What is the Lotka-Volterra model?
It is the simplest mathematical description of the relationship between a predator and its prey. Picture an ecosystem with just two species: a prey that always has plenty of food (say, hares and grass) and a predator that feeds exclusively on that prey (say, lynxes). The model tracks how both populations — prey (x) and predators (y) — change over time.
It was proposed independently by Alfred Lotka (1925), while studying autocatalytic chemical reactions, and by Vito Volterra (1926), who was trying to explain why catches of predatory fish in the Adriatic had risen during World War I. Its most striking result is that the two populations never settle down: they oscillate in a perpetual cycle, each chasing the other.
How is the cycle generated?
The rate at which each population changes depends on four rates. Prey reproduce at a rate α and die when hunted; predators multiply by consuming prey (with efficiency δ) and die naturally at a rate γ:
The β·x·y term is the key: it represents encounters between prey and predators, proportional to how many there are of each. Under these rules, the system enters a feedback loop that repeats forever:
- 1.With plenty of prey and few predators, the prey multiply.
- 2.Abundant food makes the predators grow.
- 3.Too many predators make the prey collapse.
- 4.With no prey left to hunt, the predators starve.
- 5.With few predators, the prey recover… and the cycle restarts.
A real case: the lynx and the hare
The classic empirical example is the Canada lynx (Lynx canadensis) and the snowshoe hare (Lepus americanus). For nearly a century (≈1845–1935), the Hudson’s Bay Company recorded the number of pelts traded of both species, and those records reveal sustained oscillations with a cycle of roughly 9 to 10 years, in which the lynx peaks always arrive one or two years after the hare peaks — exactly the predator-prey lag the model predicts.
With an important caveat: modern field studies show that the hare cycle is not driven by predation alone, but also by its food supply and available vegetation. The lynx, rather than driving the cycle, largely follows it. That makes the lynx-hare pair an excellent illustration of the oscillations, but not a pure Lotka-Volterra system.
Pelt data: Elton, C. & Nicholson, M. (1942), The ten-year cycle in numbers of the lynx in Canada, Journal of Animal Ecology 11(2), 215–244.
Equilibrium and period
There is a single point where both populations could stay constant — the coexistence equilibrium, where the two derivatives vanish:
(There is also the trivial equilibrium at (0, 0): total extinction.) The system never stops at this point, but orbits around it. The time it takes to complete one loop — the period — is approximated, for small oscillations near the equilibrium, by:
This formula is exact only in the limit of tiny oscillations. The wider the orbit, the more the real period departs from this estimate (always upward); that is why the calculator also reports a numerical period, measured directly from the simulation.
Since the non-linear system has no analytical solution as a function of time, the trajectory is computed by numerical integration (4th-order Runge-Kutta), which reproduces the closed orbit without accumulating drift — something simpler methods, such as Euler, fail to sustain.