Lotka–Volterra Predator–Prey Calculator

How it works

dN/dt = rN − aNP · dP/dt = baNP − mP · equilibrium N* = m ÷ (a·b), P* = r ÷ a · small-oscillation period T = 2π ÷ √(r·m)

The Lotka–Volterra equations are the simplest coupled description of a hunter and its food: prey grow exponentially when left alone and are removed in proportion to how often they meet predators, while predators multiply in proportion to what they catch and die at a constant rate. The pair has a single balance point, but any displacement from it produces a closed loop — prey rise, predators follow and overshoot, prey crash, predators starve, prey recover — with the predator peak trailing the prey peak by a quarter of the cycle, a pattern seen in lynx and hare records. The equilibrium formulas hold the famous paradox that harvesting prey benefits nobody but reduces predators, because the prey balance depends only on the predator's parameters and vice versa.

Worked example

With r = 1.0, a = 0.1, b = 0.02 and m = 0.4, the equilibrium is 200 prey and 10 predators and the cycle period about 9.9 years. Starting from 400 prey and 15 predators, prey are falling at 200 a year while predators still rise at 6 a year — the predator peak is ahead; the simulation swings between roughly 57 and 486 prey and 5 and 18 predators, with predators peaking about 2.5 years after prey.

Frequently asked questions