The Lotka-Volterra equations are the foundational mathematical model of predator-prey interaction in ecology, first proposed independently by Alfred Lotka and Vito Volterra in the 1920s. This calculator simulates the model numerically so you can see how starting populations and interaction rates shape the resulting population cycles.
How the Lotka-Volterra model works
The model couples two differential equations: prey grow exponentially at rate α absent predators but are removed at rate β with each predator encounter, while predators grow in proportion to prey consumed (efficiency δ) but die off at rate γ absent prey. Because the equations are coupled and nonlinear, there is no algebraic formula for population size at time t — this calculator numerically integrates the system forward using 4th-order Runge-Kutta (RK4) stepping, which is far more accurate than simple Euler stepping for tracking the closed, cyclical orbits this system produces.
Inputs and what they mean
Initial prey and predator populations (x0, y0) set the starting point of the simulation. The four rate parameters — α (prey growth), β (predation), δ (predator efficiency), γ (predator death) — determine both the shape of the population cycle and the location of the equilibrium point at x* = γ/δ, y* = α/β. The time horizon controls how far forward the simulation runs. Small changes to β or δ shift the equilibrium point substantially, since it depends on the ratio of death rate to efficiency (and growth rate to predation rate) rather than on the raw magnitudes.
Limits and edge cases
The classic Lotka-Volterra model is a simplified idealization: it assumes prey have unlimited food, predators eat only this one prey species, and there's no environmental carrying capacity, disease, or migration. Real ecosystems dampen or amplify cycles in ways this model can't capture — it's a teaching and intuition-building tool, not a wildlife-management forecasting instrument. All four rate parameters and both starting populations must be positive; a zero rate collapses part of the model (for example, β = 0 means prey are never predated at all), so the calculator blocks the simulation and asks for valid inputs instead of producing a nonsensical result.