Predator-Prey Dynamics

coupled predator-prey oscillations

Worked example — Predator-Prey Dynamics

Simulation span (≈ 5.3 T)	50

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The predator peak always arrives a quarter-cycle late 🖖

The predator peak always lags behind the prey peak. When prey are abundant, predators thrive and multiply — but by the time predator numbers peak, prey have already been depleted. The predator population then crashes from starvation, giving prey a chance to recover, and the cycle repeats. This quarter-cycle lag is a mathematical consequence of the Lotka-Volterra equations and matches observations in real predator-prey systems like lynx and snowshoe hare.

closed orbits, extinction, and the limits of the model 🖖

The exact Lotka-Volterra equations have a conserved quantity — H = δx − γ ln x + βy − α ln y — that stays constant along every trajectory. This means orbits are mathematically closed loops: populations oscillate forever without ever spiralling inward or outward, and without ever truly reaching zero. Start close to the equilibrium (x★, y★) and you get smooth, nearly circular loops. Start far away — say x₀ = 1, y₀ = 100 — and the orbit is a long, thin excursion that produces spiky sawtooth cycles: predators crash from starvation, prey recovers explosively, then the predator bonanza collapses prey again. The spike shape is not a numerical artefact; it is the correct large-amplitude solution. What is an artefact is survival itself. In a continuous ODE the populations approach zero but never reach it. In a real ecosystem with discrete animals, the low-point trough means a handful of individuals — and random events then drive actual extinction. The simulation sidesteps this by clamping populations at 0.01, so it shows the mathematical orbit rather than biological reality. A stochastic model would instead show that extreme initial imbalances usually end in collapse.

Volterra's paradox of the pesticide 🖖

Vito Volterra built these equations in 1926 to solve a puzzle posed by his son-in-law, biologist Umberto D'Ancona: why had the share of predatory fish in Adriatic catches risen during World War I, when fishing had nearly stopped? The answer, Volterra's principle, is that mortality applied evenly to both species raises the average prey population and lowers the predator's. It explains why broad-spectrum insecticides can backfire — killing pests and their natural enemies alike often leaves more pests than before.

Problem solved in full

Lotka–Volterra equilibrium with α = 1.1 and the measured averages 6 steps
Lotka–Volterra with α = 1.1, β = 0.4, δ = 0.1, γ = 0.4. Find the equilibrium, prove the long-run averages equal it exactly, then explain why the panel's measured averages do not.
1. Two coupled equations, and every term is a rate of encounter or a rate of birth and death. The product xy is the only nonlinearity, and it is just "how often do they meet".
  
  $\overset{x}{˙} = α x - β x y, \overset{y}{˙} = δ x y - γ y$
2. Set both derivatives to zero. Each equation determines the other species' equilibrium, which is the first sign that this system is not going to behave like two independent populations.
  
  $\overset{x}{˙} = 0, \overset{y}{˙} = 0 ⟹ x^{⋆} = \frac{γ}{δ} = \frac{0.4}{0.1} = 4.00, y^{⋆} = \frac{α}{β} = \frac{1.1}{0.4} = 2.75$
3. Now the exact averaging result. Divide the prey equation by x and integrate over one full period — the left side is the change in ln x, and over a closed orbit that is zero.
  
  $\frac{x ˙}{x} = α - β y ⟹ \int_{0}^{T} \frac{x ˙}{x} d t = [ln x]_{0}^{T} = 0 = α T - β \int_{0}^{T} y d t$
4. Rearranging gives the time average, and nothing about the orbit's size survives the algebra. A population swinging from 1 to 40 has the same mean as one barely moving.
  
  $⟨ y ⟩ = \frac{α}{β} = y^{⋆}, ⟨ x ⟩ = \frac{γ}{δ} = x^{⋆} exactly, for every amplitude$
5. The period comes from linearising at the fixed point. The Jacobian has zeros on its diagonal and a negative product off it, so the eigenvalues are purely imaginary — the cycles neither grow nor decay.
  
  $J^{⋆} = (0 δ y^{⋆} - β x^{⋆} 0) \Rightarrow λ = \pm i α γ, T = \frac{2 π}{α γ} = 9.47$
6. Finally the harvesting term. It shifts α and γ in opposite directions, and the averages follow immediately from step 4 without solving anything.
  
  $harvest both at ε : ⟨ x ⟩ = \frac{γ + ε}{δ} ↑, ⟨ y ⟩ = \frac{α - ε}{β} ↓$
Answer

The averages are exactly 4.00 and 2.75, and the panel reads 3.86 and 3.16 because 50 time units is 5.28 cycles, not 5. Average a wave over a fraction of a period and you get the fraction, not the average. What makes the exact result worth having is that it holds for every orbit, however violent — the amplitude cancels out of the integral. And it has a consequence nobody expects. Harvest both species at the same rate ε and the averages become (γ+ε)/δ and (α−ε)/β: killing predators and prey indiscriminately raises the average prey population and lowers the predators. Volterra derived exactly that in 1926 to explain a puzzle in Umberto D’Ancona’s Adriatic landings: the share of the catch made up of predatory fish rose during the First World War, when fishing had largely stopped. Spray a field with a broad insecticide and the model predicts more pests, not fewer.

$⟨ x ⟩ = \frac{γ}{δ} = 4.00, ⟨ y ⟩ = \frac{α}{β} = 2.75, T = \frac{2 π}{α γ} = 9.47$

Learning path

Growth with a limit

Leads to Epidemics

References (1)

Insight block 3 — the Adriatic catches, and the principle drawn from them: V. Volterra, "Fluctuations in the Abundance of a Species considered Mathematically." Nature 118, 558–560, 1926.

By the 42Math team · Updated 2026-05-30

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Example problems

balanced cycle - Coupled populations oscillate persistently — predators trail prey peaks by roughly a quarter cycle.
predator crash - Low conversion efficiency (δ = 0.05) means predators cannot reproduce fast enough to sustain their population.
prey crash - High predation rate (β = 0.8) suppresses prey to critically low levels — the system is near collapse.