  Predators may attack isolated or grouped prey in a cooperative, collective
way. Whether a gregarious behavior is advantageous to each species depends on
several conditions and game theory is a useful tool to deal with such a
problem. We here extend the Lett-Auger-Gaillard model [Theor. Pop. Biol. 65
(2004) 263] to spatially distributed populations and compare the resulting
behavior with their mean-field predictions for the coevolving densities of
predator and prey strategies. Besides its richer behavior in the presence of
spatial organization, we also show that the coexistence phase in which
collective and individual strategies for each group are present is stable
because of an effective, cyclic dominance mechanism similar to a well-studied
generalization of the Rock-Paper-Scissors game with four species, a further
example of how ubiquitous this coexistence mechanism is.
