dx/dt = αx - βxy, dy/dt = δxy - γy
The Lotka-Volterra equations, independently developed by Alfred Lotka (1925) and Vito Volterra (1926), describe the simplest model of predator-prey interaction. The prey equation dx/dt = αx - βxy says prey grow exponentially (rate α) but are consumed by predators (rate β). The predator equation dy/dt = δxy - γy says predators grow when they eat prey (rate δ) but die naturally (rate γ). The system produces characteristic oscillations: prey increase → predators increase → prey decrease → predators decrease → cycle repeats. Equilibrium occurs when prey = γ/δ and predators = α/β. The classic example is the Canadian lynx and snowshoe hare, whose populations cycle with ~10-year periods, documented by Hudson's Bay Company fur trading records since the 1800s. Real ecosystems are more complex, involving multiple species, spatial effects, and environmental variability, but the Lotka-Volterra model remains foundational in ecology education and conservation planning.