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

TL;DR

This paper extends cyclic competition models to four species, analyzing mean field dynamics and stochastic effects, revealing complex behaviors, alliance formations, and extinction probabilities, with implications for understanding multispecies interactions.

Contribution

It introduces a four-species cyclic competition model with alliance pairs and derives mean field equations predicting complex dynamics and survival outcomes.

Findings

Partner-pair alliances dominate in finite stochastic systems.

Survival depends on the product of species' interaction rates.

Rich extinction probabilities and population distributions are observed.

Abstract

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.