Graph topology determines coexistence in the rock-paper-scissors system

Species survival in the $(3, 1)$ May-Leonard system is determined by the mobility, with a critical mobility threshold between long-term coexistence and extinction. We show experimentally that the critical mobility threshold is determined by the topology of the graph, with the critical mobility threshold of the periodic lattice twice that of a topological 2-sphere. We use topological techniques to monitor the evolution of patterns in various graphs and estimate their mean persistence time, and show that the difference in critical mobility threshold is due to a specific pattern that can form on the lattice but not on the sphere. Finally, we release the software toolkit we developed to perform these experiments to the community.