Lotka-Volterra Equations

The predator-prey form

Let x be the prey density and y the predator density. Four terms encode the biology.

• αx — prey reproduce exponentially in the absence of predators, with intrinsic rate α.
• −βxy — prey die through predation. Rate is proportional to the encounter frequency, which is xy under the mass-action assumption (well-mixed populations). β is the per-capita per-encounter consumption rate.
• δxy — eaten prey are converted into new predators with efficiency δ. δ is always smaller than β (most prey biomass is lost to respiration, not turned into predator tissue).
• −γy — predators die at rate γ in the absence of prey.

Combining gives the canonical pair:

dx/dt = αx − βxy
dy/dt = δxy − γy

Equilibrium and stability

Set both derivatives to zero. The trivial fixed point is (0, 0). The non-trivial one comes from setting x out of dx/dt and y out of dy/dt:

x* = γ/δ,    y* = α/β

Read off the biology: prey equilibrium depends only on predator parameters, predator equilibrium only on prey parameters. Counterintuitive but right — that is why fertilizing a forest does not raise its rabbit population. It raises the predator population.

Linearize around (x*, y*). The Jacobian has trace zero and positive determinant, so eigenvalues are pure imaginary — the equilibrium is a centre, and small perturbations produce closed orbits. This is the source of LV's famous neutral stability: every initial condition launches its own closed cycle, and the system never settles.

The conserved quantity

Divide one equation by the other and integrate. The result is an invariant of motion:

V(x, y) = δx − γ ln x + βy − α ln y

V is constant along trajectories. Each level set of V is one of the closed orbits. The orbit shape depends on amplitude: small-amplitude cycles are nearly elliptical; large-amplitude cycles run close to both axes and are sharply asymmetric.

Cycle period

Linearize around (x*, y*) and the imaginary eigenvalues are ±i√(αγ). The small-amplitude oscillation period is therefore:

T ≈ 2π / √(αγ)

For α = 0.6/yr (hare reproduction) and γ = 0.4/yr (lynx mortality), T ≈ 12.8 years — close to the famous 10-year lynx-hare cycle. Large-amplitude orbits run somewhat slower because they spend longer in the low-density tails.

Three forms of Lotka-Volterra

The same skeleton handles all pairwise species interactions. Sign and sign-of-cross-term define the type:

	Predator-prey	Competition	Mutualism
Prey/sp1 equation	dx/dt = αx − βxy	dN₁/dt = r₁N₁(1 − (N₁ + α₁₂N₂)/K₁)	dN₁/dt = r₁N₁(1 − (N₁ − α₁₂N₂)/K₁)
Predator/sp2 equation	dy/dt = δxy − γy	dN₂/dt = r₂N₂(1 − (N₂ + α₂₁N₁)/K₂)	dN₂/dt = r₂N₂(1 − (N₂ − α₂₁N₁)/K₂)
Sign of cross terms	− on prey, + on predator	− on both	+ on both
Equilibrium type	Centre (closed orbits)	Stable node, unstable node, or saddle depending on αs	Stable node if αs < 1; runaway if αs > 1
Famous result	Volterra principle, 4-quarter phase lag	Gause's competitive exclusion	Mutualism trap (unbounded growth if too tight)
Empirical match	Lynx-hare, Didinium-Paramecium	Paramecium aurelia vs caudatum	Cleaner-fish coral mutualism (steady state)
Real-world use	Fisheries dynamics, biocontrol	Niche overlap analysis	Pollination network theory

The competition case and Gause's principle

Competition LV starts from two logistic populations with cross-density terms. The phase plane has four possibilities depending on whether each species' isocline lies above or below the other's K:

• Species 1 wins if α₂₁ > K₂/K₁ and α₁₂ < K₁/K₂. Trajectories converge to (K₁, 0).
• Species 2 wins if the reverse.
• Founder effect — either species wins if both αs are large. The interior equilibrium is a saddle and outcome depends on initial conditions.
• Stable coexistence if both αs are small. The interior equilibrium is a stable node and both species persist.

Coexistence requires α₁₂ < K₁/K₂ AND α₂₁ < K₂/K₁ — informally, each species must compete with itself more than with the other. That is the formal version of Gause's competitive exclusion principle: identical species (αs = 1) cannot coexist; differentiated species (αs < 1) can. Niche partitioning is the mechanism by which αs are pushed below 1 in real systems.

Worked stability analysis

Take the predator-prey form with α = 1, β = 0.1, δ = 0.075, γ = 1.5. Equilibrium (γ/δ, α/β) = (20, 10). Period 2π/√(αγ) ≈ 5.1 yr. The Jacobian at the equilibrium is:

J = [ [α − βy, −βx], [δy, δx − γ] ] = [ [0, −2], [0.75, 0] ]

Eigenvalues from det(J − λI) = λ² + 1.5 = 0, so λ = ±i·1.225. Pure imaginary — neutrally stable centre — confirms the orbit interpretation. Phase advances at angular frequency 1.225 rad/yr → period 5.13 yr, matching the closed-form result. Starting from (x, y) = (15, 5), simulation traces an asymmetric closed loop with peaks lagging by a quarter cycle — prey peak first, predator peak about 1.3 years later.

Empirical evidence and famous failures

• Hudson's Bay Company lynx-hare records (1845–1935). Pelt counts oscillate with about a 10-year period; lynx peaks follow hare peaks by 1–2 years. Often cited as the textbook LV signature, though re-analysis shows hare-vegetation cycles also matter.
• Gause's Paramecium aurelia vs caudatum. Grown together, P. aurelia drives P. caudatum extinct. With Didinium as predator, prey species cycle and crash until refugia are added.
• Adriatic fisheries 1914–1923 (Volterra's prompt). When World War I cut fishing pressure to near zero, predator-fish share (sharks, rays) rose from 12 to 36 percent. Volterra's principle accounted for it.
• Phantom-midge in salmon hatcheries. Experimental introductions show classical LV cycling for short periods before the prey (zooplankton) functional response and refugia damp the oscillations toward a stable equilibrium.

Diagram sketch

• Panel A — time series. x(t) (prey, blue) and y(t) (predator, red) plotted on the same time axis. Both oscillate; the red curve lags the blue by about T/4.
• Panel B — phase portrait. Closed loop in the (x, y) plane around the fixed point (x*, y*). Several nested loops for several initial conditions, all closed and concentric.
• Panel C — competition isoclines. Two straight lines in the (N₁, N₂) plane; the regime (winner-takes-all, founder effect, coexistence) depends on which line lies above which.

Pitfalls

• Neutral stability is unrealistic. Real cycles damp toward a limit cycle or stable equilibrium. Adding prey carrying capacity (logistic growth without predators) and a Type II functional response generates the Rosenzweig-MacArthur model, which has either a stable equilibrium or a stable limit cycle depending on parameters.
• Mass-action encounters fail at high densities. Type II/III functional responses replace −βxy with saturating forms. The resulting models can become unstable at high productivity — the paradox of enrichment.
• Predicts extinction in finite simulations. Stochastic LV cycles wander further from equilibrium each cycle; eventually one species hits zero. Real systems persist because of refugia, immigration, and stabilization the basic model lacks.
• Mutualism form runs away. If reciprocal benefits exceed self-limitation, LV mutualism predicts unbounded growth. Reality bounds it via external resource limits the bare model does not capture.
• One predator, one prey. Real food webs have hundreds of species. Generalized LV with N-by-N matrices runs into May's stability bound and needs network methods.

Variants and extensions

• Rosenzweig-MacArthur (1963). LV with logistic prey growth and a Type II functional response. Adds a Hopf bifurcation and limit cycles.
• Ratio-dependent (Arditi-Ginzburg). Predation rate scales with prey-per-predator instead of prey × predator. Resolves the paradox of enrichment but is contentious.
• Generalized Lotka-Volterra. Vector form dN/dt = N · (r + AN). Used in microbial community modelling and theoretical ecology of large food webs.
• Spatial LV. Reaction-diffusion form with diffusion of x and y across a lattice. Generates travelling waves and pattern formation.