survival
probabilities follow the law of the weakest and analyze their dependence of the
external noise intensity and switching rate. Remarkably, when, on average,
there is a finite number of switches prior to extinction, the survival
probability of the predator of the species whose reaction rate switches
typically varies non-monotonically with the external noise intensity (with
optimal survival about a critical noise strength). We also outline the
relationship with the case where all reaction rates switch on markedly
different time scales.
