Lotka-Volterra

276. Lotka-Volterra

The classical predator-prey model. Two species:

\cdot 𝑥 = 𝛼 𝑥 - 𝛽 𝑥 𝑦 prey

\cdot 𝑦 = 𝛿 𝑥 𝑦 - 𝛾 𝑦 predator

where:

$𝑥$ : prey population, $𝑦$ : predator population
$𝛼$ : prey growth rate (unlimited food, no predator)
$𝛽$ : predation rate
$𝛿$ : predator growth per prey eaten
$𝛾$ : predator death rate (no prey)

276.1. Behavior: closed orbits in the phase plane

Equilibrium: $(𝑥^{*}, 𝑦^{*}) = (\frac{𝛾}{𝛿}, \frac{𝛼}{𝛽})$ .

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

Energy-like invariant:

𝐻 (𝑥, 𝑦) = 𝛿 𝑥 + 𝛽 𝑦 - 𝛾 ln 𝑥 - 𝛼 ln 𝑦 = const

Trajectories follow contours of constant $𝐻$ .

276.2. Oscillation dynamics

Cycle structure:

Prey abundant → predators thrive (predator pop rises)
Predators abundant → prey decline (prey pop falls)
Prey scarce → predators decline (predator pop falls)
Predators scarce → prey rebound (prey pop rises)
Cycle repeats

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

276.3. 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:

\cdot 𝑥 = 𝛼 𝑥 (1 - \frac{𝑥}{𝐾}) - 𝛽 𝑥 𝑦

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

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

\cdot 𝑥 = 𝛼 𝑥 - \frac{𝛽 𝑥 𝑦}{1 + ℎ 𝑥}

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

276.4. Beyond two species

Generalizes to $𝑛$ -species:

\cdot 𝑥_{𝑖} = 𝑥_{𝑖} (𝑟_{𝑖} + \sum_{𝑗} 𝑎_{𝑖 𝑗} 𝑥_{𝑗})

with $𝑎_{𝑖 𝑗}$ = interaction matrix. Models food webs, ecological communities.

276.5. 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

276.6. See also

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