Foxes eat rabbits. More rabbits feed more foxes; more foxes eat down the rabbits; fewer rabbits starve the foxes; then rabbits rebound — forever. The Lotka–Volterra equations turn that chase into two coupled curves that OSCILLATE, the predator always lagging a quarter-cycle behind the prey. It never settles to a point. Slide time and watch the cycle turn.
The Lotka–Volterra predator–prey model: dx/dt = αx − βxy (prey grow, are eaten), dy/dt = δxy − γy (predators grow by eating, else die). The interior fixed point is a CENTRE, so trajectories are closed orbits — sustained oscillations whose amplitude is set by the initial conditions, with the predator peak trailing the prey peak by ~a quarter period. There is no stable equilibrium the system relaxes to. A fail-loud self-check throws unless the populations oscillate and stay bounded and positive. ◆ real population dynamics, node-verified.
The classic model is idealised (no prey carrying capacity, continuous populations); real systems add damping or chaos. The qualitative truth — coupled oscillation with a predator lag — is robust and is the content.