Abstract

Evolutionary game theory, where the success of one species depends on what the others are doing, provides a promising framework to investigate the mechanisms allowing the maintenance of biodiversity. Experiments on microbial populations have shown that cyclic local interactions promote species coexistence. In this context, rock-paper-scissors games are used to model populations in cyclic competition.

After the survey of some inspiring experiments, I will discuss the subtle interplay between the individuals’ mobility and local interactions in two-dimensional rock-paper-scissors systems. This leads to the loss of biodiversity above a certain mobility threshold, and to the formation of spiralling patterns below that threshold. I will then discuss a generic rock-paper-scissors metapopulation model formulated on a two-dimensional grid of patches. When these have a large carrying capacity, the model’s dynamics is faithfully described in terms of the system’s complex Ginzburg-Landau equation suitably derived from a multiscale expansion. The properties of the ensuing complex Ginzburg-Landau equation are exploited to derive the system’s phase diagram and to characterize the spatio-temporal properties of the spiralling patterns in each phase. This enables us to analyse the spiral waves stability, the influence of linear and nonlinear diffusion, and the far-field breakup of the spiralling pattern.