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 hypercubic lattices with $d=1,2,..., 6$. The interaction topology of the six species contains two loops, each of which is composed of cyclically predating three species. As the mutation rate $P$ is lowered below the well-defined phase transition point, the $Z_2$ symmetry related with the interchange of 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 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.