What happens to oscillation amplitude when you increase the predation rate (β)?

Higher β increases the amplitude of oscillations and can lead to extinction if prey crash to near zero.

At equilibrium, prey population = γ/δ. If γ = 0.8 and δ = 0.02, what is the equilibrium prey count?

X̄ = 0.8/0.02 = 40 prey at equilibrium.

Why does the predator peak always lag behind the prey peak? Explain the biological mechanism.

Predators need time to reproduce after prey becomes abundant — birth response has a delay proportional to generation time.

Pro 🔒~20 min

Population Dynamics

Lotka-Volterra predator-prey model and population oscillations

How it works

The Lotka-Volterra equations model the dynamics of biological systems with predator-prey interactions. The prey population x grows exponentially at rate α in the absence of predators, but is reduced by predation at rate βxy (proportional to encounters). The predator population y declines at rate γ without prey, but grows from successful predation at rate δxy. The system produces characteristic oscillations: prey increase → predators increase → prey decline → predators decline → cycle repeats. The amplitude and period depend on all four parameters. The equilibrium point (x̄ = γ/δ, ȳ = α/β) is a center in the phase plane — orbits are closed loops. Real ecosystems add complexity (carrying capacity, refugia, time delays) but the Lotka-Volterra model captures the fundamental mechanism of coupled oscillations.

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Step-by-step

Adjust the prey birth rate (α), predation rate (β), predator death rate (γ), and conversion efficiency (δ).
The time-series chart shows both populations over time, while the phase-plane plot shows the predator-prey trajectory.
Watch for the characteristic quarter-cycle phase lag between populations.

Key formulas

$\frac{dx}{dt} = \alpha x - \beta x y$ Prey growth rate: prey increase by birth (αx) minus predation losses (βxy)
$\frac{dy}{dt} = \delta x y - \gamma y$ Predator growth rate: gain from prey consumption (δxy) minus natural death (γy)
$\bar{x} = \frac{\gamma}{\delta}, \quad \bar{y} = \frac{\alpha}{\beta}$ Equilibrium populations: prey = γ/δ, predator = α/β

Frequently asked questions

What happens to oscillation amplitude when you increase the predation rate (β)?: Higher β increases the amplitude of oscillations and can lead to extinction if prey crash to near zero.
At equilibrium, prey population = γ/δ. If γ = 0.8 and δ = 0.02, what is the equilibrium prey count?: X̄ = 0.8/0.02 = 40 prey at equilibrium.
Why does the predator peak always lag behind the prey peak? Explain the biological mechanism.: Predators need time to reproduce after prey becomes abundant — birth response has a delay proportional to generation time.