Stable, semi-stable populations and growth potential

Abstract Starting from the definition of a Malthusian population given by Alfred J. Lotka, the author recalls how the concept of stable population is introduced in demography, first as a particular case of stable populations, and secondly as a limit of a demographic evolutionary process in which female age-specific fertility rates and age-specific mortality rates remain constant. Then he defines a new concept: the semi-stable population which is a population with a constant age distribution. He shows that such a population coincides at any point of time with the stable population corresponding to the mortality and the fertility at this point of time. In the remaining part of the paper it is shown how the concept of a stable population can be used for defining a coefficient of inertia which measures the resistance of a population to modification of its course as a consequence of changing fertility and mortality. Some formulae are established to calculate this coefficient first for an arbitrary population, and secondly for a semistable population. In this second case the formula is particularly simple. It appears as a product of three terms: the expectation of life at birth in years, the crude birth rate, and a coefficient depending on the rate of growth and for which a numerical table is easy to establish.     

Crossovers that link populations with the same vital rates

"In this paper, we consider crossovers of demographic density distributions from...populations that have the same fertility and mortality rates. We focus on observed populations and their associated stationary and stable models, and on proportional distributions of persons, births, deaths and reproductive values....Three different populations were selected to represent a range of demographic behavior. Those populations are Japan 1963, a low mortality, low fertility population; Togo 1961, a high mortality, high fertility population; and the United States 1919-1921, a population whose fertility and mortality are intermediate."     

Calculation of age-specific fertility schedules from tabulations of parity in two censuses

The mathematics of stable populations recently has been generalized to cover populations with time-varying fertility and mortality by a modification incorporating the sum of age-varying growth rates in place of the fixed growth rate of a stable population. Equations that characterize nonstable populations apply to any cohort-like phenomenon with a measurable property that cumulates gains or losses through time. In particular, the equations fit the relation between a population's average parity at a given age and age-specific fertility rates previously experienced at lower ages. Techniques devised to derive an intercensal life table from single-year age distributions in two censuses are adapted to estimate accurate intercensal fertility schedules from distributions of parity by age of woman in two censuses. Birth-order specific fertility schedules are also estimated.     

Population dynamics in Germany: the role of immigration and population momentum

The effects of changes in rates of mortality, fertility, and migration depend not only on the age-specific patterns and levels of these rates, but on the age structure of the population. In order to remove the influences of the age structure and concentrate on the effects of the demographic rates themselves, a common practice is to analyze the influences of the rates for a standard age structure. This paper analyzes current and future population changes in Germany, using a stationary population equivalent model (SPE) that shows long-term effects of current fertility, mortality, and international migration patterns. Results indicate that the German population will eventually decline because of below replacement fertility, if net immigration does not counteract this decrease. This means, for instance, that the long-term stationary population levels for Germany will decrease by approximately 6.5 million during a decade in which current fertility, mortality, and international migration levels prevail. The paper also reports how various other assumptions for mortality, fertility, and international migration affect the SPE model for Germany.     

Modeling momentum in gradual demographic

The analysis of population momentum following a gradual decline in fertility to replacement level provides valuable insights into prospects for future population growth. Here, we extend recent work in the area by applying a new form of the quadratic hyperstable (QH) model, which relates exponentially changing fertility to the resultant exponentiated quadratic birth sequence. Modeling gradual transitions from an initial stable population to an ultimate stationary population indicates that such declines in fertility increase momentum by a product of two factors. The first factor is a previously noted continuation of stable growth for half the period of decline. The second is a not previously appreciated offsetting factor that reflects the interaction between the decline in fertility, the changing age pattern of fertility, and the changing age composition of the population. Numerical examples using both hypothetical and actual populations demonstrate that for declines of any length, the product of the two factors yields momentum values that closely agree with the results of population projections. The QH model can examine monotonic transitions between any two sets of constant vital rates. As a generalization of the fixed-rate stable model, it has great potential value in numerous areas of demographic analysis.     

On the distribution of completed parities when fertility is heritable

"Over the last one hundred years, there has been, in many developed countries, a demographic convergence towards the two child family. The possible implications for population growth of such a tendency are considered in this paper in terms of both family limitation and also the intergenerational transmission of fertility. These two effects interact so that as the proportion of two-child families increases, the possible influence of mother-daughter fertility associations on population growth decreases, though even now it could override otherwise significant changes in either or both of the birth and death intensities. In particular, it is shown that according...to how fertility is transmitted through generations, it is still possible to have zero growth rates consistently with a widely dispersed stable distribution of family size as well as a typical mortality regime." (SUMMARY IN FRE)     

Stationary and stable population]

The properties of stationary and stable population models are discussed. In particular, the changes in population size and age structure that will occur under assumptions of constant fertility and mortality rates and no migration are explored.     

Population statistics] as of July 1, 1999

Reported in tabular form are population statistics published by the Institute for Population and Social Research, Mahidol University, in Thailand, as of July 1, 1999. The three objectives of the table were 1) to present the most recent estimates of relevant population numbers and vital rates twice a year, 2) to provide the most accurate demographic estimates possible through the use of standard techniques of demographic analysis, and 3) to disseminate the demographic data to Thailand, and international researchers and planners. The table shows statistics on the following elements: total population; population by sex; population in urban and rural areas; population by region; population by age group; crude birth and death rates; natural growth rate; infant mortality rate; life expectancies at birth and at 60 years and over; total fertility rate; contraceptive prevalence rate; and median age. A new basis for age estimates is also presented.     

On the distribution of completed parities when fertility is heritable

Over the last one hundred years, there has been, in many developed countries, a demographic convergence towards the two child family. The possible implications for population growth of such a tendency are considered in this paper in terms of both family limitation and also the intergenerational transmission of fertility. These two effects interact so that as the proportion of two‐child families increases, the possible influence of mother‐daughter fertility associations on population growth decreases, though even now it could override otherwise significant changes in either or both of the birth and death intensities. In particular, it is shown that according as to how fertility is transmitted through generations, it is still possible to have zero growth rates consistently with a widely dispersed stable distribution of family size as well as a typical mortality regime.     

Aging in the United Kingdom: a review of demographic trends,recent policy developments and care provision

"Although the U.K.'s population is not predicted to grow very much in the future, the population structure is changing and there is a shift towards a much older age distribution. The characteristics of demographic aging in the U.K. include a marked reduction in fertility rates, increasing rates of life expectation at birth..., increasing dependency ratios and variations in mortality and social class in old age. The U.K.'s demographic context has important implications for aged care policy and planning. This paper documents the demographic trends, reviews recent major policy changes and their impact on care provision, and discusses some of the emerging implementation issues that challenge the potential of such policies to meet the needs of an aging population."     

China's demographic dilemmas

The year 2000 marks the end of a tumultuous century in China's population history, which weathered the demographic effects of devastating famines, wars, and epidemics and population growth and change. This paper examines the effect of population policies on the demographic dilemmas of China. In the 1950s, China had seen the fastest demographic transition in history, with a dramatic decline in mortality rates, followed by a decrease in fertility rates. However, in the 1970s, revisions in population control measures, changes in age structure, and fluctuations in age at marriage resulted in lower fertility rates. The struggles encountered by China in regulating fertility are described; these include the different methods of birth control, gender preference, marriage, population aging, and minority populations. Population and development issues within the context of urbanization, employment, education, health care, economy, and environment are also discussed. Future implications of these findings indicate the need for a systematic, effective, and complete environmental clean-up, as well as fertility and population policies.     

Population statistics] as of July 1, 1998

This article presents estimates of relevant population numbers and vital rates in Thailand as of July 1, 1998. Utilizing the standard demographic techniques of analysis, the estimates provided are assured to be the most accurate demographic estimates possible. Total population was estimated at 61,143,000. Estimates by sex, locales, region, and by age group are included. In addition, the crude birth rate per 1000 population was estimated at 18.7; the crude death rate per 1000 population was 6.5. For the natural growth rate the estimate was at 1.2%, and the infant mortality rate was 25.0 per 1000 live births. In terms of life expectancy at birth, the estimate for males was 69.9 years, while for females it was 74.9 years. Additional years in life expectancy at age 60 were 20.3 years for males and 23.9 years for females. The total fertility rate per woman is 1.98, and contraceptive prevalence is 72.2%. The demographic data will be disseminated to Thai and international population researchers and planners.     

A census-based method for estimating adult mortality

Abstract A simple method is presented for converting an age distribution in any closed population into the stationary population corresponding to its current mortality conditions. The conversion only requires a set of age-specific growth rates, which will normally be available from successive censuses. From the stationary population, any life table mortality measure of interest can be computed. The index most robust to normal data errors in developing countries is life expectancy, and the paper focuses on its calculation. The sensitivity of results to various forms of data error is considered, and procedures are proposed for removing errors resulting from differential census coverage completeness and from age misstatement at older ages. Applications of the procedures are made to data from Sweden, India and South Korea. Because of the absence of a radix, estimation of life expectancy usually will begin at the fifth birthday.     

Movement toward stability as a fundamental principle of population dynamics

Although convergence to stability is typically a complex and irregular process, the Kullback distance provides a measure that moves consistently to 0 as a population becomes stable. The roots of the Kullback distance are in information theory, but it is a meaningful demographic quantity. It reflects a population's log momentum, or the amount of growth built into a population's nonstable age distribution. The rate at which the Kullback distance moves toward 0 is neither constant nor monotonic. At any point in time, however, it decreases by the covariance between a population's age-specific growth rates and its log momentum. Although the present findings are couched in terms of movement toward stability, they are generally applicable because they relate to the behavior of any population at any instant. It is a fundamental principle of population dynamics that a population is always moving toward the stable population implied by its prevailing fertility and mortality rates, and that the extent of its movement is determined by the covariance between its age-specific growth and its log momentum.     

Relations between demographic parameters

The mean life-expectancy e describes the average prospective life-time of an individual aged zero. This parameter can be explicitly described in terms of the survivorship distribution of the population. The Malthusian parameter r represents the asymptotic growth rate of a population. This parameter can be implicitly expressed in terms of the net-maternity distribution. The parameters e and r incompletely incorporate the age-specific fertility and mortality pattern of a population; distinct populations may have the same growth rate but different net-maternity functions; distinct populations may be characterized by the same mean life expectation but may have different survivorship distributions. This article analyzes a class of parameters called the entropy of a population (Demetrius, 1974a) which distinguishes between net-maternity functions with the same growth rate and also mortality distributions with the same mean life expectation. This class of parameters measures the convexity of the fertility and mortality distributions. This paper analyzes the relations between the entropy parameter and the standard demographic parameters.     

The stable equivalent population,age composition,and Fisher’s reproductive value function

This paper deals with a further analysis of the stable equivalent population for human populations. It first shows that the arithmetic difference between the size of the stable equivalent population and the actual population captures the total prospective contribution of age composition to the growth of the present population, if fertility and mortality rates are held constant at their current levels. The second part of the paper examines Fisher’s reproductive value function and investigates the location of maxima in relation to alternative values of the intrinsic growth rate. One conclusion which follows from this analysis is that, contrary to what has been discovered by others in specific examples, the reproductive value function can possess a global maximum in the first one or two years of life, provided the intrinsic rate of population growth is sufficiently negative. This principle is illustrated by reference to reproductive value functions calculated from the recent experience of U.S. females.     

The development and use of demographic models

In this review, we first examine two classical demographic models - conventional life tables and stable populations - and a modern generalization of stable population theory; we then discuss mathematical models of conception and birth. These models involve purely mathematical relations in formal demography as opposed to empirical regularities. Next we consider model age schedules of mortality, nuptialitiy, marital fertility, fertility, and migration that are explicitly based on such empirical patterns. We close this empirical section with a discussion of model stable populations, which are based on model life tables. We next examine the use of demographic models in forecasting future mortality, nuptiality, and fertility and in population projection. Following a discussion of microsimulation models, which gives us the opportunity to mention model age schedules of post partum amenorrhoea and of sterility, we close with observations about the purposes and uses of demographic models.     

The Momentum of Real Population Under Linear Fertility Transition

Population momentum is the ratio of a population's ultimate size after a demographic transition to its initial size before the transition. For stable population and instantaneous drop to replacement fertility, Keyfitz found a simple formula for the momentum. Although Keyfitz's formula has been extended to cover gradual demographic transitions, the stable initial population, however, is still an obstacle to approaching reality. Using the solution of the Lotka equation with time-varying vital rates and extending it to negative values of time, exact and accurate formulae for the momentum of any initial population that undergoes a linear fertility transition are expressed. Examples using the data from the United Nations indicate that the accurate formula works well for the age structures that are far from stable such as the more and less developed regions, as well as that close to stable such as the least developed region.     

Immigration,below-replacement fertility,and long-term national population trends

The long-term demographic effects of immigration on a population experiencing below-replacement fertility are studied by assuming that the size and age composition of the immigrant population do not change over time. The size of the first-generation immigrant population becomes stationary within a time period not greater than the human life span. Thereafter, the number dying equals the number entering over any given time interval. The stationarity of the native population, among which deaths exceed births, is maintained by the compensating number of births to the immigrant population. The limiting age distribution of the country's population, although stationary, may not decline monotonically with age and may look like a camel's back, with one or two humps.     

Effect of mortality change on stable population parameters

The stable population model is used to establish formulas expressing the effects of mortality change on population growth rates, birth rates, and age composition. The change in the intrinsic growth rate is shown to be quite accurately approximated by the average decline in age-specific death rates between age zero and the mean age at childbearing in the stable population. This change is essentially independent of the initial level of fertility in the population. Changes in birth rates and age composition are shown to be simple functions of the age pattern of cumulative changes in mortality rates relative to an appropriately defined “neutral” standard.