What are the Lotka-Volterra equations?

They are a pair of first-order differential equations describing predator-prey interactions: dx/dt = ax - bxy (prey) and dy/dt = cxy - dy (predator), where x is prey population, y is predator population, a is prey growth rate, b is predation rate, c is predator conversion rate, and d is predator death rate.

What does equilibrium mean in this model?

Equilibrium occurs when both populations stop changing. In the Lotka-Volterra model the equilibrium is x* = d/c and y* = a/b. Near this point the populations cycle endlessly around the equilibrium without settling.

Why do the populations oscillate?

Prey grows when predators are scarce, fueling predator growth. More predators consume more prey, reducing prey numbers. Fewer prey then starves predators, restarting the cycle. This produces the characteristic out-of-phase oscillations.

What happens if the predator death rate is too high?

If predators die faster than they can reproduce from eating prey, the predator population collapses to zero and prey grows without bound in this simple model.

Is this model realistic?

It is a simplified model with no carrying capacity for prey and assumes random predator-prey encounters. Real ecosystems are more complex, but the Lotka-Volterra model captures the essential qualitative behavior of predator-prey cycles.

What numerical method does this tool use?

The simulator uses the fourth-order Runge-Kutta (RK4) method with a fixed time step of 0.05 time units, which gives accurate results for typical parameter ranges.

Lotka-Volterra Calculator

Simulate predator-prey population cycles by entering Lotka-Volterra parameters and view the population dynamics chart over time.

Model Parameters

Initial Prey Population

e.g. rabbits, deer

Initial Predator Population

e.g. foxes, wolves

Prey Parameters

Prey Growth Rate (a) 0.10

Prey reproduction rate without predators

Predation Rate (b) 0.010

Rate prey are consumed per predator-prey pair

Predator Parameters

Predator Growth Rate (c) 0.010

Efficiency of converting prey to predator offspring

Predator Death Rate (d) 0.10

Natural predator mortality rate

Simulation Time (years) 200

Population Dynamics Over Time

Adjust parameters and click Simulate

Click Simulate to see the population chart

Phase Portrait (Prey vs Predator)

Phase portrait will appear after simulation

Summary

The Lotka-Volterra calculator simulates the classic predator-prey model, showing how two interacting species populations change over time. By adjusting growth rates, predation rates, and initial populations, you can explore stable cycles, population crashes, and ecological equilibria. Results are displayed as an interactive time-series chart.

How it works

Enter the initial prey population (e.g. rabbits) and predator population (e.g. foxes).
Set the prey growth rate (how fast prey reproduces without predators).
Set the predation rate (how often predators and prey encounter each other).
Set the predator death rate (how fast predators die without prey).
Set the predator growth rate from prey (how efficiently predators convert prey into offspring).
Click Simulate to generate the population chart over the selected time span.

Use cases

Visualize classic predator-prey cycles (e.g. lynx and hare).
Explore how changes in predator death rate affect ecosystem balance.
Demonstrate equilibrium populations for ecology coursework.
Test extreme parameter values to observe population collapse scenarios.
Illustrate oscillatory dynamics for biology presentations.
Compare stable vs. unstable cycle behavior by tweaking parameters.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu