Reentrant phase transition in a predator-prey model

We numerically investigate the six-species predator-prey game in complex networks as well as in d-dimensional regular hypercubic lattices with d=1,2,...,6. The food-web topology of the six species contains two directed loops, each of which is composed of cyclically predating three species. As the mutation rate is lowered below the well-defined phase transition point, the Z(2) symmetry related with the interchange in the two loops is spontaneously broken, and it has been known that the system develops the defensive alliance in which three cyclically predating species defend each other against the invasion of other species. In the Watts-Strogatz small-world network structure characterized by the rewiring probability alpha, the phase diagram shows the reentrant behavior as alpha is varied, indicating a twofold role of the shortcuts. In d-dimensional regular hypercubic lattices, the system also exhibits the reentrant phase transition as d is increased. We identify universality class of the phase transition and discuss the proper mean-field limit of the system.