Self-Organization of Mobile Populations in Cyclic Competition

TL;DR

This paper investigates how individual mobility influences the self-organized spatial patterns, such as spiral waves, in cyclic competition models like rock-paper-scissors, using stochastic PDEs to capture the complex dynamics.

Contribution

It introduces a quantitative stochastic PDE framework to analyze pattern formation in cyclic competition, linking microscopic interactions to macroscopic spiral wave structures.

Findings

Spiral waves emerge from stochastic dynamics in cyclic competition.

Wavelength and velocity of spirals are predicted by complex Ginzburg-Landau equations.

Stochastic effects lead to entangled, noisy pattern formations.

Abstract

The formation of out-of-equilibrium patterns is a characteristic feature of spatially-extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the `rock-paper-scissors' game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals' mobility on the spatial structures emerging in rock-paper-scissors games. We devise a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior - in the proper continuum limit - is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals' wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system.These findings and our methods have important applications for understanding the formation of noisy patterns, e.g., in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)-chemical reactions.