Predator–Prey Calculator
The Lotka–Volterra predator–prey model — the equilibrium populations, the period of the cycle they oscillate around, and how the populations move over a chosen time from a starting point.
Prey grow exponentially and are eaten at a rate proportional to encounters; predators starve without prey and grow with them.
How the predator–prey calculator works
Prey grow exponentially and are eaten at a rate proportional to encounters; predators starve without prey and grow with them. The result is a cycle: prey boom, predators follow, prey crash, predators starve, and it repeats.
The equilibrium is where both rates balance, and the cycle period near it is 2π over the root of the two rates' product. The model is simple to the point of caricature — no carrying capacity, no satiation — but the cycle it predicts is real, and lynx and hare data show it.
Formula: dx/dt = αx − βxy; dy/dt = δxy − γy; equilibrium x* = γ/δ, y* = α/β
Worked examples
| Inputs | Cycle period (years) | Note |
|---|---|---|
| A classic cycle | 9.472 | about 9.5-year period |
| Starting at equilibrium | 9.472 | nothing moves |
| A faster predator | 5.469 | a shorter cycle |
FAQFrequently asked questions
What is the Lotka–Volterra model?
The simplest predator–prey model: prey grow exponentially and are eaten in proportion to encounters; predators grow from eating and die at a fixed rate. It produces perpetual cycles.
Why do the populations cycle?
Because each responds to the other with a lag. Prey boom, predators follow, prey crash, predators starve. Neither settles because nothing damps the oscillation.
What sets the cycle period?
The geometric mean of the prey growth and predator death rates. Faster rates, shorter cycles — near equilibrium, 2π over the root of their product.
Is it realistic?
As a caricature. Real systems have carrying capacities, predator satiation and refuges, which damp or distort the cycle. The lynx–hare record still shows the basic pattern.
Why is the simulation deterministic?
It integrates the equations with a fixed step, so the same inputs always give the same populations. A result that changed on every keystroke would be useless.
Where these figures come from
- NCBI Bookshelf — Molecular Biology of the Cell — growth kinetics and molecular conventions
- National Human Genome Research Institute — the US genomics research institute
Last checked: September 2026. Formulas are the standard textbook forms; assumptions are stated on each page because they are where these models break.