Heteroclinic Cycles — Rock-Paper-Scissors Lotka-Volterra

Cyclic dominance creates heteroclinic orbits connecting unstable fixed points — evolutionary cycling
Simplex phase portrait (x+y+z=1)

Parameters

Rock
Paper
Scissors

State

Rock x
Paper y
Scissors z
t0

Physics

With σ=1, δ=0: trajectories spiral outward along the heteroclinic cycle — a triangle of saddle connections. Each corner is an unstable equilibrium.

Adding δ>0 stabilizes the interior coexistence point. Mutation breaks the cycle, creating limit cycles.