Cyclic competition of four species: mean field theory and stochastic evolution

Generalizing the cyclically competing three-species model (often referred to as the rock-paper-scissors game), we consider a simple system of population dynamics without spatial structures that involves four species. Unlike the previous model, the four form alliance pairs which resemble partnership in the game of Bridge. In a finite system with discrete stochastic dynamics, all but 4 of the absorbing states consist of coexistence of a partner-pair. From a master equation, we derive a set of mean field equations of evolution. This approach predicts complex time dependence of the system and that the surviving partner-pair is the one with the larger product of their strengths (rates of consumption). Simulations typically confirm these scenarios. Beyond that, much richer behavior is revealed, including complicated extinction probabilities and non-trivial distributions of the population ratio in the surviving pair. These discoveries naturally raise a number of intriguing questions, which in turn suggests a variety of future avenues of research, especially for more realistic models of multispecies competition in nature. Introduction. – Over the years evolutionary game theory and population dynamics have yielded important insights into biodiversity and the behavior of multispecies ecological systems [1–3]. The complexity of real-world systems makes a full understanding of their properties very difficult. For that reason, the study of simple model systems is extremely valuable, as the complete knowledge of these systems allows to identify generic features valid in the more realistic but also more complex situations. In this context multispecies models with cyclic competition constitute some of the simplest cases where coexistence and species extinction can be studied using techniques from statistical mechanics and from non-linear dynamics [3, 4]. Many recent investigations revealed a rich and complex behavior. In particular, for systems with three species (a.k.a. rock-paper-scissors game) [5–25], the results range from surprising survival/extinction probabilities in models with no spatial structure to pattern formation and mobility effects in one-and two-dimensional lattices. By contrast, far less is known for systems with more than three species 1. Frachebourg et al. [5, 6, 26] consid