the Lotka-Volterra predator-prey model
/ loht-kah vol-TAIR-uh /
Picture an island with just two kinds of animal: rabbits, who eat grass, and foxes, who eat rabbits. When rabbits are plentiful the foxes feast and multiply; but as the foxes grow numerous they eat the rabbits down; with fewer rabbits the foxes starve and decline; and with fewer foxes the rabbits bounce back — and the whole story repeats. The Lotka-Volterra model is the simplest pair of differential equations that captures this endless chase, this rise and fall of hunter and hunted.
Let x be the prey (rabbits) and y the predators (foxes). The model is the pair x' = a x - b x y and y' = -c y + d x y. Read each term as a story: a x means prey breed on their own at rate a (unlimited grass); -b x y means prey are lost in proportion to how often a rabbit and a fox meet (the product x y counts encounters); -c y means predators die off at rate c with no food; and +d x y means predators gain from those same encounters. The two species are coupled precisely through the meeting term x y, the heart of the model.
Solve it and you find the populations do not settle to a steady value — they cycle, going round and round a closed loop in the rabbit-fox plane, predators peaking just AFTER prey, forever out of step. Remarkably, the quantity d x - c ln x + b y - a ln y stays exactly constant along every cycle, so the model has a conserved 'energy' and the orbits are nested closed curves. This was one of the first demonstrations that ecology obeys mathematics, and it explains the puzzling out-of-phase oscillations seen in real fur-trapping records.
Be honest about what it leaves out: with no grass limit the prey would explode without the foxes, the cycles are structurally fragile (a tiny change to the equations can spiral them in or out), and real predators get full and stop eating. The model is a first, idealized sketch — illuminating precisely because it is simple, not because it is realistic.
Take x' = x - x y and y' = -y + x y. The equilibrium where neither population changes is x = 1, y = 1 (set both right sides to zero). Start nearby, say at (1.5, 1), and the pair circles forever around (1, 1): prey rise, then predators rise, then prey fall, then predators fall.
A closed loop around the coexistence point — populations cycle but never settle.
The neat closed cycles are special to this exact form; they are not robust. Adding even a small carrying-capacity term for the prey usually turns the cycle into a spiral that winds in to a steady coexistence, so do not over-read the perpetual oscillation as a universal ecological law.