Lotka–Volterra

The model

Alfred Lotka was a statistician and physical chemist working in the United States. In 1925, he published Elements of Physical Biology, in which he proposed a pair of coupled differential equations to describe the interaction between a predator species and its prey. Independently and almost simultaneously, the Italian mathematician Vito Volterra derived the same equations in 1926, motivated by his son-in-law Umberto D’Ancona’s observation that predatory fish in the Adriatic Sea had increased in proportion during World War I, when fishing pressure dropped. The equations are: dx/dt = αx − βxy for the prey, and dy/dt = δxy − γy for the predator. Prey grow exponentially at rate α in the absence of predators; predators consume prey at a rate proportional to the product of both populations (βxy); predators convert consumed prey into offspring with efficiency δ; and predators die at rate γ in the absence of prey. The model is deliberately stripped of every complication — no carrying capacity, no immigration, no age structure, no time delays — and this is precisely its value. It isolates the core feedback loop: more prey feeds more predators, more predators reduce prey, fewer prey starves predators, fewer predators allows prey to recover. The result is perpetual oscillation, with the predator population lagging behind the prey in a quarter-cycle phase shift.

Why the orbits close

Unlike most nonlinear systems, the Lotka–Volterra equations possess a conserved quantity — an integral of motion analogous to energy in classical mechanics. Volterra himself identified it: H = δx − γ ln(x) + βy − α ln(y). This function remains constant along every trajectory. In the phase portrait, each initial condition determines a value of H, and the resulting orbit is a level curve of this function. Since H is conserved, the system returns exactly to its starting point after each cycle. The orbits are closed curves, not spirals. This is mathematically exceptional. Most predator-prey models with even slightly more realistic assumptions — logistic prey growth, predator satiation, time delays — lose this conservation law, and their orbits spiral inward toward a stable equilibrium or outward toward extinction. The Lotka–Volterra system sits on a knife edge: structurally unstable, in the language of dynamical systems. Any perturbation to the equations destroys the closed orbits. This is why numerical integration with Euler’s method fails spectacularly — the discretization error acts as a perturbation that pumps energy into the system, causing orbits to spiral outward. The simulation above uses fourth-order Runge–Kutta integration precisely to preserve H as accurately as possible.

Real populations

The most famous empirical test of the Lotka–Volterra equations comes from the Hudson’s Bay Company fur-trading records, which tracked pelts of snowshoe hares and Canadian lynx across nearly a century (1845–1935). The data show striking oscillations with a period of roughly nine to eleven years, and the lynx population clearly lags behind the hare — qualitatively matching the model’s prediction. But the fit is imperfect. The oscillations vary in amplitude, the period is not perfectly regular, and both populations sometimes crash far lower than the simple equations predict. Similarly, the Kaibab Plateau deer irruption of the 1920s — in which the removal of predators led to a population explosion followed by mass starvation — illustrates predator-prey coupling, but the dynamics involved carrying capacity, overgrazing, and habitat degradation that the basic model ignores. The discrepancies are instructive. Real ecosystems involve stochastic variation (weather, disease), spatial structure (prey refuges, migration corridors), multiple trophic levels (the hare also has food plants that can be depleted), time delays (gestation periods, maturation), and density-dependent effects that the Lotka–Volterra equations deliberately omit. The model does not fail because it is wrong; it fails because it is too simple — and knowing exactly how it is too simple has driven a century of theoretical ecology.

Beyond the basic model

The most immediate refinement is to replace the linear predation term βxy with a saturating functional response. C. S. Holling proposed three types in 1959: Type I (linear, as in the basic model), Type II (hyperbolic saturation, modeling handling time — a predator cannot eat faster than it can process prey), and Type III (sigmoidal, modeling prey switching or learning). The Type II response, βxy / (1 + βh x) where h is handling time, produces qualitatively different dynamics: the closed orbits are replaced by a stable limit cycle or a stable spiral, depending on parameters. A separate lineage of models extends Lotka–Volterra to competition between species sharing a resource. The Lotka–Volterra competition equations, formalized by Gause in the 1930s, predict competitive exclusion or coexistence depending on the ratio of intraspecific to interspecific competition coefficients. And in 1972, Robert May published his landmark paper showing that large, randomly assembled food webs are less stable than small ones — the complexity-stability paradox that overturned decades of ecological intuition and launched the modern study of food-web architecture.

What the simulation cannot show

The Lotka–Volterra equations are deterministic and continuous. In a real population of discrete individuals, stochastic fluctuations can push either species to zero — and zero is absorbing. Stochastic extinction is the rule rather than the exception for small populations governed by predator-prey dynamics; the smooth, eternal oscillations of the ODE are an idealization that finite populations cannot sustain. The simulation also ignores spatial structure. In reality, prey may have refuges where predators cannot reach them; predator-prey waves can travel across landscapes, producing spatial patterns invisible to a well-mixed model. Age structure matters: juvenile prey are often more vulnerable than adults, and predator fecundity depends on recent feeding history, not instantaneous prey density. Multi-species food webs introduce indirect effects — trophic cascades, apparent competition, intraguild predation — that are entirely absent from a two-species model. And evolution is silent: prey are not evolving faster running speeds, predators are not evolving sharper teeth. The Red Queen hypothesis, the co-evolutionary arms race that shapes real predator-prey systems over evolutionary time, lies entirely outside the model’s scope.

Connection to epidemiology

The structural similarity between predator-prey and epidemiological models runs deeper than analogy. In the SIR model formulated by Kermack and McKendrick in 1927, susceptible individuals (S) are “consumed” by infected individuals (I) through transmission, much as prey are consumed by predators. The infected population grows when transmission exceeds recovery, just as predators grow when consumption exceeds mortality. The recovered class (R) acts as a sink, analogous to dead predators. The Kermack–McKendrick threshold theorem — that an epidemic can only invade when the basic reproduction number R₀ exceeds one — has a direct parallel in ecology: a predator can only invade when its per-capita growth rate at the prey equilibrium is positive. Both systems exhibit the same qualitative behavior: damped oscillations toward equilibrium (in the SIR case, herd immunity; in predator-prey, the coexistence equilibrium of more realistic models). The mathematical toolkit — stability analysis, bifurcation theory, stochastic simulation — is shared almost entirely between the two fields. This is not a coincidence. Both are instances of a broader class of systems in which one population exploits another, and the exploitation depletes the resource on which the exploiter depends. The feedback loop is universal; only the biological details differ.