Population Growth: Why the Curve Always Bends

More than just births minus deaths

A population's size at any moment is driven by three flows: births, deaths, and migration (people moving in or out). Demographers track birth and death rates per 1,000 people per year, but the more predictive number for long-run growth is the fertility rate — the average number of children born per woman during her reproductive years, roughly ages 15-49. In developed countries the replacement rate — the fertility level needed for a population to hold steady generation to generation — sits at about 2.1 births per woman.

Underlying all of this is the demographic transition model, which describes how countries typically move through four stages as they develop: high birth and death rates with little growth (pre-industrial), death rates falling while birth rates stay high (rapid growth), birth rates then falling as education and urbanization rise (growth slows), and finally low birth and death rates together (near-zero growth). Some demographers now propose a fifth stage where extremely low fertility actually shrinks a population.

The logistic growth equation

The simulation on this page models population with the classic logistic differential equation, which builds resource limits directly into the growth rate:

dP/dt = r · P · (1 − P/K)

P = population,  r = intrinsic growth rate,  K = carrying capacity

When P is small relative to K, the term (1 − P/K) is close to 1 and growth looks almost exponential. As P approaches K, that term shrinks toward zero and growth slows to a crawl — the population approaches but never exceeds its carrying capacity, the maximum size an environment can sustainably support given its food, water, space and other resources. Solved analytically, this gives a closed-form S-shaped curve:

P(t) = K / (1 + ((K − P0)/P0) · e^(−rt))

Exponential growth, for contrast

Plotted alongside the logistic curve on this page is the simpler exponential model, P(t) = P0·e^(rt), which assumes resources never run out. It is a reasonable approximation only while a population is far below its carrying capacity — every real population eventually hits some limiting factor, whether that is food, disease, predation, or in human terms, economic and social constraints. The two curves start identically, then visibly diverge once the logistic curve begins bending toward K.

Migration, urbanization and the human picture

Real human population change layers migration on top of birth and death rates. Push factors (poverty, conflict, lack of opportunity) drive emigration; pull factors (economic opportunity, stability, education) drive immigration. At the same time, urbanization — the shift of population from rural areas to cities — has reshaped where people live, driven by jobs, services and reduced demand for rural agricultural labor, though it brings its own pressures: megacities of over 10 million people, informal settlements lacking basic infrastructure, and urban sprawl into surrounding farmland.

Where the world is actually headed

The United Nations projects world population reaching roughly 9.7 billion by 2050 and 10.4 billion by 2100 — with growth increasingly concentrated in Africa and South Asia, even as the global growth rate itself keeps declining. That declining global rate is exactly the logistic curve's signature: a population that once grew close to exponentially is now visibly bending over, not because of some external catastrophe, but because fertility rates are falling worldwide as education, urbanization and family-planning access spread.

Frequently asked questions

Try it live

Everything above runs in your browser — open Population Growth Model and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.