Population Dynamics — Lotka-Volterra

Lotka-Volterra Equations

Prey growth rate — increases with α, decreases with predation βxy

Predator growth — gains from predation δxy, loses to death γy

Equilibrium: prey at γ/δ, predators at α/β

How It Works

The Lotka-Volterra model captures one of nature's most elegant cycles. When prey is abundant, predators thrive and multiply. Rising predator numbers reduce prey. Fewer prey means predators starve. Recovery of prey restarts the cycle.

The phase portrait (x vs y) traces a closed loop around the equilibrium point — a signature of neutral stability. The system never settles but orbits forever.

Classic Example

The lynx-snowshoe hare data from Hudson Bay Company (1845–1935) shows ~10-year oscillations remarkably consistent with this model.

Common Misconception

Populations don't "try" to reach equilibrium — these oscillations are stable cycles, not damped spirals.

AP Biology Connection

AP Bio Unit 8: Ecology — community interactions, predation as a density-dependent regulating factor.

Phase Portrait (x vs y)

Prey →← Predators

POPULATION STATUS

Prey (x)—

Predators (y)—

Equil. prey—

Equil. pred—

Sim time—

Quick Quiz

1. In the Lotka-Volterra model, what happens to prey population when predators are at their peak?

It immediately increases due to less competition It is near its minimum and declining It stays at equilibrium

2. If you increase α (prey growth rate), what happens to the equilibrium predator count?

It increases (y* = α/β) It decreases It stays the same

3. The phase portrait of Lotka-Volterra forms a closed loop. This indicates:

The system spirals to extinction Neutrally stable cycles — populations oscillate indefinitely Populations converge to a fixed point