Predator-Prey Dynamics

The predator peak always lags behind the prey peak. When prey are abundant, predators thrive and multiply — but by the time predator numbers peak, prey have already been depleted. The predator population then crashes from starvation, giving prey a chance to recover, and the cycle repeats. This quarter-cycle lag is a mathematical consequence of the Lotka-Volterra equations and matches observations in real predator-prey systems like lynx and snowshoe hare.

The exact Lotka-Volterra equations have a conserved quantity — H = δx − γ ln x + βy − α ln y — that stays constant along every trajectory. This means orbits are mathematically closed loops: populations oscillate forever without ever spiralling inward or outward, and without ever truly reaching zero. Start close to the equilibrium (x★, y★) and you get smooth, nearly circular loops. Start far away — say x₀ = 1, y₀ = 100 — and the orbit is a long, thin excursion that produces spiky sawtooth cycles: predators crash from starvation, prey recovers explosively, then the predator bonanza collapses prey again. The spike shape is not a numerical artefact; it is the correct large-amplitude solution. What is an artefact is survival itself. In a continuous ODE the populations approach zero but never reach it. In a real ecosystem with discrete animals, the low-point trough means a handful of individuals — and random events then drive actual extinction. The simulation sidesteps this by clamping populations at 0.01, so it shows the mathematical orbit rather than biological reality. A stochastic model would instead show that extreme initial imbalances usually end in collapse.

Vito Volterra built these equations in 1926 to solve a puzzle posed by his son-in-law, biologist Umberto D'Ancona: why had the share of predatory fish in Adriatic catches risen during World War I, when fishing had nearly stopped? The answer, Volterra's principle, is that mortality applied evenly to both species raises the average prey population and lowers the predator's. It explains why broad-spectrum insecticides can backfire — killing pests and their natural enemies alike often leaves more pests than before.