Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species.

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.