Population interaction structure and the coexistence of bacterial strains playing 'rock-paper-scissors'

The simplest example of non-transitive competition is the game rock-paper-scissors (RPS), which exhibits characteristic cyclic strategy replacement: paper beats rock, which in turn beats scissors, which in turn beats paper. In addition to its familiar use in understanding human decision-making, rock-paper-scissors is also played in many biological systems. Among other reasons, this is important because it potentially provides a mechanism whereby species- or strain coexistence can occur in the face of intense competition. Kerr et al. (2002, Nature 418: 171-174) use complementary experiments and simulations to show that RPS-playing toxic, resistant, and susceptible E. coli bacteria can coexist when interactions between the strains are spatially explicit. This raises the question of whether limited interactions associated with space are sufficient to allow strain coexistence, or whether space per se is crucial. I approach this question by extending the Kerr et al. model to include different (aspatial) population network structures with the same degree distributions as corresponding spatial lattice models. I show that the coexistence that occurs for some parameter combinations when simulated bacterial strains compete on lattices is absent when they compete on random regular graphs. Further, considering small-world networks of intermediate 'quenched randomness' between lattices and random regular graphs, I show that only small deviations from pure spatial interactions are sufficient to prevent strain coexistence. These results emphasize the explicit role of space, rather than merely limited interactions, as being decisive in allowing the coexistence of toxic, resistant, and susceptible strains in this model system.