Lotka-Volterra: The Maths of Predator-Prey Cycles

Two species, one uneasy truce

Alfred Lotka (1920) and Vito Volterra (1926) independently arrived at the same pair of equations while trying to explain something odd: Adriatic fishermen had reduced their catch during the First World War, yet the proportion of sharks and rays landed had gone up. Volterra's biologist correspondent, Umberto d'Ancona, wanted to know why less fishing pressure seemed to help predators more than prey. The answer turned out to require only two coupled, nonlinear differential equations — and it produces a result nobody expects on first encounter: the populations never settle down. They orbit forever.

The equations

Let x(t) be the prey population (rabbits, hares, fish) and y(t) the predator population (foxes, lynx, sharks). Four assumptions drive the whole model: prey grow exponentially without limit when predators are absent; predators starve exponentially without prey; encounters between the two are proportional to the product x·y (mass-action, like molecular collisions); and the ecosystem is closed — no immigration, no seasons, nothing else going on.

dx/dt = α·x − β·x·y     // prey: grows alone, eaten at rate β·x·y
dy/dt = δ·x·y − γ·y     // predator: starves alone, fed at rate δ·x·y

// α: prey birth rate        (typ. 0.5–2.0 yr⁻¹)
// β: predation rate         (typ. 0.01–0.1)
// γ: predator death rate    (typ. 0.2–1.0 yr⁻¹)
// δ: predator conversion efficiency (typ. 0.001–0.02, usually δ < β)

The only nonlinear term is x·y. Delete it and the two equations decouple into independent exponentials — no interaction, no interesting dynamics. Keep it, and the system couples tightly enough to oscillate indefinitely without any external forcing.

Nullclines and the fixed points

Setting dx/dt = 0 gives the prey nullcline: either x = 0, or y = α/β. Setting dy/dt = 0 gives the predator nullcline: either y = 0, or x = γ/δ. Where the non-trivial nullclines cross is the interior equilibrium E* = (γ/δ, α/β) — a prey level that exactly holds predators steady, paired with a predator level that exactly holds prey steady. The origin (0,0) is a second, trivial equilibrium: both species extinct, an unstable saddle that no realistic trajectory ever approaches.

Linearising at E* gives a Jacobian with purely imaginary eigenvalues, λ = ±i√(αγ). Imaginary eigenvalues mean a centre: trajectories neither spiral in toward the equilibrium nor spiral away from it. They just go around, on a period of approximately T ≈ 2π/√(αγ) for small oscillations near E*.

Why the orbits close: a conserved quantity

The reason nothing spirals is that the system has a first integral — a quantity conserved exactly along every trajectory, playing the same role that total energy plays in a frictionless pendulum:

V(x, y) = δ·x − γ·ln(x) + β·y − α·ln(y)

// dV/dt = (δ − γ/x)(dx/dt) + (β − α/y)(dy/dt)
//       = (δ − γ/x)(αx − βxy) + (β − α/y)(δxy − γy)
//       = 0   for all t  (exact cancellation)

Because V is fixed by the initial conditions and never changes, every trajectory is trapped forever on the closed contour line of V it started on. Orbits close to E* have low V and small amplitude; orbits far from E* have high V and swing closer to extinction on each cycle before recovering. Crucially, prey peaks a quarter-period before predators do: abundant prey feeds a predator boom, the boom over-hunts, prey crash, predators then starve back down, and prey recover to start again.

Does it actually happen in nature?

The best-known real-world test is the Hudson's Bay Company fur-trading records: pelt counts for Canada lynx and snowshoe hare stretching from 1845 to 1935. The data show remarkably regular 9–11 year cycles, with hare abundance consistently leading lynx abundance by roughly a year or two — qualitatively exactly the quarter-period lag the model predicts. It remains one of ecology's most-cited pieces of evidence for cyclic predator-prey dynamics, even though modern analysis attributes part of the hare cycle to vegetation over-grazing rather than predation alone, meaning the real system needs more structure than the bare two-species model provides.

Where the idealisation breaks

The perfect centre at E* is a mathematical knife-edge — structurally unstable in the technical sense that almost any added realism destroys it. Give prey a carrying capacity (logistic growth α·x·(1 − x/K) instead of pure exponential α·x) and the neutral centre becomes a genuine stable spiral or, for the right K, a Hopf-bifurcation limit cycle that trajectories converge to regardless of starting point — a much more robust and biologically plausible picture than the fragile closed orbits of the original model. Adding a saturating (Type II) functional response, time delays, or a third trophic level (a predator that eats the predator) can all push the system toward genuinely chaotic dynamics, as shown in the Hastings-Powell three-species food-chain model. The unmodified Lotka-Volterra equations remain the right starting point for teaching population dynamics — just not the final word on any real ecosystem.

Frequently asked questions

Try it live

Every term above runs live in Lotka-Volterra — Predator-Prey Population Dynamics. Drag α, β, γ, δ and watch the phase portrait redraw its orbit instantly, entirely in your browser.