Kill both species at the same rate and you get more of the pest
Kill the pest and its predator at the same rate, and the long-run average number of pests goes up. Not in a contrived example — in the standard model, exactly, with a one-line proof.
The predator–prey simulator opens on a stable-looking cycle: prey growth 1.1, predation rate 0.4, conversion 0.1, predator death 0.4. It prints two equilibrium values — 4.00 prey and 2.75 predators — and the populations swing round and round without settling on either.
They never settle, and it does not matter. The long-run average of each population is exactly its equilibrium value, whatever the swing looks like.
An average you can prove in two lines
The prey equation is ẋ = αx − βxy. Divide by x and it becomes ẋ/x = α − βy, and the left side is the derivative of ln x. Integrate over one complete cycle:
The left side is ln x at the end minus ln x at the start, and after a full cycle the population is back where it began, so that is zero. The right side is αT minus β times the integral of y. Rearranged, the mean of y over the cycle is α/β — 2.75, the equilibrium value. Do the same to the predator equation and the mean of x is γ/δ, which is 4.00.
Nothing about the size of the swing survives that algebra. A population oscillating gently between 3 and 5 and one crashing from 40 to 0.1 have the same mean. That is unusual enough to be worth a moment: the average of a wild oscillation is normally a fact about the oscillation, and here it is a fact about the four rate constants alone.
The panel measures 3.86 and 3.16 rather than 4.00 and 2.75, and the reason is on the panel too. It averages over 50 time units, and the cycle period is 2π/√(αγ) = 9.47, so the window holds 5.28 cycles. Average a wave over a fraction of a period and you get the fraction.
Now kill both of them
Add a harvest that removes the same fraction ε of each population per unit time: a net that catches both, a spray that kills both, a bad winter. The prey equation loses ε from its growth rate and the predator equation gains ε on its death rate. Nothing else changes.
Run the same two-line argument on the modified equations and the averages come out as (γ + ε)/δ for the prey and (α − ε)/β for the predators. Both constants moved, in opposite directions.
Put ε = 0.1 in. The average prey population goes from 4.00 to 5.00 — up 25% — and the average predator population goes from 2.75 to 2.50, down 9%. Kill both species and you get more of the one you were trying to kill.
The mechanism is not mysterious once the algebra has pointed at it. The prey average is set by what the predators need to break even, and the predator average by what the prey can support. Harvesting makes it harder to be a predator, so it takes more prey to sustain one, so the prey level the system settles around rises. Each species' average is controlled entirely by the other one's arithmetic.
Push it further and the model is blunt about the endpoint: at ε = α = 1.1 the predator average hits zero and the pest is alone.
Where this came from, and where it went
Umberto D'Ancona was going through Adriatic fish-market records from before, during and after the First World War, and found something odd. While the war had suspended most fishing, the share of the catch made up of predatory fish — sharks, rays, skates — went up. When the boats returned, it went back down. Less fishing had favoured the predators; more fishing favoured the prey.
He could not explain it, so he asked Vito Volterra, who was his father-in-law and one of the best analysts in Europe. Volterra wrote down the two equations, ran the averaging argument, and published the answer in 1926. Fishing is a harvest of both species at once, and the algebra says exactly which way each average moves.
Half a century later the same result turned up in rice paddies. Spraying a broad-spectrum insecticide kills the planthoppers and the spiders and wasps that eat them, and fields have repeatedly come back with more planthoppers than untreated ones. The effect is well enough documented to have a name, insecticide-induced resurgence, and a literature going back to the 1970s. It is the fish market again, with a different net.
What to distrust
The model deserves some suspicion. Lotka–Volterra has no carrying capacity, so its prey grow without limit if the predators vanish, and its cycles are neutrally stable: every orbit is closed and a nudge moves you permanently to a different one, which no real population does. You can watch that fragility directly in population growth, where a ceiling changes the behaviour completely.
What survives the caricature is the direction of the effect, and it survives because it does not depend on the cycle shape at all, only on which species is the bottleneck for which. That is also the practical version: if a pest is being held down by something that eats it, a treatment that hits both is a treatment that helps the pest. The arithmetic said so in 1926, and the fields kept saying so afterwards.
References (3)
- the short English announcement of the result Volterra (1926). Fluctuations in the Abundance of a Species considered Mathematically. Nature 118, 558–560.
- where the Adriatic fish-market data came from, and what Volterra did with it Scudo (1971). Vito Volterra and theoretical ecology. Theoretical Population Biology 2, 1–23.
- the same effect measured in a rice paddy half a century later Chelliah and Heinrichs (1980). Factors Affecting Insecticide-Induced Resurgence of the Brown Planthopper, Nilaparvata lugens, on Rice. Environmental Entomology 9(6), 773–777.