Intransitivity and coexistence in four species cyclic games

TL;DR

This paper explores how intransitivity in four-species cyclic games influences species coexistence, revealing that invasion rates and graph structure critically determine whether all species persist or some go extinct.

Contribution

It introduces a four-species cyclic game model on a lattice, showing how invasion rates affect coexistence and intransitivity, extending understanding beyond the classic three-species Rock-Paper-Scissors model.

Findings

Transition from coexistence to extinction depends on invasion rates.

Interaction graph alone does not predict outcomes.

Intransitivity level influences species coexistence.

Abstract

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.