  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.
