Predator–Prey

More hares mean more food for lynx; more lynx mean fewer hares; fewer hares mean hungry lynx … and round it goes. Set the growth, hunting and death rates and watch both populations rise and fall in never-ending cycles.

EcologyPopulationsSimulation

Inputs

Input examples

N is the number of prey, P the number of predators; time is in years. Without predators the prey grow by r per year; without prey the predators die off at rate m. The attack rate a sets how often a predator catches a given prey, and e how many new predators each prey eaten produces. The model is solved step by step (Runge–Kutta).

Prey growth rate r — per year

Attack rate a — per predator per year

Conversion efficiency e — new predators per prey eaten

Predator death rate m — per year

Prey at the start N₀

Predators at the start P₀

Years to simulate

How Predators and Prey Cycle

The Lotka–Volterra model is the simplest description of predators and prey. Prey multiply on their own and are eaten in proportion to how often predators and prey meet. Predators grow from the prey they eat and die at a constant rate.

The result is a cycle: prey peak first, the predators follow a quarter of a cycle later, eat the prey down, then starve, which lets the prey recover. In the phase plane (prey against predators) each cycle is a closed loop around the equilibrium. The fur records of the Hudson’s Bay Company famously show such cycles in hares and lynx.

prey at equilibriumN* = m / (e·a)

predatorsP* = r / a

period≈ 2π / √(r m)

predatorspeak after prey

Populations Over Time

years

prey N (left scale)

predators P × 10 (same scale)

Phase Plane

cycle

path of the two populations

equilibrium

start

Result

Enter the values above

Worked Techniques

The Equilibrium

Remarkably, the number of prey at equilibrium depends only on the predators’ properties, and the other way round.

Example: m = 0.6, e = 0.1, a = 0.02: N* = 300 prey, P* = 40 predators.

Period of the Cycles

For small swings around the equilibrium. Large swings take a little longer.

Example: r = 0.8, m = 0.6: T ≈ 9.1 years.

A Conserved Quantity

This combination stays the same along each cycle, which is why the paths in the phase plane close up into loops.

Example: a start further from the equilibrium gives a bigger loop.

Sources & Verification

References

N. J. Gotelli, A Primer of Ecology, 4th ed., Sinauer, 2008 — chapter on predation (Lotka–Volterra predator–prey model).