Population Ecology

In 1924 Charles Elton, a young Oxford zoologist who had been on expeditions to Arctic Spitsbergen, pointed out that many animals rise and fall in regular cycles. The most spectacular record came from the Hudson's Bay Company, which had counted the furs brought in by Canadian trappers for over a century. Lynx numbers peaked about every ten years, just after the snowshoe hares they hunt. In his 1927 book Animal Ecology Elton also set out the idea of the food chain, and of each species' niche, its role in the community.

At the same time two scientists wrote down the mathematics. Alfred Lotka, an American chemist interested in the physics of living systems, and Vito Volterra, one of Italy's leading mathematicians, each described a predator and its prey with a pair of equations. Prey multiply unless eaten. Predators multiply in proportion to the prey they catch, and die without them. The solutions go round in endless cycles, with predators lagging about a quarter of a cycle behind their prey.

Volterra had been set the problem by his son-in-law, the marine biologist Umberto D'Ancona. During the First World War, when fishing in the Adriatic was reduced, sharks and other predatory fish had made up a larger share of the catch. The equations showed why: reduced fishing helps the predators more than the prey. The same logic explains why a pesticide that kills pests and their natural enemies alike can leave the pests more numerous than before.

The equations were easy to write and hard to test in the wild. In Moscow in the early 1930s, Georgii Gause, still in his early twenties, tested them with microbes. He grew two species of Paramecium on the same bacterial food. Each alone grew along a logistic curve. Together, one always drove the other extinct. When he paired species that fed in different parts of the tube, they coexisted. The rule that complete competitors cannot coexist became known as the competitive exclusion principle.

In the field, ecologists argued about what really controls numbers. Alexander Nicholson in Australia held that crowding does: as a population grows, competition for food and space raises deaths and cuts births. His blowflies, kept in cages on a fixed ration of meat, cycled in number for years. Herbert Andrewartha and Charles Birch, also working in Australia, replied in 1954 that most insects never get crowded, because the weather kills them first. The argument ran for two decades and ended with both sides partly right.

Then, in 1974, Robert May, a physicist who had turned to ecology, showed that crowding alone can produce wild fluctuations. In a population with separate generations, if numbers overshoot strongly when crowded, the simplest density-dependent rule gives steady numbers, then two-year cycles, then four-year cycles, and then chaos, fluctuations that never repeat and cannot be predicted far ahead. With Michael Hassell and John Lawton he fitted the model to real insect populations, and found most of them in the stable range. Whether many wild populations are chaotic is still debated. What changed for good was the assumption that erratic numbers need an outside cause.

The logistic model says that a population of size $N$ grows at the rate

$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right),$$

where $r$ is the growth rate when the population is small and $K$ is the carrying capacity. Take a fish stock with $K = 1{,}000{,}000$ tonnes and $r = 0.4$ per year. Its yearly growth, the surplus a fishery could remove without shrinking the stock, depends on how big the stock is:

Stock (tonnes)	Growth per year (tonnes)
100,000	36,000
250,000	75,000
500,000	100,000
750,000	75,000
900,000	36,000

Growth is small when the stock is small, because there are few fish to breed. It is also small near $K$, because the fish are crowded. It peaks at half the carrying capacity, and the peak is the maximum sustainable yield:

$$\text{MSY} = \frac{rK}{4} = \frac{0.4 \times 1{,}000{,}000}{4} = 100{,}000 \text{ tonnes per year}.$$

The danger is in what happens below the peak. Suppose managers set a fixed quota of 100,000 tonnes, but the stock has already fallen to 400,000 tonnes. It now grows by only $0.4 \times 400{,}000 \times 0.6 = 96{,}000$ tonnes a year, less than is taken. The stock shrinks, so it grows even more slowly, and the gap widens. In this model the fishery removes the last fish in about twenty years. A quota of 110,000 tonnes, just 10% above the maximum, also empties the stock within about twenty years, even starting from the best possible size.

A smaller fixed quota is safer, but only down to a point. A quota of 80,000 tonnes has two balance points, at about 724,000 tonnes and 276,000 tonnes. The upper one is stable, and a stock near it recovers from a bad year. Below 276,000 tonnes the stock can no longer keep up, and it collapses even under this modest quota. Real stocks add uncertainty about $r$ and $K$, random bad years and fish counted from the catch itself, which is why fixed quotas near the maximum led to collapses, and why modern management aims well below it.

The models of Lotka, Volterra and Gause dealt with one species, or two. Real habitats hold hundreds, competing, preying on each other and depending on each other. How so many species manage to live together, and how they are organised, became the subject of community ecology.

The equations also travelled the other way. The Lotka–Volterra system became a standard example in the theory of differential equations, and May's population model became a founding example of chaos theory.