Lotka-Volterra Calculator
Calculate predator-prey population dynamics using Lotka-Volterra equations.
Why the Equilibrium Is Counterintuitive
Setting both derivatives to zero gives the equilibrium point, and the result surprises people: each population's equilibrium is set entirely by the other species' parameters.
Prey abundance depends on predator death rate and conversion efficiency, not on prey growth rate. Predator abundance depends on prey growth rate, not on predator biology. Increasing prey birth rate raises the number of predators, not prey.
The Paradox of Pesticides
This produces a genuinely important prediction. Applying a pesticide that kills both pest and its predator raises the pest's death rate and the predator's equally — but because prey equilibrium depends on predator mortality, killing predators raises the pest equilibrium.
Broad-spectrum insecticide application has repeatedly produced pest resurgence for exactly this reason, and the model predicted it before it was widely documented.
| Intervention | Effect on N* | Effect on P* |
|---|---|---|
| Increase predator death rate d | Raises prey | No direct change |
| Increase prey growth rate r | No direct change | Raises predators |
| Increase predation rate α | Lowers prey | Lowers predators |
Neutral Cycles and Why They Are Unrealistic
The classic model produces neutrally stable cycles — the amplitude depends entirely on initial conditions and never damps or grows. Any perturbation shifts the system permanently to a new orbit.
Real systems do not behave this way, and the model's assumptions explain why: unlimited prey growth without a predator, constant predation rate at all densities, no time delays, no spatial structure and no age classes.
| Refinement | What it adds |
|---|---|
| Logistic prey growth | Damped oscillations toward a stable point |
| Type II functional response | Predator saturation at high prey density |
| Time lags | Can destabilise into larger cycles |
| Spatial structure | Refuges permit persistence |
The Rosenzweig–MacArthur model adds logistic prey growth and a saturating functional response, producing the paradox of enrichment: increasing prey carrying capacity destabilises the system into larger oscillations that can drive extinction.
Worked Examples
Common Mistakes
It produces neutrally stable cycles whose amplitude depends on starting conditions. Logistic prey growth is needed for damped oscillations.
It depends on prey growth rate and predation rate. Each species' equilibrium is set by the other's parameters.
Real predators saturate. A type II functional response levels off at high prey density and changes the dynamics substantially.
Killing both pest and predator raises the pest equilibrium, because prey abundance is set by predator mortality. Broad-spectrum application has caused repeated resurgence.