Lotka-Volterra Predator-Prey

Prey (x)

—

Predator (y)

—

Time

Model α (prey growth) = 1.0 β (predation) = 0.5 γ (predator death) = 0.75 δ (predator growth) = 0.25

■ Prey ■ Predator
— x-nullcline — y-nullcline
Equilibrium: (γ/δ, α/β)
Conservative H(x,y) = δx−γlnx+βy−αlny

The classic Lotka-Volterra equations ẋ = αx−βxy, ẏ = δxy−γy produce neutral cycles around the equilibrium (γ/δ, α/β). The conserved quantity H = δx−γln x+βy−αln y foliates phase space into closed orbits — not a stable limit cycle. With logistic prey or Holling type II functional response, the system can have a stable spiral or limit cycle (Hopf bifurcation). 3-species extensions can exhibit deterministic chaos on strange attractors.