May-Leonard System (1975)
ẋ₁ = x₁(1 - x₁ - αx₂ - βx₃)
ẋ₂ = x₂(1 - βx₁ - x₂ - αx₃)
ẋ₃ = x₃(1 - αx₁ - βx₂ - x₃)
Heteroclinic cycle when α+β=2
(May-Leonard, 1975)
Saddle equilibria: e₁=(1,0,0), e₂=(0,1,0), e₃=(0,0,1)
Connection: e₁→e₂→e₃→e₁ (RPS cycle)
Heteroclinic cycles arise when each species beats the next cyclically. Trajectories approach the boundary simplex in an intermittent pattern — spending increasingly long time near each vertex before being repelled. This "winner takes all temporarily" dynamic is seen in ecology, evolutionary game theory, and neuroscience.