Populations: Growth and Its Limits
Nothing grows exponentially for long. What stops it — and what happens when the ceiling itself moves.
Builds on: 6.1 Energy Flow: Why Food Chains Are Short5.1 Natural Selection
The exponential trap
Populations grow by multiplication, not addition: every individual can produce more individuals, who can produce more still. Unchecked, that gives an exponential curve — slow at first, then explosively steep.
Darwin liked the elephant example: the slowest breeder known, yet from one pair, with every calf surviving, you would have millions within a few centuries. You do not see elephants filling continents, which means almost every elephant that is ever born dies before reproducing. That gap between potential and reality is exactly where natural selection operates.
Carrying capacity
Real populations run into limits: food, water, space, nesting sites, predators, disease. The maximum an environment can sustain indefinitely is its carrying capacity, written K.
The result is an S-shaped (logistic) curve with four phases:
- Lag — few individuals, slow absolute growth.
- Exponential — resources plentiful, growth accelerating.
- Slowing — competition bites as the population nears K.
- Plateau — births ≈ deaths, fluctuating around K.
The structure of that equation is worth a moment. When N is small the bracket is close to 1 and growth is nearly exponential. As N approaches K the bracket approaches zero and growth stops. The population does not need to know anything about K — the limitation emerges from competition.
Two kinds of limit
Density-dependent factors bite harder as the population gets denser: competition for food, disease transmission, predation, accumulating waste. These are what actually produce the plateau — they are a feedback loop.
Density-independent factors hit regardless of numbers: a hard frost, a flood, a wildfire. A drought kills the same fraction of a sparse population as a dense one, so these cause crashes rather than smooth regulation.
Overshoot
Populations can exceed K temporarily, especially when they respond slowly — and the correction can be brutal. Reindeer introduced to St Matthew Island in 1944 grew from 29 to 6,000 by 1963, stripped the slow-growing lichen they depended on, and crashed to about 42 animals in a single winter. The carrying capacity had not just been exceeded; it had been damaged, so the population could not recover to its old level.
That is the general lesson, and it is why K is not simply a fixed number. It changes with conditions, and a population that overshoots badly enough can lower it.
Human population growth has looked exponential for centuries because we keep raising our own carrying capacity — agriculture, sanitation, fertiliser, medicine. Each is a genuine increase in K, not an exemption from it. The open question is not whether K exists but where it currently is, and whether we are drawing it down the way the reindeer drew down their lichen.
⚗️ Lab — Growth Curves
One population, two models. Compare what they predict.
- Run the exponential model. It never stops — which should tell you it is wrong past a point.
- Switch to logistic and watch the curve bend over as it approaches K.
- Raise the growth rate with K fixed. It gets there sooner, but it does not get further.