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A stable population is one that has experienced constant age-specific fertility and mortality rates (and zero migration) long enough for its age distribution to become fixed. In a stable population, every age group grows at the same constant rate — the intrinsic rate of natural increase (r) — and the proportion of the population in each age group remains constant over time even as the total population grows or shrinks. Alfred Lotka proved that any population subject to constant vital rates will eventually converge to a stable age distribution regardless of its initial age structure — a result known as the ergodic theorem. When the intrinsic rate equals zero (NRR = 1), the population is stationary: constant size and age structure. Stable population theory provides the mathematical foundation for demographic estimation, projection, and the analysis of population momentum.
Compare the actual age distribution of a country with the stable-equivalent population implied by its current fertility and mortality rates. The difference reveals how far the population is from stability and what its trajectory would be if current rates persisted indefinitely.
You have built life tables (converting mortality rates into survivorship), computed fertility measures (TFR, NRR), and studied population dynamics through the balancing equation. Stable population theory integrates these into a single mathematical framework that reveals the long-run implications of any given set of vital rates.
Imagine a hypothetical population that has experienced exactly the same age-specific fertility rates, age-specific mortality rates, and zero migration for a very long time — centuries, say. Alfred Lotka proved in the 1920s that such a population converges to a unique, fixed age distribution determined entirely by the vital rates, regardless of what the initial age distribution looked like. The proportion of the population in each age group becomes constant, and every age group grows (or shrinks) at the same rate: the intrinsic rate of natural increase (r). This is the ergodic property — the system "forgets" its initial conditions and is governed only by the vital rates.
The intrinsic rate r is related to the net reproduction rate (NRR) by the relationship r = ln(NRR) / T, where T is the mean generation length. When NRR > 1 (each woman has more than one surviving daughter), r is positive and the stable population grows exponentially. When NRR < 1, r is negative and the population shrinks. When NRR = 1, r = 0, and the population is stationary — constant in both size and age distribution. A stationary population is a special case of a stable population.
The practical value of stable population theory is not descriptive — no real population has constant vital rates. Its value is analytical. First, stable population models are the foundation of indirect demographic estimation. In countries with incomplete vital registration (much of sub-Saharan Africa, parts of South Asia), demographers compare observed age distributions to model stable populations to estimate birth rates, death rates, and life expectancy. Second, comparing a population's actual age distribution to its stable-equivalent (the stable population implied by current rates) reveals population momentum — built-in future growth or decline that would occur even if vital rates remained constant. A young population with below-replacement fertility has a stable-equivalent that is smaller and older; the gap represents the momentum for continued growth as the large young cohorts pass through their reproductive years. This concept, which you will study next, is one of the most important applications of stable population theory for policy and projection.
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