Lotka Volterra Calculator

What Are the Lotka-Volterra Equations?

The Lotka-Volterra model describes predator-prey population dynamics in ecology. It captures how prey grow exponentially in the absence of predators, predators decline without prey, and the two populations oscillate when both interact. The model is a foundation of theoretical ecology and population biology.

Predator-prey differential equations

The coupled system is:

$$\frac{dx}{dt} = \alpha x - \beta x y \quad \text{(prey)}$$

$$\frac{dy}{dt} = \delta x y - \gamma y \quad \text{(predator)}$$

Where x is prey population, y is predator population, α is prey intrinsic growth rate, β is predation rate, γ is predator death rate, and δ is predator growth efficiency from consuming prey.

Equilibrium and oscillations

At equilibrium, both population growth rates are zero:

$$x^* = \frac{\gamma}{\delta}, \quad y^* = \frac{\alpha}{\beta}$$

Near equilibrium, populations oscillate with an approximate period:

$$T \approx \frac{2\pi}{\sqrt{\alpha \gamma}}$$

This calculator integrates the system numerically using fourth-order Runge-Kutta (RK4) in the browser, reporting final populations and the peak and trough values observed during the simulation.

Example with default parameters

With α = 1, β = 0.1, γ = 1.5, δ = 0.075, initial prey x₀ = 40, predator y₀ = 9, and simulation time 50, equilibrium prey is 20 and equilibrium predator is 10. The populations cycle around these values as predators lag behind prey booms and busts.

For related ecology tools, see the Carrying Capacity Calculator or the Fish Mercury Calculator.