The Lotka-Volterra predator-prey model is one of the oldest and most studied systems in mathematical ecology. Formulated independently by Alfred Lotka (1925) and Vito Volterra (1926), it describes the oscillating relationship between two populations: a prey species that grows freely when predators are absent, and a predator species that depends entirely on prey for sustenance. The model correctly predicts that both populations oscillate in perpetual cycles with the predator cycle lagging the prey cycle by roughly a quarter period.
The mathematical properties of the system are unusual: the Lotka-Volterra equations have a conserved quantity H = δ·x − γ·ln(x) + β·y − α·ln(y) that is constant along any trajectory. This conservation means orbits are closed and neutrally stable — perturbations shift the trajectory to a new orbit rather than returning it to the old one. Real ecosystems break this conservation through resource limitations, spatial effects, or age structure, which is why more realistic models add logistic damping or additional species.
This visualization integrates the Lotka-Volterra equations interactively and plots both the time series (populations versus time) and the phase portrait (predators versus prey). Adjusting birth rates, death rates, and initial conditions lets you explore how oscillation amplitude and period change, and what happens when you push the system far from its equilibrium. You can clearly see the quarter-phase lag between predator and prey that is the hallmark signature of the Lotka-Volterra cycle.