Lotka-Volterra equations

The Lotka–Volterra equations are the classic pair of ordinary differential equations that describe how a prey population and a predator population change together over time. Prey grow in the absence of predators and decline in proportion to encounters; predators starve without prey and grow from those same encounters. Students, ecologists, fisheries analysts, and modelers need a deterministic population time series from initial counts and the four rate coefficients, not a screenshot of a plot that cannot be replayed. This Lotka–Volterra calculator integrates dx/dt = αx − βxy and dy/dt = δxy − γy with a fixed-step fourth-order Runge–Kutta method and returns the trajectory, peaks, troughs, and the interior equilibrium when it exists.

What the Lotka–Volterra equations say about predator and prey

Alfred Lotka and Vito Volterra independently wrote the same two-species model in the 1920s: a prey stock x that would grow exponentially on its own, and a predator stock y that would decline exponentially without food. Encounters are written as the product x times y. Prey therefore follow dx/dt equals αx minus βxy, where α is the intrinsic prey growth rate and β is the predation rate coefficient.

Predators follow dy/dt equals δxy minus γy, where δ is the conversion of eaten prey into predator growth and γ is the predator death or starvation rate. Those four coefficients plus the two initial populations fully determine the vector field in the first quadrant.

When α, β, γ, and δ are all positive, the system has an interior equilibrium at x-star equals γ over δ and y-star equals α over β; orbits around that point are closed in the exact continuous model, which is why textbook plots show the famous out-of-phase cycles: prey rise, predators follow, prey crash, predators crash, and the loop repeats. The model is not a full ecosystem: it ignores carrying capacity, age structure, refuge, and stochastic extinction. It is the first-order teaching and planning tool for coupled growth and predation.

Units stay yours: if x and y are animals per square kilometre and time is in years, the rates are in reciprocal years and the returned series uses the same units. Zero populations and zero rates are allowed and produce degenerate but well-defined paths, such as exponential prey growth when predators start at zero. Negative counts or rates are not biological in this contract and are rejected rather than silently clamped at the input boundary.

Who uses predator–prey simulations

Secondary-school and university ecology courses repeatedly ask students to sketch Lotka–Volterra cycles from a table of x(t) and y(t). A deterministic RK4 oracle lets a learner check that prey peak before predators, that the discrete orbit stays near the predicted equilibrium, and that a negative rate is illegal instead of a silent NaN in a spreadsheet. Fisheries and wildlife planners sometimes need a first-order coupled forecast: given current counts and assumed growth and predation rates, how do the two stocks move over a planning horizon of T time units?

The bounded step count keeps the isolate cheap while still resolving a few cycles. Sensitivity work is a few extra calls: hold initials fixed and vary α or β to see whether the predator peak grows or collapses.

Keep all eight parameters non-negative and finite, choose steps large enough to resolve the orbit you care about (ten to forty is typical for a homework plot; two hundred is the hard cap), and treat the series as a numerical trajectory of the classic equations—not as a fitted field census, a stochastic extinction model, or a logistic prey-only growth curve.