What this calculator does
The Lotka-Volterra equations are the simplest model of two populations that depend on each other. Prey grow on their own and are eaten at a rate set by how often the two meet, which is taken as proportional to the product of the two populations. Predators do the reverse: they gain from those meetings and die off at a steady rate without them.
The model produces endless cycles rather than a settled state, with predator numbers lagging prey. What this calculator gives is the instantaneous rate at one moment, not a projection forward, which is the right tool for asking whether each population is currently rising or falling and how close the pair is to balance.
The formula
Two expressions are evaluated at the populations given. The prey rate is αPrey minus βPrey·Predator, so growth less losses to predation. The predator rate is δPrey·Predator minus γPredator, so gains from feeding less natural deaths. Both are rates at this instant, and either can be zero, which marks an equilibrium for that population.
| Term | Meaning |
|---|---|
| α (alpha) | Prey growth rate in the absence of predators. |
| β (beta) | Predation rate: how effectively encounters remove prey. |
| δ (delta) | Predator growth efficiency: how much predator increase each encounter produces. |
| γ (gamma) | Predator death rate in the absence of prey. |
| Equilibrium | The pair of populations at which both rates are zero, found at prey = γ/δ and predators = α/β. |
The inputs explained
| Field | What to enter |
|---|---|
| Prey population | The current prey population. |
| Predator population | The current predator population. |
| Prey growth rate α (/year) | Prey growth rate α, per unit time. |
| Predation rate β | Predation rate β. Small values are usual, since it multiplies two populations together. |
| Predator growth efficiency δ | Predator growth efficiency δ, also small for the same reason. |
| Predator death rate γ (/year) | Predator death rate γ, per unit time. |
When to use it
Finding the equilibrium of a modelled pair
The equilibrium populations depend only on the four rates, not on where the populations currently are. Varying one population until its counterpart rate reaches zero locates it directly.
Checking which way a system is heading
The two trend lines answer whether each population is currently rising or falling, which is the immediate question when a model is being set up or checked.
Teaching the predator-prey cycle
Stepping the populations around a cycle by hand shows why the two peaks are out of phase, with predators peaking after prey rather than with them.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
At what predator population do the prey stop growing?
A fixed prey population of 1,000 against a rising number of predators.
| Predators | Prey growth rate dPrey/dt | Predator growth rate dPredator/dt | Prey trend |
|---|---|---|---|
| 10 | 300.0 /year | 97.0 /year | growing |
| 25 | 0.0 /year | 242.5 /year | growing |
| 50 | -500.0 /year | 485.0 /year | declining |
| 100 | -1,500.0 /year | 970.0 /year | declining |
At what prey population do the predators stop growing?
A fixed predator population of 50 against a rising number of prey.
| Prey | Predator growth rate dPredator/dt | Prey growth rate dPrey/dt | Predator trend |
|---|---|---|---|
| 20 | -5.0 /year | -10.0 /year | declining |
| 30 | 0.0 /year | -15.0 /year | growing |
| 50 | 10.0 /year | -25.0 /year | growing |
| 100 | 35.0 /year | -50.0 /year | growing |
Questions
Where is the equilibrium of a Lotka-Volterra system?
At prey = γ/δ and predators = α/β. The curious feature is that each equilibrium depends on the other species parameters rather than its own: the prey equilibrium is set by the predator rates and the predator equilibrium by the prey rates. With the defaults here that gives 30 prey and 25 predators.
Why do the populations cycle rather than settle?
Because the equilibrium is neutrally stable in this model. A system displaced from it orbits the equilibrium forever at whatever amplitude the displacement set, rather than spiralling in or out. That is a known artefact of the model being so simple, and real systems damp or are driven by outside factors.
Why does the predator population peak after the prey?
Because predators respond to prey abundance rather than causing it. Prey rise first, predators then grow on the surplus, prey fall under the increased predation, and predators fall afterwards through lack of food. The lag is a quarter of a cycle and is the model most recognisable prediction.
How realistic is this model?
As a picture of the cycling mechanism, useful. As a quantitative tool, limited. It assumes prey grow without any ceiling of their own, that predators have no other food, that encounters scale with the simple product of the populations, and that nothing else varies. Adding a carrying capacity for the prey is the usual first correction.
For a single population under a fixed ceiling, see the logistic growth calculator. For measuring the diversity of a whole community, see the Shannon diversity index calculator.