Perturbative Field-Theoretical Analysis of Three-Species Cyclic Predator-Prey Models

We apply a perturbative field-theoretical analysis on the symmetric Rock-Paper-Scissors (RPS) model and the symmetric May-Leonard (ML) model, in which three species compete cyclically. Compared to the two-species Lotka-Volterra predator-prey (LV) model, according to numerical simulations, these cyclical models appear to be less affected by intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although as yet not observed in numerical simulations, we find that the RPS model acquires significant fluctuation-induced renormalizations in the perturbative regime, similar to the LV model. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS model, whereas the spontaneous emergence of spatio-temporal structures features prominently in the LV and the ML models.