Three-Species Lotka-Volterra

Plant → Herbivore → Predator food chain dynamics
Plant (V₁)
Herbivore (V₂)
Predator (V₃)
ODEs:
dV₁/dt = r₁V₁ − a₁₂V₁V₂  |  dV₂/dt = e₁a₁₂V₁V₂ − a₂₃V₂V₃ − d₂V₂  |  dV₃/dt = e₂a₂₃V₂V₃ − d₃V₃
The phase portrait (right) shows the 3D trajectory projected onto herbivore–predator axes, tracing quasi-periodic or chaotic orbits. Cascading trophic control means removing the predator can cause herbivore explosions and plant collapse — a trophic cascade.