We study a variant of the cyclic Lotka-Volterra model with three-agent
interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors
game, the model describes an ideal ecosystem in which cyclic competition among
three species develops through cooperative predation. Its rate equations in a
well-mixed environment display a degenerate Hopf bifurcation, occurring as
reactions involving two predators plus one prey have the same rate as reactions
involving two preys plus one predator. We estimate the magnitude of the
stochastic noise at the bifurcation point, where finite size effects turn
neutrally stable orbits into erratically diverging trajectories. In particular,
we compare analytic predictions for the extinction probability, derived in the
Fokker-Planck approximation, with numerical simulations based on the Gillespie
stochastic algorithm. We then extend the analysis of the phase portrait to
heterogeneous rates. In a well-mixed environment, we observe a continuum of
degenerate Hopf bifurcations, generalizing the above one. Neutral stability
ensues from a complex equilibrium between different reactions. Remarkably, on a
two-dimensional lattice, all bifurcations disappear as a consequence of the
spatial locality of the interactions. In the second part of the paper, we
investigate the effects of mobility in a lattice metapopulation model with
patches hosting several agents. We find that strategies propagate along the
arms of rotating spirals, as they usually do in models of cyclic dominance. We
observe propagation instabilities in the regime of large wavelengths. We also
examine three-agent interactions inducing nonlinear diffusion.Comment: 22 pages, 13 figures. v2: version accepted for publication in EPJ
