Four- and three-state rock-paper-scissors games with long-range selection

Considering the effect of a long-range selection, we investigate the crossover to extinction of species and the oscillation behaviors in the four- and three-state models of the rock-paper-scissors (RPS) game extensively on a two-dimensional lattice. It is found that the selection in a long-range leads to the destruction of species coexistence in the four-state RPS game model but it cannot induce the extinction of species nearly and even promotes the coexistence of three species when a long-range migration occurs in the three-state RPS game model. Through the simulations of spatiotemporal patterns, it is found that there are two different mechanisms to result in the extinction of species in two models. In the four-state model, a phase separation occurs and there are some vortices where three species coexist and dominate each other cyclically. Through a long-range selection, the vortices are eliminated to cause the destruction of species coexistence. In the three-state model, the spreading of species takes a key role in the crossover to extinction. When one of the species spreads rapidly, a giant cluster forms and the system becomes unstable and sensitive to the stochastic noise and the destruction of the species coexistence takes place. As the selection range is not too large, it can inhibit the formation of the giant cluster and then promotes the coexistence probability in the three-state model