Renewal Theory and the Stable Population Model

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Deaths deplete a population and births add new individuals, with the overall effect being a renewal of population numbers. A mathematical analysis of this process is called, accordingly, a theory of renewal. At its core, this theory is a bookkeeping scheme to describe changes in a population over time, a goal achieved by tracking the time course of births. Tracking births over time is the same as tracking every cohort, that is, every group of individuals born at the same time. Because the number of individuals of a given age in the population is just the cohort of that age, the theory also tracks the age composition of a population. Finally, the theory combines the age composition (the result of past births) with fertility rates by age to obtain current births, thereby completing the accounting process. As will be seen below, this theory provides a powerful tool for the analysis of population composition and change.

Population renewal theory has origins in the work of the Swiss mathematician Leonhard Euler (1707–1783) in 1760 and the German demographer Johann Peter Süssmilch (1707–1767) in 1761, but its modern form was largely developed by the American demographer and biometrician Alfred Lotka (1880–1949) between 1907 and 1913. Renewal theory has obvious relevance for any collection of things that are created in the manner of a birth and eliminated in the manner of a death, for example, a collection of machinery parts in a factory. A broader theory of renewal for such processes, not discussed here, has been developed for other applications.

Lotka's work provides the basic framework of mathematical demography and is the primary focus of this article. Many applications and extensions of his work have been made as the subject of demography has blossomed since the 1950s. Some of the more important elements of this newer work as it is used by demographers are summarized below.

Births, Cohorts, and Lotka's Equation

To track births, researchers record the rate at which new individuals are born into a population at each instant of time. They simplify by considering only female births, males being accounted for separately later. At a time denoted by t let the rate at which females are born be denoted by B(t). As time advances these newborns will age, so the cohort of females aged at time t consists of the survivors among the B(t - a) individuals who were born at the earlier time (t - a). Using the notation l(a) to indicate the fraction of newborn females that survives from birth to age a, one sees that at time t there must be l(a) B(t - a) females aged a. This expression links past births to cohorts and thus to the age composition of the female population.

To obtain current births at time t researchers need the per-capita fertility rate, denoted by m(a) and defined as the rate of female births to a mother at age a. Mothers aged a will together produce female births at a rate m(a) l(a) B(t - a). The total birth rate B(t) in the population is a sum of births to mothers of all ages. Using an integral to indicate the sum yields the first form of Lotka's equation for population renewal:

The limits on the integral here are the limits on the reproductive ages of mothers. For human populations, this would typically range from 15 to 45 years. An unfortunate aspect of equation [1] is that to obtain today's birth rate one must know the past birth rate going back at least as far as the oldest reproductive age. This is information researchers rarely have. Instead, in most situations one can specify only the age composition of the population at some starting time (which one labels as time = 0) and would like to use the renewal argument to obtain birth rates B(t) for all later times. Researchers deal with such cases by noting that at time t (after the starting time), births into the population are a sum of (1) births to individuals who were present before the starting time–call these f(t); and (2) births to individuals who were born at or after the starting time. The latter births must clearly follow from the argument that produced equation [1], so one can write:

In this equation the limits on the integration indicate a sum of the individuals born after the starting time, which implies that the limits are 0 and t. A mathematical expression for f(t) may be written out by applying to the initial population the logic that was used to derive equation [1].

Between the two forms given, Lotka's equation translates a population's past composition into its future. The equations above are written for female births: What about males? In most countries (with some striking exceptions) human births occur with a sex ratio of about 105 male births to 100 female births, and this ratio can be used to infer male births from female. In addition, if male survival rates are available, male cohorts can be tracked just as was shown above for female cohorts.

The Stable Population

Suppose that a population has for many generations followed the renewal process of births and deaths with some fixed schedule of age-specific birth and survival rates. It is reasonable to expect that the population as a whole, and the birth rate in particular, should then experience a steady rate of growth r per unit time. Such steady growth would imply that the birth rate B(s) at some time and the birth rate B(s + t) at a later time (s + t) would be in the ratio:

Such an exponential ratio of births at different times satisfies equation [1] only if the growth rate r satisfies what is called Lotka's characteristic equation:

This equation shows how age-specific fertility and survival determine the unique steady rate of population growth that these rates can support. Among the many important results that flow from this equation, two will be mentioned here. First, the equation implies that r will be positive (population increase) or negative (population decline) depending on whether the net reproduction rate (NRR), defined by NRR = ∫m(a)l(a)da, is larger or smaller than one. An NRR equal to one characterizes and defines a population in which r = 0. This is called a stationary population. The characteristic equation for this case also defines what is commonly called replacement fertility. Second, in populations whose NRR is not far from one, an expansion of the exponential term in the equation yields the useful approximate result that:

in which the quantity T is the average age at child bearing of mothers. This result provides ready insight into the impact of changes in the level and timing of fertility, and of changes in survival rate, on the steady population growth rate.

When births over time grow at a steady rate r, if the total birth rate is B(t) at time t, the population at the same time at any age can be inferred to be B(t)l(a)e-ra. Hence a population in a steady state growing at rate r will have an age composition in which the fraction of the population at age a is proportional to l(a)e-ra. The age composition of such a population does not change over time and is known as the stable population. The stable population is basic to demographic analysis. It underlies the use and interpretation of population pyramids. For example, the well-known difference between the shapes of the pyramids for rapidly growing (broad-base) versus slowly growing (narrow-base) populations is attributable to the differences in growth rates and survival rates. In population analysis, the stable population provides a ready standard of comparison that yields insights into observed population structures that are rarely stable. (A mathematical note: This discussion glosses over the fact that B(t) is a rate, not a number, and the stable structure as it is written above is a density. Correctly, the density should be integrated over an age interval, such as a calendar year, to obtain the number of individuals in that interval; but there is little danger of error, and greater clarity, in the exposition without introducing this refinement.)

Dynamics of Births

As was shown above, a fixed set of age-specific fertility and survival rates define a stable population. Suppose, however, that at a time t = 0 a population has an age composition that is not the stable one, but the fertility and survival rates still remain fixed. Lotka proved that over time the population's age composition will converge toward the stable composition–this is the reason for the adjective stable. The mathematical demonstration of this convergence starts with equation [2], in which an arbitrary initial age composition can be used to determine the term f(t). Lotka showed that the birth rates B(t) obtained by solving that equation have the form:

In this series of terms, the first simply represents the stable population, with r being the stable growth rate (as can be seen by omitting all the other terms). The second and later terms (the "etc." in the equation stands for a possibly infinite sequence of similar terms) contain exponents such as r1 and corresponding sinusoidal terms that are computed as additional solutions–called the roots–of the characteristic equation. These additional roots have the property that their exponential parts are smaller than the first exponential r, which implies that stable population dominates the solution as time increases. Each additional term has an oscillatory character indicated by the sine and cosine terms; the second and third terms above have a period of (2π/ω). These decaying cycles are transient aspects of the population that are observed en route to the stable population.

These cycles have real demographic consequence in populations that are far from stability. The many countries (e.g., Japan, China) that experienced rapid fertility declines in the second half of the twentieth century provide striking examples of this phenomenon. After the decline, fertility rates were relatively stable at their new low levels, but the population structure still reflected years of high pre-decline fertility and rapid growth. Age compositions in these populations in subsequent years clearly show Lotka's damped oscillations–they consist of population booms, busts, and echoes. The Lotka solution in this case is also the basis for Nathan Keyfitz's calculation of momentum–the effect on long-run numbers of a sudden transition to replacement fertility. The tendency of a population with fixed fertility and survival rates to gravitate to the stable population structure is an example of demographic ergodicity–the convergence of age composition to a stable structure whereby the history of a population's age structure is gradually obliterated by the process of population renewal.

Reproductive Value

The discussion thus far has largely ignored the population's starting age composition except as the term f(t) in equation [2]. But the starting composition is surely relevant to the future population, which leads to the question: If a single female of age a is added to the population, what contribution does she make to the future population? That contribution can be specified by keeping count of her children, her grandchildren, and so on. The issue here is not the total number of these descendants, which might well be infinite if all future generations are counted; rather, the question is how the future contributions of a female depend on her age. In other words, what is the relative contribution to the future population made by females of different ages? This relative contribution was called reproductive value by the great statistician and geneticist R. A. Fisher (1890–1962), who introduced the notion in his work on the evolution of reproductive characteristics.

How is reproductive value, denoted by V(a) at age a, to be determined? To make a sensible comparison between ages, the effect of population growth must be discounted and then the contributions made by the female at her current and future ages until death must be added. Doing so produces the expression:

with the limits on the integral running from current age to the maximum reproductive age (45). The reproductive value plays a key role in population momentum, in evaluating the dynamics of a population subject to disturbance, and in evolutionary theory.

Applications of Lotka's Theory

The logic and mathematics of Lotka's theory underlie several methodological developments. The cohort-component method of projecting a population, first used by P. K. Whelpton and now in regular use by forecasters, is an expression of equation [2]. Although equation [2] applies only to populations that are closed to migration, it is easily modified to incorporate births gained or lost via migration. If the net migration flow is constant over time and exhibits some specified age pattern that is also constant over time, the stability properties of equation [2] apply with suitable modification to the equation that includes migration. Such modified equations are commonly used to project and analyze the dynamics of populations subject to significant migration flows.

The mathematical analysis outlined above treats time and age as continuous, hence the integrals. In practice, demographers must work with events observed over discrete intervals, such as one year or five years. The equations carry over fairly directly to the discrete case. For example, in equation [1], if m(a) and l(a) are the rates for a discrete interval with a length of one year, the equation remains the same except that a sum over discrete age intervals replaces the integral.

The tracking of cohorts in Lotka's analysis extends directly to cases in which the growth rate of births or other population segments changes with time and has made possible methods of demographic estimation for unstable populations as developed by Ansley J. Coale, Samuel H. Preston, and others. The characteristic equation highlights the central demographic importance of the age schedules of mortality and survival, which have therefore been the subject of considerable independent study. Coale, Paul Demeny, and James Trussell developed their model demographic schedules of mortality and fertility in the context of renewal theory. Numerous methods of direct and indirect demographic estimation originate in the logic and mathematics of Lotka's analysis.

Demographic Ergodicity and Time-Varying Rates

Perhaps the most obvious limitation of Lotka's equation is that the fertility and survival rates are taken as fixed and known. But derivation of the equation does not require the rates to be fixed: The equations remain valid with time-dependent birth and death rates. Solving them in such cases is harder because Lotka's analysis does not apply, but progress has been made on several fronts. A key feature of Lotka's theory is demographic ergodicity, and, as Álvaro López (1926–1972) first showed, ergodicity also holds for many situations in which rates vary over time. In other words, given a time-dependent sequence of fertility and survival rates, the process of population renewal causes the population's age composition to converge toward a particular stable (but time-varying) composition. This phenomenon is called weak demographic ergodicity and has been shown to hold for many types of temporal change in the rates. Joel Cohen has shown that ergodicity holds for many kinds of random change over time in fertility and survival rates. Aside from this general property, progress has been made in finding some useful explicit solutions to Lotka's equation for time-varying rates, including work by Robert Schoen and Young Kim for cyclically changing rates and by Shripad Tuljapurkar and Nan Li for populations undergoing demographic transitions from old to new rates. The latter work is also the basis for calculations of population momentum when fertility or mortality transitions take place gradually over some years.

Sex and Marriage

Lotka's theory greatly simplifies the real-world situation by using fertility rates for females by age as a proxy for a more realistic accounting of marriage (and, in modern times, cohabitation or simply mating) as a precursor to reproduction. This simplification is serviceable and is surely correct in some aggregate average sense. But demographers have naturally been interested in adding an explicit accounting of marriage to the theory of population renewal. Efforts to do this have not yet been successful in producing what demographers call a two-sex theory of population renewal. Some useful work has been done in terms of defining the mathematical structure of a marriage function that translates male and female age compositions into an age composition of married couples. Robert Pollak has established stability properties for some particular forms of two-sex renewal. But a generally useful theory remains elusive. In addition, much of this two-sex theorizing makes little contact with the far more successful theory of the proximate determinants of fertility or of fecundability (the probability of conception). In recent years, this type of theory has attracted new interest among epidemiologists who analyze sexually transmitted diseases and are therefore concerned with tracking the rate of sexual interactions.

Feedback and Nonlinearity

The renewal theory discussed above assumes that fertility and mortality rates are exogenously determined (i.e., by factors such as economics or culture that are not explicitly included in the present analysis), even if they vary over time. Yet there are persuasive arguments that one or both of these rates may be at least partly determined by the state of the population itself. One such line of thought begins with the English economist T. R. Malthus (1766–1834) and continues through more recent and considerably more general arguments for the necessity of homeostasis–a self-regulatory force in human population growth. A different line of thought is exemplified by Richard Easterlin's work on the relationship between cohort size, expectations, and eventual cohort fertility. In either type of argument, a relationship is posited between fertility or mortality and the past time sequence of births; this is called a feedback relationship. If researchers introduce a feedback relationship into equation [2], for example, they make the equation nonlinear in the birth rates. A general analysis of the dynamics of such nonlinear equations poses many mathematical challenges. The goal of such an analysis is to produce understanding that parallels the understanding of Lotka's theory–to determine what sustained population trajectories can be maintained by particular types of nonlinearity. Intuitively, negative feedback, in which increasing birth rates act to depress future fertility rates, should lead to population cycles. Various studies show how feedback can maintain population cycles, but they unfortunately also suggest that it is hard to find good specifications of feedback that are consistent with observed population cycles. More complex types of feedback can generate interestingly complex, even chaotic, dynamics in nonlinear models, but the practical usefulness of this work for human demography remains uncertain.

See also: Actuarial Analysis; Age Structure and Dependency; Coale, Ansley Johnson; Cycles, Population; Euler, Leonhard; Keyfitz, Nathan; Life Tables; Lotka, Alfred; Momentum of Population Growth; Multistate Demography; Projections and Forecasts, Population; Stochastic Population Theory; Süssmilch, Johann.


Caswell, Hal. 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sunderland, MA: Sinauer.

Coale, Ansley. 1972. The Growth and Structure of Human Populations: A Mathematical Investigation. Princeton, NJ: Princeton University Press.

Keyfitz, Nathan. 1968. Introduction to the Mathematics of Population. Reading, MA: Addison-Wesley.

Preston, Samuel H., Patrick Heuveline, and Michel Guillot. 2001. Demography: Measuring and Modeling Population Processes. Malden, MA: Blackwell.

Smith, David P., and Nathan Keyfitz. 1977. Mathematical Demography: Selected Papers. Berlin: Springer-Verlag.

Shripad Tuljapurkar