How this calculator works
The Lotka-Volterra model was developed independently by Alfred Lotka in 1925 and Vito Volterra in 1926. It assumes prey grow exponentially when predators are absent, predators starve exponentially when prey are absent, and each encounter between them kills one prey and produces new predators at a fixed rate. Those assumptions give two equations that cycle forever: prey rise, predators rise, prey fall, predators fall, and the cycle repeats. The equilibrium is where both populations stop changing: prey at gamma divided by delta, predators at alpha divided by beta. Linearising around that point gives a period of about two pi divided by the square root of alpha times gamma, which is good near small cycles and less exact for large ones. The calculator integrates the equations with a fourth-order Runge-Kutta method, so the chart shows the actual trajectory rather than a sine wave.
Formula
dPrey/dt = alpha x prey - beta x prey x predators. dPredators/dt = delta x prey x predators - gamma x predators. Equilibrium: prey = gamma / delta, predators = alpha / beta. Approximate period = 2 pi / sqrt(alpha x gamma).
Worked example
With alpha = 1, beta = 0.05, gamma = 0.5 and delta = 0.02, the equilibrium is 0.5 / 0.02 = 25 prey and 1 / 0.05 = 20 predators. The period estimate is 2 pi / sqrt(1 x 0.5) = 8.89 years. Starting from 50 prey and 10 predators, the prey peak is well above 50 and the predator peak follows it a few years later.
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Frequently asked questions
Does the model predict real populations?
It captures the mechanism behind many cycles, including the classic lynx and hare records, but real populations are affected by seasonality, territory, other predators and disease. Use it to understand the interaction, not to forecast a field population.
Why do the cycles not die out?
The classic equations conserve a quantity, so a disturbance moves the system to a larger or smaller closed loop rather than dampening back to equilibrium. Adding density dependence in the prey or a predator that saturates would make the cycles damp.
What happens if I start exactly at equilibrium?
Nothing: both populations stay constant and the chart is two flat lines. That is a perfect balance that does not occur in nature, and any small disturbance starts a cycle.
What do the four rates mean?
Alpha is the prey birth rate, beta the chance of a predator encounter, gamma the predator death rate and delta the conversion of eaten prey into predators. Only the ratios matter for the equilibrium and period, which is why different sets of numbers can produce the same cycle.