dx/dt = αx − βxy (prey: growth minus predation)
dy/dt = δβxy − γy (predator: gains from prey, natural death)
The equations produce perpetual oscillations where prey peaks precede predator peaks. The equilibrium point is x* = γ/δβ, y* = α/β. These are neutrally stable cycles — energy is conserved in the pure Lotka-Volterra system, though real populations have damping from environmental stochasticity.
Classic empirical example: Canadian lynx and snowshoe hare data from Hudson's Bay Company fur records (1845–1935) show clear ~10-year oscillations. Phase space shows closed orbits around the equilibrium — the hallmark of Hamiltonian structure.