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Globally synchronized oscillations in complex cyclic games

The Rock-Paper-Scissors (RPS) game and its generalizations with ${\cal S}>3$ species are well studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one predator, one prey symmetry. Lütz {\it et al} (J. Theor. Biol. {\bf 317} (2013) 286) have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food web subloops (if any). Here we show that upon replacing a minimum fraction $Q_c$ of the short range interactions by long range ones, there is a Hopf bifurcation and global oscillations become stable. Surprisingly, to build such long distance, global synchronization, the four species coexistence phase requires less long range interactions than the three species phase, while one would naively expect the contrary. Moreover, deviations from highly homogeneous conditions ($χ=0$ or 1) increase $Q_c$ and the more heterogeneous is the food web, the harder the synchronization is. By further increasing $Q$, while the three species phase remains stable, the four species one has a transition to an absorbing, single species state. The existence of a phase with global oscillations for ${\cal S}>3$, when the interaction graph has multiple subloops and several possible local cycles, lead to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs.