Stability and multi-population in a 3-state rock-paper-scissors game

Statistical physics provides many tools to treat ecological models far from equilibrium. One of them, mean field theory, is appropriate as an initial step to a more realistic investigation of a complex system. In particular, Lotka-Volterra predator-prey system was one of the first strategies employed to study a biological community, and later, inspired and originated the rock-paper-scissors game. In this paper, the game is analysed in the context of the mean field theory, in the pair approximation and also through the Monte Carlo technique. The emphasis is on the stability condition and the number of species in the stationary regime. According to the pair approximation approach there are three possibilities concerning the kind of stability: marginal stability, asymptotic stability and instability, shown in a 1D and 2D phase diagram. Moreover, the rock-paper-scissors game in a lattice is studied via Monte Carlo simulations. The main achievements are coarsening in 1D and a 2D phase diagram indicating the edges between single species concentration and multi-population states.