Lotka-Volterra Equations

The Lotka-Volterra Equations Calculator simulates the classic predator-prey model developed independently by Alfred J. Lotka (1925) and Vito Volterra (1926). It uses a pair of first-order nonlinear differential equations to model the cyclical interaction between two species: a prey population (such as rabbits) that grows exponentially and a predator population (such as foxes) that depends on prey for food. The model is foundational in mathematical biology and ecology.

The simulation uses Euler's method with a small time step (Δt = 0.01) to numerically integrate the equations over time. The prey population changes according to dx/dt = αx − βxy, where α is the prey growth rate and β is the predation rate. The predator population changes according to dy/dt = δxy − γy, where δ is the predator growth efficiency per prey consumed and γ is the predator death rate. The results include a time series chart showing both populations oscillating over time, a phase portrait showing the cyclic relationship, peak and final populations, and the theoretical equilibrium point (γ/δ, α/β) where both populations stabilize.

The model makes several simplifying assumptions: prey have unlimited food, predators have limitless appetite, the environment does not change, and there is no age or spatial structure. Despite these limitations, the Lotka-Volterra model successfully captures the fundamental oscillatory nature of predator-prey systems observed in nature, such as the classic lynx-snowshoe hare cycles recorded by the Hudson's Bay Company. Numerous extensions — including the Rosenzweig-MacArthur model with a Holling functional response and prey carrying capacity — have been developed to address the model's limitations while retaining its core insights.