After the First World War, the Italian marine biologist Umberto D'Ancona went through the fish catches of the Adriatic ports and saw something odd. During the war years, 1914 to 1918, the share of predatory fish (sharks, rays and the like) in the catch had gone up. Fishing had dropped because of the war, so he expected the gentle prey fish to benefit most. The opposite happened.
He asked the mathematician Vito Volterra, who published a pair of equations in 1926 that explained it. The American chemist Alfred Lotka had written down the same equations earlier (in 1910 for chemical reactions, then in his 1925 book on biology), so they carry both names.
The model has only two rules. Prey, left alone, multiply. Every meeting of a predator and a prey removes a prey and helps the predator. Predators, with nothing to eat, die out. That is all. The number of meetings is proportional to prey times predators, which is the only maths you need to know.
Below, the lime axis is prey and the red axis is predators. Each point on the plane is a possible state of the pond. Drag the dot anywhere and release it: the pond starts from there and the trail shows where it goes.
Drag in the top panel to set the starting numbers. The bottom panel shows both populations over time.
Start with plenty of prey and few predators. Prey boom. Predators find food everywhere and boom a little later. Now the predators are many and the prey are being eaten faster than they can breed, so the prey crash. The predators, starving, crash after them. With few predators left, the prey recover, and the loop starts again. The two curves chase each other, with the predators always lagging behind the prey, which you can see in the bottom panel.
In the pure model these loops never decay and never grow. Each starting point stays on its own closed loop for ever. Bigger loops swing closer to zero and take longer to go round. Try dragging the dot far from the centre: the cycle length in the readout grows.
In the middle sits a balance point where nothing moves. Its location holds a surprise. The number of prey at balance depends on the predators' habits: the more food a predator needs, the more prey the pond must hold. The number of predators at balance depends on the prey's traits: faster-breeding prey support more predators. Each species' balance is set by the other one.
Fishing is the part Volterra cared about. Nets catch both species, so fishing lowers the prey's growth rate and raises the predators' death rate. Because the balance of each species is set by the other's traits, the balance moves in a direction that is not obvious: prey go up and predators go down. Drag the fishing slider and watch the dashed lines and the balance point move.
The averages over a full cycle in this model are exactly the balance values, so this applies to the average numbers, not to any single year. Switch fishing off, as the war did, and the model's average predator number rises while the average prey number falls. That matches what D'Ancona saw.
The model is far too neat for real water. Real predators get full, real prey run out of food, and with small numbers a chance bad year can wipe a species out. Records of lynx and snowshoe hare pelts from the Hudson's Bay Company do show boom-and-bust cycles of roughly a decade, but a real cycle is never as clean as a closed loop. Still, the lesson that solved a puzzle in 1926 holds up: in a chase, hitting both sides does not help the prey.