Biology · Introductory biology · Worked example
Find the equilibrium of a predator–prey model
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.
Hold the prey steady
dX/dt = X(α − βY) is zero for X > 0 when Y = α/β.
Hold the predators steady
dY/dt = Y(δX − γ) is zero for Y > 0 when X = γ/δ.
Check the starting point
At 300 prey and 20 predators both rates are positive: prey are above 200, so the predators grow, and predators are below 30, so the prey grow too.
Read the simulated cycle
In the simulation the prey peak at about 330 within the first year, and the predators peak at about 53 some 1.8 years later. The prey then bottom out near 110 and the predators near 15. The cycle repeats every 9.3 years or so, and over each cycle the populations average 200 prey and 30 predators.
Result
The equilibrium is 200 prey and 30 predators. From 300 prey and 20 predators both populations rise at first; in the simulation they then cycle about every 9.3 years, the prey between about 110 and 330 and the predators between about 15 and 53.
Your turn
With α = 0.5, β = 0.01, δ = 0.002 and γ = 0.4, find the equilibrium.
Show the answer and explanation
200 prey and 50 predators.
X* = γ/δ = 0.4/0.002 = 200 and Y* = α/β = 0.5/0.01 = 50.
Keep exploring
The predator–prey tool opens with these parameters and simulates 30 years; its summary lists the equilibrium, 200 prey and 30 predators.
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Authored study material. Tool results depend on the stated inputs and model assumptions.
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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.