  Intransitivity is a property of connected, oriented graphs representing
species interactions that may drive their coexistence even in the presence of
competition, the standard example being the three species Rock-Paper-Scissors
game. We consider here a generalization with four species, the minimum number
of species allowing other interactions beyond the single loop (one predator,
one prey). We show that, contrary to the mean field prediction, on a square
lattice the model presents a transition, as the parameter setting the rate at
which one species invades another changes, from a coexistence to a state in
which one species gets extinct. Such a dependence on the invasion rates shows
that the interaction graph structure alone is not enough to predict the outcome
of such models. In addition, different invasion rates permit to tune the level
of transitiveness, indicating that for the coexistence of all species to
persist, there must be a minimum amount of intransitivity.
