Lotka-Volterra Equations
The Lotka-Volterra equations model predator-prey population dynamics, showing how two species cycle in size over time.
Includes worked examples.
The Formula
The Lotka-Volterra equations, developed independently by Alfred Lotka in 1925 and Vito Volterra in 1926, describe the cyclical relationship between a predator population and its prey. The two coupled differential equations show how the size of each population changes over time depending on the other.
The first equation describes prey (X): the prey population grows at rate α when predators are absent, but is reduced by encounters with predators at rate β. The more predators there are, the faster prey numbers fall.
The second equation describes predators (Y): predators increase when prey is abundant (rate δ based on successful hunts), but die off at rate γ when prey becomes scarce. Together the equations produce a repeating cycle — prey numbers rise, then predator numbers rise and crash the prey, then predators decline too, allowing prey to recover again.
This elegant model captures the oscillating boom-and-bust cycles observed in real ecosystems, such as the famous 90-year record of Canadian lynx and snowshoe hare populations tracked by the Hudson Bay Company.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| X | Prey population size | individuals |
| Y | Predator population size | individuals |
| α | Prey birth rate (growth rate without predators) | per unit time |
| β | Predation rate (rate at which predators kill prey) | per individual per unit time |
| δ | Predator reproduction rate per prey consumed | per individual per unit time |
| γ | Predator death rate (mortality without prey) | per unit time |
| t | Time | years, months, etc. |
Example 1
Find the equilibrium populations for a predator-prey system with α = 0.4, β = 0.02, δ = 0.01, γ = 0.3.
At equilibrium, both dX/dt = 0 and dY/dt = 0.
From dX/dt = 0: αX − βXY = 0 → X(α − βY) = 0 → Y* = α/β = 0.4/0.02 = 20 predators
From dY/dt = 0: δXY − γY = 0 → Y(δX − γ) = 0 → X* = γ/δ = 0.3/0.01 = 30 prey
Equilibrium: 30 prey and 20 predators — the system oscillates around these values indefinitely
Example 2
Determine how adding more predators affects the equilibrium prey count, given the same parameters as above.
Note that the equilibrium prey count X* = γ/δ does NOT depend on α or β — it only depends on predator parameters.
Adding more predators does not reduce the long-run equilibrium prey count in this model.
This counterintuitive result, known as Volterra's principle, means that harvesting both species proportionally actually increases predator numbers relative to prey — a key insight for fisheries management.
When to Use It
Use the Lotka-Volterra equations when:
- Modeling the cyclical dynamics between a predator species and its prey
- Analyzing how hunting or fishing policies affect ecosystem balance
- Studying coevolution and arms races between competing species
- Teaching mathematical biology, ecology, and differential equations
- Building simulations for population ecology research