Lotka Volterra Calculator

The intricate dance between predators and their prey is one of the most fundamental interactions in ecology. The Lotka-Volterra equations, developed independently by Alfred J. Lotka and Vito Volterra in the 1920s, provide a mathematical model to describe these dynamics. Our Lotka-Volterra Calculator allows you to simulate these interactions over time.

The Predator-Prey Equations

The model consists of a pair of first-order, non-linear differential equations that describe the rate of change of two populations:

Prey Population Growth:

dx/dt = αx - βxy

This equation states that the rate of change of the prey population (x) is determined by:

• αx: The natural growth of the prey population (exponential growth, where α is the prey growth rate), assuming no predators.
• -βxy: The rate at which prey are consumed by predators. This term depends on the rate of encounters between prey (x) and predators (y), with β representing the predation rate.

Predator Population Growth:

dy/dt = δxy - γy

This equation describes the rate of change of the predator population (y) based on:

• δxy: The growth of the predator population, which depends on the rate of prey consumption (xy). The parameter δ represents the efficiency of converting consumed prey into new predators.
• -γy: The natural death rate of the predator population (where γ is the predator death rate), assuming a lack of food.

Our calculator numerically solves these equations over a specified time period to plot the population cycles.

Understanding the Cyclical Relationship

When you run the simulation, you will typically observe an oscillating pattern:

1. An abundance of prey leads to an increase in the predator population.
2. The growing predator population consumes more prey, causing the prey population to decline.
3. With fewer prey available, the predator population begins to starve and decline.
4. The reduction in predators allows the prey population to recover and grow again, starting the cycle anew.

A key feature of this model is that the predator population's cycle lags behind the prey population's cycle.

This calculator is a fantastic educational resource for students of biology, ecology, and applied mathematics. It offers a hands-on way to explore how predator-prey relationships shape the structure and dynamics of ecosystems.