Rock-paper-scissors played within competing domains in predator-prey games

Abstract

We consider (N, r) games of prey and predation with N species and r < N prey and predators, acting in a cyclic way. Further basic reactions include reproduction, decay and diffusion over a one-or two-dimensional regular grid, without a hard constraint on the occupation number per site, so in a `bosonic' implementation. For special combinations of N and r and appropriate parameter choices we observe games within games, that is different coexisting games, depending on the spatial resolution. As a concrete and simplest example we analyze the (6,3) game. Once the players segregate from a random initial distribution, domains emerge, which effectively play a (2,1)game on the coarse scale of domain diameters, while agents inside the domains play (3,1) (rock-paper-scissors), leading to spiral formation with species chasing each other. The (2,1)-game has a winner in the end, so that the coexistence of domains is transient, while agents inside the remaining domain coexist, until demographic fluctuations lead to extinction of all but one species in the very end. This means that we observe a dynamical generation of multiple space and time scales with emerging re-organization of players upon segregation, starting from a simple set of rules on the smallest scale (that of the grid) and changed rules from the coarser perspective. These observations are based on Gillespie simulations. We discuss the deterministic limit derived from a van Kampen expansion. In this limit we perform a linear stability analysis and numerically integrate the resulting equations. The linear stability analysis predicts the number of forming domains, their composition in terms of species; it explains the instability of interfaces between domains, which drives their extinction; spiral patterns are identified as motion along heteroclinic cycles. The numerical solutions reproduce the observed patterns of the Gillespie simulations including even extinction events, so that the mean-field analysis here is very conclusive, which is due to the specific implementation of rules.