274. Lotka-Volterra

The classical predator-prey model. Two species:

where:

• : prey population, : predator population
• : prey growth rate (unlimited food, no predator)
• : predation rate
• : predator growth per prey eaten
• : predator death rate (no prey)

274.1. Behavior: closed orbits in the phase plane

Equilibrium: .

Around the equilibrium, the system oscillates in closed orbits (in space). Pre-mathematical solution by Lotka (1925) and Volterra (1926).

Energy-like invariant:

Trajectories follow contours of constant .

274.2. Oscillation dynamics

Cycle structure:

1. Prey abundant → predators thrive (predator pop rises)
2. Predators abundant → prey decline (prey pop falls)
3. Prey scarce → predators decline (predator pop falls)
4. Predators scarce → prey rebound (prey pop rises)
5. Cycle repeats

Lynx-hare population data (Hudson Bay records) shows this pattern beautifully.

274.3. Limitations and extensions

The pure Lotka-Volterra has issues:

• Closed orbits (mathematically) → no damping. Real systems usually damp out or destabilize.
• Unlimited prey if no predator. Add a carrying capacity:

This logistic version has stable spiral or limit cycle depending on parameters.

• Functional response (Holling Type II): predator becomes saturated:

with = handling time. Can produce limit cycles or even chaos.

274.4. Beyond two species

Generalizes to -species:

with = interaction matrix. Models food webs, ecological communities.

274.5. Outside biology

The model is structurally fundamental — appears wherever two stocks interact with the “predator gains by consuming prey” pattern:

• Sales force vs prospect pool: salespeople convert prospects (prey) and decline without them
• Consumer behavior: cyclical fashion (predator) consumes available trends (prey)
• Marketing budgets vs customer base

274.6. See also

• Feedback Loops — oscillation from coupled feedback
• Phase Plane — visualization of trajectories
• SIR/SEIR — related compartmental dynamics