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

TL;DR

This paper investigates how randomly switching environmental conditions influence species survival in the cyclic Lotka-Volterra model, revealing non-monotonic survival probabilities and the interplay of external and internal noise.

Contribution

It introduces a model with environmental noise switching reaction rates and analyzes its impact on species survival and extinction times, extending understanding of stochastic effects in ecological models.

Findings

Survival probability of the predator varies non-monotonically with external noise intensity.

The 'law of the weakest' holds under certain noise conditions.

Optimal survival occurs at a critical noise strength.

Abstract

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.