Three- and four-state rock-paper-scissors games with diffusion

TL;DR

This paper explores how diffusion affects species coexistence in generalized three- and four-state rock-paper-scissors models, revealing conditions for spiral pattern formation and extinction dynamics.

Contribution

It extends previous RPS models by analyzing the effects of diffusion and reaction rate variations on pattern formation and species coexistence.

Findings

Spiral formation occurs only without conservation of total density.

Fast diffusion can lead to species extinction.

Reaction rate variations do not significantly alter pattern formation.

Abstract

Cyclic dominance of three species is a commonly occurring interaction dynamics, often denoted the rock-paper-scissors (RPS) game. Such type of interactions is known to promote species coexistence. Here, we generalize recent results of Reichenbach et al. (e.g. Nature 448, 1046 (2007)) of a four-state variant of RPS. We show that spiral formation takes place only without a conservation law for the total density. Nevertheless, in general fast diffusion can destroy species coexistence. We also generalize the four-state model to slightly varying reaction rates. This is shown both analytically and numerically not to change pattern formation, or the effective wave length of the spirals, and therefore does not alter the qualitative properties of the cross-over to extinction.