Predator-Prey Models: Lotka-Volterra Equations

Population Dynamics Models: Predator-Prey

One of the most beautiful demonstrations of nonlinear dynamics is the periodic fluctuations in the population numbers of predators and prey. The data on lynx and snowshoe hare in Canada show stable cycles lasting about 10 years: no central control, no plan—pure interaction dynamics.

Lotka-Volterra Equations

Alfred Lotka (1925) and Vito Volterra (1926) independently derived a system of ODEs describing the dynamics of two interacting populations:

dx/dt = αx − βxy

dy/dt = δxy − γy

Deciphering each term:

• x — prey population (hares, lemmings); y — predator population (lynxes, wolves)
• αx — exponential growth of prey in the absence of predators; α — specific growth rate
• −βxy — decrease in prey due to encounters with predators; β — predation rate (proportional to the number of encounters x·y)
• +δxy — growth of predators due to consuming prey; δ — conversion efficiency
• −γy — natural mortality of predators in the absence of prey

Equilibrium Analysis

Trivial equilibrium: x=0, y=0 (extinction of both).

Nontrivial equilibrium: dx/dt = 0, dy/dt = 0:

x* = γ/δ, y* = α/β

Deciphering x*: the equilibrium population of prey depends on the predator's parameters (γ, δ), but not the prey's growth parameter α. Volterra’s paradox: improving conditions for the prey (increasing α) leads to an increase in predator numbers, not prey!

Stability: Jacobian at the point (x*, y*):

J = [[0, -βx*], [δy*, 0]]

Eigenvalues: ±i√(αγ) — purely imaginary. Equilibrium is a center: the system oscillates periodically without damping. Oscillation period T ≈ 2π/√(αγ).

First integral: H(x,y) = δx − γln(x) + βy − αln(y) = const. Trajectories are closed curves in phase space (x,y).

Model Extensions

Logistic prey growth: dx/dt = αx(1 − x/K) − βxy. With carrying capacity K → stable spiral equilibrium instead of a center.

Predator functional response (Holling): type I: g(x) = βx (linear). Type II: g(x) = βx/(x + K_h) — saturation at high prey numbers. Type III: g(x) = βx²/(x² + K_h²) — sigmoidal, with threshold switching.

Spatial structure: reaction-diffusion equations add spatial relocation → waves, spirals, Turing spots.

Numerical Example

Parameters: α=0.8 (prey), β=0.02, δ=0.01, γ=0.5. Equilibrium: x* = 0.5/0.01 = 50, y* = 0.8/0.02 = 40.

Oscillation period T = 2π/√(0.8·0.5) ≈ 9.9 (years, given appropriate units). Initial conditions (x₀=80, y₀=20) — above-normal prey, below-normal predator. Forecast: predator numbers will rise, then prey numbers will fall, then predator numbers will fall, then prey numbers will recover. Cycle length ≈10 years. Matches lynx-hare observation data in Canada!

Real Applications

Fisheries management: LV with harvesting: dx/dt = f(x,y) − Ex, where E is fishing effort. Optimal E ≈ α(1 − E_MSY/α). Over-harvesting → population collapse.

Biological invasions: introduction of a new predator → temporary collapse of prey population, then recovery.

Pharmacology: “drug-bacteria” models for optimizing antibiotic dosing.

Assignment: Solve the Lotka-Volterra system numerically (scipy.integrate.odeint) with α=1, β=0.1, δ=0.075, γ=1.5, x₀=10, y₀=5. (1) Plot the time series x(t) and y(t). (2) Plot the phase portrait (x,y). (3) Add logistic restriction (K=100). How does the phase portrait change? (4) Add “harvesting” of prey (E=0.2·x). At what E does the system collapse?