Lotka–Volterra equations

Based on Wikipedia: Lotka–Volterra equations

The Mathematics of Hungry Foxes

During World War One, something strange happened in the Adriatic Sea. With most of Europe's fishermen off fighting in the trenches, fishing nearly stopped. You'd expect the fish populations to boom—and they did. But here's the puzzle that captivated a young marine biologist named Umberto D'Ancona: the percentage of predatory fish in the catches actually went up during the war years, not down.

This seems backwards. Shouldn't fewer fishing boats mean more of everyone's favorite catch—the tasty prey fish that people actually want to eat?

D'Ancona brought this mystery to his future father-in-law, a mathematician named Vito Volterra. And Volterra, working independently from an American scientist named Alfred Lotka who'd been exploring similar ideas, developed a mathematical model that explained not just the Adriatic paradox, but something fundamental about how predators and prey dance together through time.

The result is now called the Lotka-Volterra equations. They're elegant, counterintuitive, and they reveal why making life better for prey sometimes just means more predators—not more prey.

Rabbits and Foxes: A Mathematical Fable

Let's think about rabbits and foxes. Imagine a square kilometer of meadowland with a population of each. The rabbits eat grass, which we'll assume is always plentiful. Left alone, they'd multiply exponentially—more rabbits making more baby rabbits making even more baby rabbits, population exploding toward infinity.

But rabbits aren't left alone. Foxes are hungry.

The foxes hunt rabbits. When a fox encounters a rabbit, sometimes that rabbit becomes dinner. The more foxes there are, and the more rabbits there are, the more of these deadly encounters occur. It's simple probability: double the foxes, double the rabbit deaths. Double the rabbits too, and you've quadrupled the killing.

Now consider life from the fox's perspective. Foxes don't photosynthesize. They don't graze. Without rabbits to eat, foxes starve. Their population naturally declines through death and emigration. But every rabbit they catch gives them energy to survive, to reproduce, to make more little foxes.

Here's where it gets interesting. These two populations are locked in a feedback loop that creates waves.

The Eternal Oscillation

Watch what happens over time. Start with lots of rabbits and few foxes. The foxes feast. Their population booms. But as fox numbers climb, rabbit deaths accelerate. Soon there are too many foxes chasing too few rabbits.

Now the foxes begin to starve.

Fox numbers crash. With fewer predators around, the surviving rabbits breathe easy. They multiply. Rabbit numbers climb back up. And with all those rabbits hopping around? The remaining foxes eat well again. Fox numbers start rising.

The cycle repeats. Forever.

This isn't just mathematical abstraction. Ecologists have documented these oscillations in real populations for over a century. The most famous example comes from the Hudson's Bay Company in Canada, which kept meticulous records of fur trapping going back to the 1800s. The data shows lynx and snowshoe hare populations rising and falling in beautiful, almost synchronized waves—the lynx population trailing the hares by about a year, just as the mathematics predicts.

Similar patterns appear in the wolves and moose of Isle Royale National Park, a remote island in Lake Superior where scientists have tracked the two species for decades.

The Paradox That Explains Everything

Now we can return to D'Ancona's puzzle about the Adriatic fish.

The Lotka-Volterra equations reveal something deeply counterintuitive: the equilibrium population of prey depends on characteristics of the predators, while the equilibrium population of predators depends on characteristics of the prey.

Read that again. It's strange.

If you somehow make life easier for rabbits—say, by providing more food or better shelter—you might expect more rabbits. But the math says otherwise. Better conditions for rabbits means rabbits reproduce faster. More baby rabbits. Fox food becomes more plentiful. Fox population grows. More foxes eat more rabbits. The system settles into a new equilibrium with more foxes but the same number of rabbits as before.

The benefit flows entirely to the predators.

This explains the Adriatic. During World War One, the reduction in fishing effectively reduced predation pressure on the prey fish. In the language of the model, this lowered the "death rate" parameter for prey. The equilibrium shifted—but it shifted toward more predatory fish, not more prey fish.

Ecologists call this the "paradox of enrichment," and it has practical consequences far beyond abstract mathematics.

Iron in the Ocean

In the 1990s and 2000s, scientists conducted a series of ambitious experiments. They dumped iron—thousands of kilograms of iron sulfate—into patches of open ocean. The idea was climate engineering. Iron is a limiting nutrient for phytoplankton, the microscopic plants that form the base of the marine food web. More iron, the reasoning went, would mean more phytoplankton. More phytoplankton would absorb more carbon dioxide from the atmosphere. Problem solved.

It didn't work out that way.

The iron did trigger phytoplankton blooms—brief explosions of microscopic plant life. But almost immediately, zooplankton and small fish moved in to feast on the bounty. The extra phytoplankton was consumed before it could sink to the ocean floor, taking its captured carbon with it.

The Lotka-Volterra equations predicted exactly this outcome. Enriching the environment for the "prey" (phytoplankton) didn't produce more prey at equilibrium. It produced more predators. The phytoplankton boom was eaten almost as fast as it appeared, and the carbon stayed in circulation.

Pesticides and Perversity

This same logic underlies what's sometimes called the "paradox of pesticides." Farmers spray chemicals to kill insect pests that eat their crops. But those pests have natural predators—ladybugs eating aphids, parasitic wasps laying eggs in caterpillars, spiders catching everything in their webs.

Pesticides often kill the predators along with the pests. And predator populations, being smaller and reproducing more slowly, recover more slowly than pest populations. The result can be worse infestations than before.

The math explains why. Remove predators, and you've changed the prey's equilibrium conditions. The pest population can now settle at a higher level. Sometimes the cure really is worse than the disease.

From Biology to Economics

In 1965, an economist named Richard Goodwin noticed something peculiar about the relationship between wages and employment. When unemployment is low, workers have bargaining power. They demand higher wages. But higher wages squeeze profits. Companies invest less, hire less, and unemployment rises. With more workers competing for fewer jobs, wages stagnate or fall. Profits recover. Investment picks up. Unemployment drops.

Wages and employment oscillate together, just like foxes and rabbits.

Goodwin mapped the Lotka-Volterra equations directly onto this economic dynamic. Workers play the role of predators, consuming profits the way foxes consume rabbits. Capital plays the role of prey, growing when wages are low but suppressed when workers take too large a share.

The parallel to Marxian class conflict was not lost on Goodwin or his readers. Here was a mathematical model showing how capitalism might contain inherent cycles of boom and bust, not because of policy mistakes or external shocks, but because of the fundamental structure of the relationship between labor and capital.

The same equations have since been applied to competition between firms, where companies effectively "prey" on each other's market share. They've been used to model the dynamics of sharing economies, platform competition, and market disruption. The mathematical structure is flexible enough to capture any situation where two populations feed off each other in asymmetric ways.

The Atto-Fox Problem

For all their elegance, the Lotka-Volterra equations have a serious limitation that becomes obvious if you simulate them on a computer.

Imagine our meadow with its foxes and rabbits. Start the simulation and watch the populations oscillate. In each cycle, the rabbit population crashes to very low numbers before recovering. In the mathematical model, this is fine. The equations are continuous—they deal with population densities, not individual animals. A density can be any positive number, including extremely small fractions.

But real animals come in discrete units. You can have one rabbit or zero rabbits, not 0.0000001 rabbits.

In many parameter regimes, the model drives the prey population down to absurdly small densities—the equivalent of one-billionth of a rabbit per square kilometer. Mathematically, this population bounces back. In reality, the rabbits are extinct. And shortly thereafter, so are the foxes.

Ecologists have given this the wonderfully evocative name "the atto-fox problem." An atto is the metric prefix for ten to the negative eighteenth power. An atto-fox is a quintillionth of a fox—roughly five hundred-millionths of a fox across the entire surface of the Earth.

In practical terms, that's no foxes at all.

The lesson is that deterministic mathematical models capture average behavior, not the randomness inherent in small populations. When numbers get small enough, chance dominates. A few unlucky encounters, a harsh winter, a disease outbreak—any random fluctuation can push a real population across the extinction threshold that the smooth mathematical curves glide right through.

The Scientists Behind the Equations

Alfred Lotka was an unusual figure—a physical chemist and statistician born in what is now Ukraine to American parents, educated in Europe, and employed for most of his career by an American insurance company. His path to predator-prey dynamics was circuitous. In 1910, he was studying autocatalytic chemical reactions—reactions where the product catalyzes its own formation, creating feedback loops. The mathematics led him naturally to population biology.

Lotka recognized that the same differential equations governing molecular reactions could describe species interactions. A population reproducing exponentially is mathematically identical to a chemical reaction where the product accelerates its own production. Predation is just another rate equation.

By 1925, Lotka had published these ideas in a book on mathematical biology. But the equations that bear his name achieved their fame through a different route.

Vito Volterra was a titan of Italian mathematics—a senator, a public intellectual, and one of the few Italian academics to refuse the oath of loyalty to Mussolini's Fascist regime. His interest in population dynamics came from that family connection: his daughter's boyfriend puzzling over Adriatic fish data.

Volterra developed the model independently in 1926, unaware of Lotka's earlier work. When he learned of it, he credited Lotka properly, and posterity has remembered them together.

Extensions and Arguments

The original Lotka-Volterra model makes assumptions that no real ecosystem satisfies. Prey have unlimited food. Predators have unlimited appetites. Neither species has age structure or spatial distribution. The environment never changes. No evolution occurs.

Yet the model's core predictions—oscillation and the counterintuitive equilibrium properties—turn out to be robust. They appear in more sophisticated models that relax these assumptions.

The Rosenzweig-MacArthur model, developed in the 1960s, adds density-dependent prey growth and a more realistic "functional response"—the recognition that predators can only eat so fast, no matter how abundant prey become. A fox can only digest so many rabbits per day. This model still oscillates, though the oscillations can destabilize in interesting ways.

A more radical departure came in the 1980s with the Arditi-Ginzburg model, which proposed that predation rates depend not on absolute prey abundance but on the ratio of prey to predators. This "ratio-dependent" approach generated intense debate among theoretical ecologists—one of those scientific controversies that outsiders find baffling but that touches on deep questions about how to mathematically represent biological reality.

The argument continues. Different models fit different systems better. The original Lotka-Volterra equations remain valuable not because they're correct in every detail, but because they capture something essential about predator-prey dynamics in a form simple enough to analyze and understand.

The Conservation Law Hidden in Ecology

There's one more mathematical surprise buried in the Lotka-Volterra equations. Physicists will recognize it immediately.

In classical mechanics, many systems have conserved quantities—values that remain constant as the system evolves. Energy is the famous example. A pendulum swings back and forth, converting potential energy to kinetic and back again, but the total energy never changes.

The Lotka-Volterra system has a similar conserved quantity. As fox and rabbit populations cycle through their endless oscillations, a particular combination of their densities remains constant. Plot the populations against each other—foxes on one axis, rabbits on the other—and the system traces closed curves, never spiraling inward or outward, forever repeating the same loop.

This conserved quantity functions mathematically like energy. It implies that the ecological system, despite all its biological complexity, obeys a kind of conservation law. The oscillations aren't damped by friction or amplified by energy input. They're truly perpetual, at least in the idealized mathematical world where atto-foxes can exist.

Reality, of course, has friction. Real populations experience random shocks, environmental changes, evolution, and all the messy complications the model ignores. But the mathematical structure suggests something profound: ecology, like physics, may have deep symmetries and conservation principles waiting to be discovered.

Why This Matters

The Lotka-Volterra equations are over a century old. They describe no real ecosystem with quantitative accuracy. Their assumptions are violated everywhere we look.

And yet.

They explain why protecting prey doesn't always help prey. They explain why enriching ecosystems can sometimes destabilize them. They explain business cycles and class conflict and the failure of iron fertilization. They reveal that predator and prey are locked in an eternal dance where neither can win permanently, where boom follows bust follows boom in a rhythm older than humanity.

The equations remind us that complex systems often behave in ways that violate common sense. Intuition fails when feedback loops dominate. The obvious solution—more food for prey, fewer predators for prey, pesticides for pests—can make things worse.

This is the gift of mathematics to ecology, to economics, to anyone trying to understand systems where everything affects everything else. The equations don't tell us what to do. But they warn us that what seems obviously right may be precisely wrong.

Somewhere in the Adriatic, a century after Volterra first scribbled his equations, predatory fish still swim. Their populations rise and fall in waves that a mathematician and his lovesick future son-in-law first noticed in the catch records of a world at war.