Biology · Introductory biology · Concept
Predator–prey cycles: the Lotka–Volterra model
Predator–prey models describe two populations that drive each other. In the Lotka–Volterra model, prey grow exponentially when there are no predators, predators die off when there is no prey, and each encounter removes prey and feeds predator growth. The populations cycle around an equilibrium, with each predator peak following a prey peak.
The equations
X is the prey population and Y the predator population. Prey grow at the per-capita rate α and are eaten at a rate proportional to encounters, βXY. Predators gain δXY from those encounters and die at the per-capita rate γ.
The equilibrium
Both populations hold steady when both rates are zero. Apart from extinction, that happens at X* = γ/δ and Y* = α/β. The prey level at equilibrium is set by the predators’ parameters, and the predator level by the prey’s.
Why the populations cycle
Plentiful prey feed a growing predator population. More predators eat prey faster, and the prey decline. With less food the predators decline, which lets the prey recover, and the cycle starts again. Each predator peak lags behind a prey peak.
Simulated, not observed
These equations have no simple formula for X(t) and Y(t), so the predator–prey tool solves them numerically, in small time steps, and its curves are simulations of the model. Near the equilibrium a cycle lasts about 2π/√(αγ) time units, and over one full cycle the populations average exactly X* and Y*.
What the model leaves out
The prey have no carrying capacity, individuals have no ages, nothing moves in or out, and the parameters never change. Real cycles, such as those of lynx and snowshoe hares, also depend on the prey’s own food supply. The model explains why cycles can arise, not the exact numbers in a real ecosystem.
Common mistakes
- Expecting predators to peak together with prey: in the model their peaks come later.
- Reading the equilibrium as where the populations settle: in this model they cycle around it without approaching it.
- Swapping the equilibrium values: the prey level γ/δ comes from the predator parameters.
- Treating simulated cycles as a forecast for a real ecosystem.
Key terms
- Lotka–Volterra model
- A pair of equations for how predator and prey populations affect each other: prey grow and get eaten, and predators need prey to grow. It predicts repeating cycles, which real populations may not follow.
- Predation
- One organism, the predator, catching and eating another, the prey. In a model, the term for their meetings is an assumed pattern, not an observed mechanism.
- Population
- All the individuals of one species living in a given area at a given time. Population size counts them; density divides the count by the area or volume.
- Per-capita rate
- A rate per individual rather than for the whole population, such as births per person per year. A constant per-capita rate adds more individuals as the population grows.
- Exponential population growth
- Growth at a constant per-capita rate, dN/dt = rN, so the population grows faster and faster. It assumes unlimited resources, which never lasts.
Work through an example
In a Lotka–Volterra model, prey grow at α = 0.6 per year, β = 0.02, δ = 0.004, and predators die at γ = 0.8 per year. Find the equilibrium, decide how both populations change from 300 prey and 20 predators, and describe the simulated cycle.
Find the equilibrium of a predator–prey model →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Open the predator–prey tool Open worked example on a board Lotka–Volterra equations in Biology ReferenceYour existing work stays on this device. Examples open as editable copies.