In 1944 the US Coast Guard put 29 reindeer on St. Matthew Island, a treeless island in the Bering Sea with no predators, as an emergency food source. By summer 1963 there were about 6,000. The next winter or two the herd collapsed to 42 animals, and by the 1980s there were none. Nobody shot them. The island simply could not feed the number that compounding growth had produced.
A wolf pack, a coral reef, a cornfield and a cod fishery are all the same kind of puzzle: living things that multiply, eat each other, compete for the same food, and pass the same atoms around. Ecology is the course that gives you the few equations that explain why numbers boom, why they crash, and why they sometimes balance for centuries.
A population grows the way money does at compound interest. If each individual leaves on average more than one surviving offspring, the next generation is bigger, and every extra individual adds to the next increase. Ecologists write the per-capita growth rate as r. With nothing in the way, the population follows a curve that doubles every ln 2 / r time units, about 0.69 / r.
But food, space and disease do not scale up. Add a brake that gets stronger as the population nears what the place can support, the carrying capacity K, and you have the logistic equation, written by Pierre-François Verhulst in 1838.
The curve is an S. Growth looks exponential at first, then slows, then flattens at K. The fastest absolute growth is at half of K, where the population adds r K / 4 individuals per unit time. That one fact runs fisheries, which we meet below. The St. Matthew reindeer show the limit of the model: they overshot K and crashed, because food recovers slowly. A study by Klein linked the crash to scarce food combined with an exceptionally severe winter. Real populations lag behind their own brakes.
Illustrative numbers, not a species. Logistic closed form N(t) = K / (1 + ((K − N0)/N0) e−rt). The dashed line is unchecked exponential growth, clipped at the top of the chart.
With r = 0.8 per year, K = 1,000 and a start of 10, the population takes about 5.7 years to reach half of K (ln 99 ÷ 0.8) and adds about 200 individuals a year at that moment. The same animals with no brake would pass 1,000 in about 5.8 years, then keep going. The two lines are almost indistinguishable at first. Early growth tells you nothing about the ceiling.
Add a second species that eats the first. Prey multiply when predators are few; predators multiply when prey are plentiful; then predators over-eat, prey crash, predators starve, and prey recover. Alfred Lotka (1925) and Vito Volterra (1926) wrote this down independently, Volterra after fish-catch records from the Adriatic during the First World War.
The model predicts something odd and testable: endless loops, with the predator peak lagging the prey peak by about a quarter cycle. Near equilibrium the period is about 2π / √(αγ). The famous real-world match is the Hudson's Bay Company's fur returns for snowshoe hare and Canada lynx, which cycle every ten years or so. But the model is a cartoon. It lets prey grow without limit and predators eat without limit, and its loops are fragile: small random dips can wipe a species out (the "atto-fox problem"). Ecologists still argue how much of the hare cycle is predators and how much is food and other factors.
Lotka–Volterra with β = 0.04, δ = 0.02, solved by fourth-order Runge–Kutta for 40 time units; starting predators 9. Illustrative units.
With α = 0.8 and γ = 0.6 the equilibrium is 30 prey and 20 predators, the loops orbit it, and a full cycle takes 9 to 10 time units (the small-loop formula gives 9.1; big loops run a little longer). Raise the predator death rate to 0.9, for instance by culling them, and the prey equilibrium climbs from 30 to 45 while the predator equilibrium stays at 20. The counter-intuitive lesson: in this model, the average number of predators depends on the prey's growth rate, and the average number of prey on the predators' death rate.
Two species that need the same limited thing cannot both stay for ever. In the early 1930s Georgii Gause grew two Paramecium species, alone and together, in steady conditions: together, P. aurelia eventually drove P. caudatum out, though changing the conditions let the loser survive. The idea became the competitive exclusion principle: complete competitors cannot coexist. What lets many species share a forest is niches: each uses the resource differently enough that it limits itself more than it limits the others.
Extend the logistic with competition coefficients: α12 is how much one individual of species 2 counts against species 1's room, and the reverse for α21. The rule falls out of the algebra:
K1 = 100, K2 = 80, r = 0.5 for both, starting at 10 each; Euler steps of 0.05 to time 100. Illustrative.
With the starting values the two settle at about 90 and 17: the weaker competitor survives only because each species crowds itself more than the other. Push either coefficient above its limit and one disappears. Push both over and the winner depends on who got there first.
Sunlight enters through plants, algae and some bacteria. Each time something is eaten, most of the energy is spent on living (respiration, heat), left as waste, or lies uneaten. The fraction passed from one level to the next is the trophic transfer efficiency. The textbook figure is "about 10%", from Raymond Lindeman's 1942 study of a Minnesota lake. Real values vary widely, from well under 1% to about 37% in his data, and marine plankton systems are often higher, near 15 to 20%.
Because the energy shrinks geometrically, a food chain runs out after only a handful of links, and top predators are rare and need large territories. Webs are tangled, not chains, and removing one species can ripple: sea otters eat sea urchins that eat kelp (Estes and Palmisano, 1974), so otter loss opened the way for urchin-grazed kelp loss. The Yellowstone wolf story is famous but contested: wolves returned in 1995 and 1996, browsing of some trees fell in places, but bears, cougars, hunting and a changed stream channel also matter, and "wolves changed the rivers" overstates what the evidence shows.
Illustrative units. A level counts as viable if it gets at least 1 unit. Bar width is on a log scale, the labels give the true energy.
At 10% and 10,000 units from plants, herbivores get 1,000, small carnivores 100, bigger predators 10 and a top predator 1: five levels. At 5% the chain loses a level. That is the arithmetic behind "why are there no predators the size of a house".
Energy flows through an ecosystem once and leaves as heat. Atoms of carbon, nitrogen, phosphorus and water go round and round. Every cycle has pools (the atmosphere, plants, soil, ocean) and fluxes (photosynthesis, respiration, decay, runoff). The average time an atom stays in a pool, its residence time, is simply pool ÷ flux. A pool that is small compared with the flux through it turns over fast: the carbon in the atmosphere cycles through plants and oceans within a few years, while carbon in deep sediments and fossil fuels sits for millions.
Humans are now a big flux. Burning fossil carbon moves carbon from a slow pool to a fast one; that is the climate story. Nitrogen is the other: the Haber process (met in Equilibrium & Kinetics, 35/37) lets factories turn atmospheric N2 into fertiliser, feeding billions, while the excess washes into rivers and coasts and fuels algal blooms that leave low-oxygen "dead zones". Which pool, which flux, and which way the arrow points answers most questions about a cycle.
Biodiversity is the variety of life at three levels: genes, species and ecosystems. About 8.7 million species is one estimate for all eukaryotes, most still unnamed, and diversity peaks in the tropics, which hold about half of species on less than a fifth of the land. Why does it matter? Three reasons, and a careful ecologist separates them. Function: long-running grassland experiments such as those at Cedar Creek, Minnesota, found that more diverse plots tend to be more productive and steadier, an "insurance" effect because different species do well in different years. Services: pollination, water filtering, fisheries. Value in itself: a claim about ethics, not data.
How many species live somewhere depends strongly on area. Plotted on log axes, species number rises as a power of area, S = c Az, with z in textbooks usually quoted between about 0.15 and 0.35. A tenfold area gives 10z times the species: with z = 0.25, about 1.8 times. Run it backwards and cutting habitat to a tenth eventually loses about 44% of species. Some ecologists argue this method overstates extinctions from habitat loss, and extinctions lag behind the loss, an "extinction debt".
In 1967 Robert MacArthur and E. O. Wilson explained islands with a balance: immigration of new species falls as an island fills up (fewer newcomers are new), extinction rises as it fills (more species, each with fewer individuals). Where the lines cross is the equilibrium number, and it is not static: species keep arriving and vanishing. Near islands get more immigrants; small islands lose species faster. Simberloff and Wilson tested it on tiny Florida Keys mangrove islands in 1969.
Mainland pool P = 100 species; immigration rate 1·e−d/400·(1 − S/P) per year and extinction rate 0.25·(A/100)−0.5·S/P per year: illustrative shapes, not measured. Habitat lost uses Safter/Sbefore = (1 − lost)z.
Try the preset-free defaults: a 100 km² island 200 km out settles near 70 of 100 mainland species; shrink it or move it far and the equilibrium falls. And with z = 0.25, losing 90% of the habitat leaves (0.1)0.25 = 56% of the species in the end.
Headlines about nature are full of enormous numbers, so read the denominator. The WWF and the Zoological Society of London's Living Planet Report 2024 says monitored wildlife populations fell on average 73% between 1970 and 2020. That is an average change across a sample of vertebrate populations, not 73% of animals or species gone, and critics note the data lean toward birds and Europe and that a few populations with huge changes can drive the average. The direction is not in doubt; the exact size is. Insects are similar: a 2017 German study of 63 nature reserves found about three-quarters less flying-insect biomass over 27 years, and it is a real and alarming result for one region, not a global count.
| Place | What happened |
|---|---|
| North America | Passenger pigeons, once in the billions, were hunted out; the last one died in 1914. Atlantic cod off Newfoundland collapsed and Canada closed the fishery in 1992. |
| South America | Amazon forest clearing is measured every year by Brazil's satellite programme (INPE); the rate has swung with policy. |
| Europe | Farmland birds and insects have fallen; the Krefeld result above. |
| Africa | The Nile perch, introduced to Lake Victoria in the 1950s, was followed by the loss or collapse of many of the lake's cichlid fish species; the numbers are debated. |
| Asia | The Aral Sea shrank from the 1960s as rivers feeding it were diverted to irrigate cotton. |
| Oceania | The Great Barrier Reef has had repeated mass coral bleaching since 2016 as ocean heat waves hit. |
| Antarctica | 20th-century whaling cut Southern Ocean whale populations to small fractions of their original size; some are slowly recovering. |
It is not all one-way. The 1987 Montreal Protocol cut ozone-damaging chemicals, and several whale populations are rebuilding. Management works when the model is right. The simplest case is a harvest: if you take a fraction h of the population each year from a logistic population, it settles at N = K (1 − h/r) and you take h N a year. That yield is highest at h = r/2, which gives the maximum sustainable yield r K / 4. Fish harder than r and the population falls to zero.
Schaefer-style surplus-production model with K = 1000 (illustrative). Equilibrium N = K(1 − h/r) and catch = h·N, valid for h < r; above r the stock goes to zero. Real fisheries add noise, age structure and uncertain stock estimates.
At r = 0.5 and h = 0.25 the stock sits at 500 and the catch is 125 a year, the maximum. Push h to 0.4 and the catch is 80 while the stock is only 200: you work harder for less, and a bad year could tip it over. That is why managers target below the peak.
Numbers here describe how ecologists think; they are not a forecast for any real place.
Numbers of living things are set by a few ratios: how fast they can grow, how much room there is, who eats whom, and how much energy gets passed up. Learn those and you can read most ecological news. It also explains why saving a species is mostly about habitat and harvest, not good intentions.
Try this: this week, find one population near you (pigeons, ants on a path, weeds in a crack) and ask what its ceiling K is and what sets it. You will usually find the answer is food, space or something that eats it.
Sources. Exponential and logistic growth (Verhulst, 1838), Lotka–Volterra (Lotka 1925, Volterra 1926), competition (Gause, early 1930s) and island biogeography (MacArthur and Wilson, 1967) are standard textbook ecology (Begon et al.; Ricklefs). St. Matthew Island reindeer, trophic transfer efficiency (Lindeman 1942), the Yellowstone dispute, the Living Planet Report 2024 figure and its criticisms, and the Krefeld insect study were checked against encyclopaedia summaries of the primary papers. Regional examples are well-documented cases given in outline. All interactive numbers were computed in node; model parameters are illustrative.