Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

The rock–paper–scissors game is one of the main theoretical models in evolutionary game theory and has been used successfully to explain some observed phenomena in biology, economics and social science. In order to explore the influence of environmental noise on cyclic dominance in the rock–paper–scissors game, stochastic stability in a continuous-time dynamics of the game with random payoffs in pairwise interactions is investigated by using the stochastic stability theory of Itô’s stochastic differential equations. After deducing a stochastic replicator equation for the strategy frequencies in a symmetric version of the game, stochastic stability conditions for constant equilibrium states are obtained and stochastic simulations of the global dynamics are performed. The main results are that (1) none of the fixation states of the system can be stochastically stable; and (2) an increase in the noise level (or stochastic fluctuation intensity) can result in the loss of stochastic stability of the constant interior equilibrium. More importantly, the simulation results not only match the theoretical predictions but also show the appearance of a noise-induced quasi-heteroclinic cycle when the constant interior equilibrium loses its stochastic stability as the noise level increases.