Predator and Prey: The Mathematics of a Chase That Never Ends

An economist’s puzzle about fish

After the First World War, an Italian biologist named Umberto D’Ancona noticed something odd in the fish-market records of the Adriatic. During the war years, when fishing had largely stopped, the catches that did come in contained a higher proportion of predatory fish, sharks and rays, and a lower proportion of the prey fish they ate. Less fishing had somehow favored the predators over their prey. D’Ancona could not explain it, so he took the problem to his father-in-law, the mathematician Vito Volterra. Volterra answered it in 1926 with a pair of equations, and the American biophysicist Alfred Lotka had arrived at nearly the same equations independently the year before. The result, now called the Lotka-Volterra model, is the foundation of mathematical ecology.

The model is built from a few simple assumptions about a predator population and a prey population sharing a habitat. Left alone with plenty of food, the prey breed and their numbers grow. The predators, by contrast, cannot live without prey to eat, so left without food their numbers decline. The two are coupled: predators eat prey, which hurts the prey population and feeds the predator population. Written as rates of change, the prey grow on their own but are eaten at a rate that depends on how many predators are around to eat them, while the predators die off on their own but are sustained at a rate that depends on how much prey is available.

The cycle that falls out

The interesting thing is what these assumptions produce when you let them run. Neither population settles to a steady value. Instead they chase each other around a perpetual cycle, and the logic of the cycle is worth walking through because it is the heart of the whole subject.

Start with abundant prey. With so much food, the predators thrive and their numbers climb. But a growing army of predators eats prey faster than the prey can reproduce, so the prey population, despite its early abundance, begins to crash. Now there is a glut of predators and not enough prey to feed them, so the predators begin to starve, and their numbers fall in turn. With few predators left, the surviving prey are released from pressure and begin to recover, breeding back toward abundance, and once they are plentiful again the predators rebound to feast on them, and the whole cycle repeats. The populations rise and fall forever, out of step with each other.

 population
    |    prey                          prey
    |   .-''-.                        .-''-.
    |  /      \                      /      \
    | /        \      predator      /        \
    |/          \    .-''-.        /          \
    |            \  /      \      /            \  /
    |   .-''-.    \/        \    /     .-''-.    \/
    +-/--------\-/--\---------\/-----/--------\-/----> time
       predator     '._.'   (predator peak lags the prey peak)

Notice the lag. The predator peak always comes after the prey peak, because the predators are responding to a food supply that has already started to change. This phase shift, the predators trailing the prey by a quarter cycle, is the model’s signature, and it is exactly what lets the cycle sustain itself rather than collapsing to a fixed point.

The model also answers D’Ancona’s fish puzzle. Volterra showed that fishing, which removes both predators and prey at once, actually shifts the long-run average in favor of the prey, because the predators, being the more fragile population, are hurt more by the extra mortality. Stop fishing, as the war did, and the balance tilts back toward the predators. The higher proportion of sharks in the wartime catches was the prediction of the equations, recovered from a market ledger.

How seriously to take it

The honest caveat is that the basic Lotka-Volterra model is a caricature. It assumes the prey would grow without limit in the absence of predators, ignores everything else in the ecosystem, and produces cycles whose exact size depends awkwardly on the starting conditions. Real populations face food limits, weather, disease, and many interacting species, and ecologists have spent a century adding those complications back in. The most famous real-world cycle that resembles the model, the roughly ten-year boom and bust of snowshoe hares and the lynx that hunt them, recorded over nearly two centuries in the fur-trading ledgers of the Hudson’s Bay Company, turns out to be driven by more than just the two species, even though the hare and lynx numbers do rise and fall in the lagged dance the model predicts.

None of that diminishes the achievement. Before Lotka and Volterra, the idea that two populations could generate sustained oscillations purely through their interaction, with no external driver, was not obvious and perhaps not even imaginable. The model showed that cycling can be an internal property of a predator-prey relationship, a consequence of the coupling itself. That insight, that simple rules of interaction can produce endless dynamic behavior rather than a quiet equilibrium, reaches far beyond ecology, and it started with a question about why there were more sharks in the market during a war.