Space promotes the coexistence of species: Effective medium approximation for rock-paper-scissors system

- Lattice model of rock-paper-scissors game evolves to a stable state.
- The previous theories cannot explain the stable dynamics over a long time.
- We apply effective medium approximation in a self-consistent way.
- This theory well predicts the stable dynamics and aggregated formation.

Stochastic cellular automata for rock-paper-scissors games are related to Lotka-Volterra model. Simulations are usually performed by two methods local and global interactions. It is well known that the population dynamics with local interaction is stable, where all species coexist. In contrast, global interaction leads to extinction. So far, theories such as mean-field theory and pair approximation have been presented, but they never explained the stable dynamics in local simulation. In the present article, we apply effective medium approximation (EMA) which has been developed in Physics. The effective medium is determined in a self-consistent way. The EMA theory well predicts the stability of population dynamics. Moreover, it fairly explains the aggregation of each species observed in the stationary state.