Stochastic evolution of four species in cyclic competition

TL;DR

This paper investigates the stochastic dynamics of four species in cyclic competition, revealing diverse extinction scenarios and analyzing how system size, reaction rates, and initial conditions influence outcomes.

Contribution

It provides exact results and simulations to understand extinction processes in multi-species cyclic competition models, highlighting deviations from mean-field predictions near extinction events.

Findings

Stochastic evolution closely follows mean-field results for large N.

Extinction scenarios range from single species dominance to coexistence.

Deviations from mean-field are significant near extinction events.

Abstract

We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios, from single species domination to coexistence between non-interacting species. Using exact results and numerical simulations we discuss the temporal evolution of the system for different values of $N$, for different values of the reaction rates, as well as for different initial conditions. As expected, the stochastic evolution is found to closely follow the mean-field result for large $N$, with notable deviations appearing in proximity of extinction events. Different ways of characterizing and predicting extinction events are discussed.