Lotka-Volterra Predator-Prey Model Calculator

The Lotka-Volterra Equations Developed independently by Alfred Lotka in 1925 in the United States and Vito Volterra in 1926 in Italy, these coupled differential equations model the dynamics of two interacting species, one predator and one prey. They are the foundation of mathematical ecology.

The Two Equations Prey: dN/dt = alphaN - betaNP. Predator: dP/dt = deltaNP - gammaP. Where N = prey population, P = predator population, alpha = prey growth rate (births in absence of predators), beta = predation rate (how effectively predators catch prey), delta = predator growth efficiency (how well predators convert prey into offspring), gamma = predator death rate (natural mortality).

The Population Oscillations The model produces characteristic cycles: Prey increase when predators are scarce. More prey leads to more predator food, so predators increase. More predators reduce prey numbers. Fewer prey leads to predator starvation, so predators decrease. Fewer predators allows prey to recover. The cycle repeats. The predator cycle lags behind the prey cycle, so predator peaks follow prey peaks with a delay.

The cycles are exactly closed, and that is fussy to compute. Lotka-Volterra conserves a quantity, V = δN − γ·ln(N) + βP − α·ln(P), so a real orbit returns to precisely where it started, forever, at the same amplitude. Step the equations forward with the crude method (take the slope where you are, move a little, repeat) and the orbit spirals outward instead: the amplitude grows every cycle and the model appears to predict an ecosystem tearing itself apart. That is the integrator failing, not the biology. This page uses fourth-order Runge-Kutta, which holds the amplitude steady over the whole run, and it reports how far the conserved quantity drifted so you can see for yourself that it did not.

Small-amplitude period. Near the equilibrium the cycle time settles at 2π / √(αγ), independent of how many animals there are. With the default α = 1 and γ = 0.5 that is about 8.9 time units. Bigger loops swing wider and take longer, so the number this page reports for a large orbit is legitimately above that floor.

Real-World Example The most famous example is the Canadian lynx and snowshoe hare populations, tracked by the Hudson’s Bay Company fur records from the 1840s to the 1930s. The data shows remarkably regular 9-11 year oscillation cycles that match the Lotka-Volterra predictions. Hare populations peak first, followed by lynx populations approximately 1-2 years later.

Model Limitations The basic model assumes no carrying capacity for prey, no prey immigration, constant predator efficiency, and no other species interactions. Real ecosystems are far more complex. Extensions of the model include carrying capacity (competition among prey), predator satiation (type II/III functional responses), and multiple interacting species.