Predator and Prey: The Lotka-Volterra Oscillation

Two coupled equations, one oscillation

The Lotka-Volterra equations (Alfred Lotka, 1925; Vito Volterra, 1926, discovered independently) are the simplest mathematical model of a predator and a prey species interacting through consumption alone, with no other limiting factor:

dPrey/dt     = a*Prey - b*Prey*Predator
dPredator/dt = c*Prey*Predator - d*Predator

a = prey growth rate         b = predation rate
c = predator growth per prey eaten     d = predator death rate

Without predators, prey grow exponentially (a·Prey); without prey, predators starve exponentially (−d·Predator). The coupling terms b·Prey·Predator and c·Prey·Predator model encounters between the two populations — more of both means more meetings, so predation and predator growth both scale with the product of the two populations, the same mass-action assumption used for chemical reaction rates.

Why the populations chase each other in circles

Solve the equations and, away from the trivial all-zero equilibrium, the two populations trace a closed, repeating cycle rather than settling to a fixed point: prey grow while predators are scarce, the resulting abundance of prey lets predators grow, the resulting abundance of predators crashes the prey population, and the resulting scarcity of prey then crashes the predator population — back to the start. Plotted over time, prey and predator curves both oscillate with the same period, but the predator peak lags visibly behind the prey peak, because predators only grow after prey have already become abundant.

Plotted instead as prey versus predator on a phase plane (rather than each against time), the system traces a closed loop around a single equilibrium point where dPrey/dt = dPredator/dt = 0 — a centre, in the language of dynamical systems, meaning nearby trajectories neither spiral into it nor away from it, they just orbit it at a fixed amplitude set entirely by the initial conditions. This is a delicate, structurally unstable property: the basic Lotka-Volterra model has no damping, so unlike almost every other coupled oscillator in physics or engineering, its cycles do not shrink toward equilibrium over time.

From two numbers to hundreds of particles

The two differential equations describe population totals directly, assuming everyone in each species can instantly interact with everyone in the other — a well-mixed, spatially uniform population. The simulation on this page instead gives each prey and predator individual position, velocity and simple local rules: prey wander and reproduce at some rate, predators chase nearby prey and reproduce when they catch one, and both die (of old age for prey, of starvation for predators that catch nothing for too long). No global equation is solved anywhere in the code — the same boom-and-bust oscillation you would get from the differential equations emerges purely from thousands of local, particle-level interactions, plus the ability to actually see the spatial structure an ODE model completely hides: prey clusters getting hunted down locally, predators concentrating temporarily where prey is dense, and empty regions recovering while the action moves elsewhere.

Why real ecosystems don't oscillate forever

The pure model is famously fragile: any added realism — a carrying capacity limiting how many prey the environment can support even with zero predators, a saturating (Type II) functional response where each predator's kill rate plateaus instead of growing linearly with prey density, or demographic noise from small population sizes — typically damps the oscillation toward a stable equilibrium or, in other parameter regimes, destabilises it into extinction, rather than preserving the neutral, undamped cycles of the idealised equations. Lotka-Volterra is therefore best understood as a minimal, deliberately oversimplified starting point that demonstrates why predator-prey systems oscillate at all, not as a quantitatively accurate model of any specific real ecosystem.

Frequently asked questions

Try it live

Everything above runs in your browser — open Prey-Predator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.