Three species play non-transitive competition (R beats S, S beats P, P beats R). The replicator dynamics on the simplex:
form a heteroclinic cycle — the trajectory spirals through each pure-strategy vertex, spending increasingly long near each before moving on. The cycle is neutrally stable under replicator dynamics but can be asymptotically stable under weak noise (population N < ∞) or certain perturbations. This contrasts with the Lotka-Volterra system where interior fixed points cycle forever. With asymmetry α ≠ 0, one species gains an edge and the cycle breaks — either spiraling inward (coexistence) or outward (extinction). Spatial RPS models produce traveling waves and spirals; the non-transitive competition maintains biodiversity in microbial communities (e.g., E. coli strains).