Predator–prey simulator (Lotka–Volterra)

They are a pair of differential equations: prey change as dx/dt = αx − βxy and predators change as dy/dt = δxy − γy. Prey grow at rate α when left alone and are removed in proportion to how often they meet predators (βxy); predators gain from those meetings (δxy) and die at rate γ when food is scarce. The simulator integrates them with a fourth-order Runge–Kutta method and plots both populations over time.

Predators lag behind their food. When prey are plentiful, predators breed and their numbers climb; heavy predation then drives prey down, predators starve and decline, and the prey recover to start the next cycle. With the default rates the populations orbit an equilibrium of x* = γ/δ = 30 prey and y* = α/β = 12 predators, with a period of about 15 time units, close to the small-swing estimate 2π/√(αγ) = 14.8. The lynx and snowshoe hare records kept by the Hudson’s Bay Company show roughly 10-year cycles of this kind.

It is a simplification: prey have unlimited food, predators eat nothing else, there is no carrying capacity, and the size of the cycle depends entirely on where the populations start. Real systems have refuges, alternative prey, and density limits that damp the swings, and small numbers can go extinct by chance. The model still captures the key insight that the two populations peak out of step and cycle around a fixed point.