Self-Organization of Mobile Populations in Cyclic Competition

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 `rockpaper-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 in uence 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.