Stability and Evolution in Rock-Paper-Scissors Ecologies

Abstract

Four aspects of rock-paper-scissors ecosystems are considered. In the first, the effects of variations in the population's spatial structure are described. With a lattice-based spatial structure, three-species coexistence is unstable when dispersal is long-range, but becomes stable at a critical threshold as the dispersal distance is gradually reduced. A continuous-space model is constructed and is shown to reproduce the three-species stability of the lattice model when interactions are local. But unlike the lattice model, three-species coexistence can be stable in the continuous space model even when dispersal is long-range. In the second investigation, the pair approximation technique is applied to the rock-paper-scissors system. The resulting equations fail to predict the stability of the three-species state. A local structure approximation based on four neighbouring sites is shown to produce a more accurate result. The third contribution is the presentation of two models in which rock-paper-scissors is able to evolve from a simple two-species system. In both cases, a two-species competitive cellular automaton is augmented with individual variation in a phenotypic trait. In the first model, the trait determines the individual's investment in interspecific versus intraspecific competition, and in the second model, the trait determines the individual's investment in the production of an interspecific toxin versus its growth rate. When interactions are local, selection can cause a divergence in the values of both traits such that only individuals with extreme levels of the trait survive, while those with intermediate levels of the trait die out. After this divergence, the resulting polymorphic community competes in an intransitive rock-paper-scissors cycle. Finally, an assessment is made of the claim that competitive restraint in rock-paper-scissors systems is caused by selection for stable subcommunities in a process of community-level selection. It is argued that members of unstable subcommunities may be as fit or fitter than identical individuals in stable subcommunities, and that the invocation of the higher-level selective force does not add to the understanding of the process of competitive restraint. Measurements of community stability in cellular automata models of rock-paper-scissors are undertaken and these provide no evidence for a community-level selective force promoting restraint.