The momentum of population growth with time dependent net maternity function

The momentum of population growth problem of Keyfitz is generalized to contain a gradual change of the age-specific birth rate ro the level of bare replacement. Assuming a time dependence for the net maternity function of the form (formula: see text) R being the net reproductive rate, we show that for the Malthusian model the asymptotic birth rate is increased by exp (r/lambda), where r is the rate of increase of the population before t = 0. A numerical method for obtaining the asymptotic birth rate for a general net maternity function with the same time dependence is outlined.     

The stable births resulting from a time dependent change between two net maternity functions

The asymptotic birth rate for a one-sex population in which the net maternity function changes to one of bare replacement was first discussed by Keyfitz and has since been studied by several authors. The present generalization allows for a time dependent transition from any net maternity function to another and, thus, includes all previous models.     

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.     

A Taylor series expansion for Lotka’s r

A. J. Latka and F. R. Sharpe published their classical deterministic population model in 1911, and since that date, numerous numerical methods have been suggested for solving the integral equation inherent in the model. We consider the familiar integral equation (Equation (1)) and derive a Taylor series for ro, the unique real solution. The series is no more advantageous from the calculation point of view than many previous solutions, but it may be useful in a theoretical context. The effects on the intrinsic rate of increase of the population due to changes in the cumulants of the net maternity function are readily discernible.     

The cohort feedback model with symmetric net maternity

"In this paper we consider the simplest and most widely used demographic feedback model, the birth-response cohort feedback model. In the case of symmetric net maternity, we put the model into a form in which one of the rare global bifurcation theorems in the mathematical literature can be brought to bear. As a consequence, we prove that the model has solutions with period exactly twice the mean age of net maternity for at least a specified range of parameter values which include cycles of non-infinitesimal amplitude." (SUMMARY IN FRE)     

Stabilization,birth waves,and the surge in the elderly

This study examines the transition of a population to stability following a shift to a new fixed set of vital rates. Specifically, the authors develop a simple discrete population model and use it to derive an explicit solution for the birth trajectory. "The new vital rates interact with the population's initial age composition and generate birth waves whose amplitude and attenuation depend on the ratio of ultimate to initial growth and on the new pattern of stable net maternity. A greater change in growth and a later stable net maternity pattern produce larger fluctuations in the number of births. Stabilization begins at the youngest ages and proceeds upward. Sixty years after the shift, the birth waves have largely disappeared and the proportion under age 15 approximates the stable level implied by the new rates. Those patterns are manifest in the stabilization of both observed and Coale-Demeny model stable populations." (SUMMARY IN FRE)     

Migration, fertility, and aging in stable populations

Fertility is below replacement level in all European countries, and population growth is expected to decline in the coming decades. Increasing life expectancy will accentuate concomitant aging of the population. Migration has been seen as a possible means to decelerate aging. In this article, I introduce a stable, open-population model in which cohort net migration is proportional to births. In this case, the migration-fertility trade-off can be studied with particular ease. I show that although migration can increase the growth rate, which tends to make the age distribution younger, it also has an opposite effect because of its typical age pattern. I capture the effect of the age pattern of net migration in a migration-survivor function. The effect of net migration on growth is quantified with data from 17 European countries. I show that some countries already have a level of migration that will lead to stationarity. For other countries with asymptotically declining population, migration still provides opportunities for slowing down aging of the population as a whole.     

Stabilization,birth waves,and the surge in the elderly*

Substantial regularities characterize the transition to stability that follows a shift from one set of vital rates to another. The new vital, rates interact with the population's initial age composition and generate birth waves whose amplitude and attenuation depend on the ratio of ultimate to initial growth and on the new pattern of stable net maternity. A greater change in growth and a later stable net maternity pattern produce larger fluctuations in the number of births. Stabilization begins at the youngest ages and proceeds upward. Sixty years after the shift, the birth waves have largely disappeared and the proportion under age 15 approximates the stable level implied by the new rates. Those patterns are manifest in the stabilization of both observed and Coale‐Demeny model stable populations.

When fertility falls, the new stable population has a larger fraction at all ages above (approximately) 30, with greater changes characterizing the extremes of life. Fifteen years after the fall, there is a trough in the number at ages 0–14. Sixty years after the fall, when the largest pre‐decline cohort is age 60–74 and the smallest post‐decline cohort is age 45–59, there is a surge in the relative size of the elderly population. Thus after two generations, the birth waves produced by a rapid decline in fertility accentuate the effects of population aging.     

Pre‐procreative ages in population stability and cyclicity

Age‐specific models of population renewal (with and without feedback) which imply convergence to a stable state for some levels of fertility or feedback may imply the presence of long‐term cycling around a constant or exponentially changing equilibrium for other levels of fertility or feedback. The switch from one regime to the other is a “bifurcation.”; The conditions for bifurcation involve the roots of an analogue of Lotka's Equation.

Typically bifurcation is induced by raising the strength of feedback or the level of fertility. It has been known since the early 1980s, however, that this is sometimes impossible. It is sometimes impossible even with the linear renewal equation itself and with the most basic of non‐linear models, Lee's cohort feedback model.

Here it is proved that this typical route to bifurcation does not fail for these basic models in the presence of a condition which always holds for realistic applications with higher organisms: the existence of a span of ages before the onset of fertility.

Specifically, a strictly positive lower bound on ages of procreation is proved to be sufficient to guarantee the existence of a rescaling of Lotka's Equation for which the real part of some complex root vanishes. This result holds for absolutely Lebesgue‐integrable (signed) net maternity functions on the positive real line and for absolutely summable (signed) net maternities on the positive integers.

It follows that Coale's rescaling device for the analysis of approach to stability in stable population theory can be implemented for all realistic human net maternity schedules. It also follows that the many special cases of the cohort feedback model throughout population biology will all generate persistent cycling instead of stability if feedback is sufficiently strong.     

The multiregional net maternity function and multiregional stable growth

This paper extends recent efforts to generalize Lotka's integral equation to the case of a multiregional population that experiences internal migration. It develops the concept of a multiregional net maternity function and considers the relationships between a multiregional population's schedule of fertility, mortality, and migration and its stable growth properties.     

The Vital Statistics Method of Estimating Net Migration By Age Cohorts

The focus of this paper is the development and testing of a method of estimating deaths which occur during a decade to aging birth and death cohorts, so that it may be possible to estimate net migration by the vital statistics (VS) method for age cohorts. Until now the VS method has been used only in making estimates of total net migration.The results obtained by using the VS method for age cohorts show that (1) the average census survival rate (CSR) method generally yields algebraically lower estimates of net migration than does the VS method; but (2) there are some striking exceptions which are apparently associated with errors in census enumeration by age, sex, and color. Comparisons between the average CSR and the VS methods are shown, by age, for both the North Carolina and the coterminous United States populations.A cursory examination of these comparisons suggests that the exclusive use of the VS method in estimating net migration for age cohorts may lead to substantial error. Finally, the magnitude of these errors in estimating net migration, as well as in census enumeration, can be roughly approximated if it is assumed that the use of the CSR method yields reasonably accurate estimates of net migration.     

Indigenous labor supply,sustenance organization,and population redistribution in nonmetropolitan america: An extension of the ecological theory of migration

The ecological theory of migration asserts that change in sustenance organization, to the extent that it produces changes in the opportunities for living, necessitates a change in population size. Migration may thus be viewed as a demographic response to the population’s need to reestablish a balance between its size and sustenance organization, thus attaining its best possible living standard. However, the levels of net in- or out-migration needed to restore the balance should be affected by the degree of positive or negative growth of the indigenous labor force population. We thus test the hypothesis that changes in opportunities for living will be balanced by net changes in the number of persons in the labor force, where this is a function of both indigenous labor supply and net migration.     

Developing strategies for deriving small population fertility rates

Many agencies require population estimates and projections by ethnic group. These projections need ethnic-specific, age-specific fertility rates (ASFRs) but their inclusion is challenging since ethnicity is not recorded at birth registration. In this paper maternity data are used in a case study of electoral wards in Bradford, West Yorkshire, to develop fertility rates for small populations for a 1991 based projection. The challenge is to capture local variations in fertility by ethnic group when data are sparse. Small areas were grouped together using cluster analysis to define combinations with similar sociodemographic and fertility experiences so that sparse data could be aggregated to estimate reliable ethnic-specific fertility rates. For comparison, the data were aggregated into the 1991 Office for National Statistics area type classification. Fertility rates by single year of age for all area types were smoothed using the Hadwiger function. For the White ethnic group there were sufficient births to create ethnic-specific, ward-level ASFRs. For other ethnicities grouping of areas was necessary. The accuracy of the ASFRs in predicting births was assessed using mean absolute percentage error. Results show that for some minority groups district-level ethnic-specific fertility rates produced the most accurate birth estimates even though they were based on a larger area. This implies that rates created may be informative about the local area for White ethnic type but not in the same way for smaller ethnic groups. In terms of grouping strategies we recommend that existing classifications are assessed to determine how well variations in rates are stratified before embarking on a custom scheme. Where population sub-groups are small in some areas, it may be more reliable to use rates derived for larger areas and apply these to local populations. Inevitably, the rates used in a projection are a compromise but hopefully will still capture important dimensions of population change.     

A Decision Aid for midwifery continuity of care: Development and pilot acceptability testing

BackgroundMany maternity services in Australia offer women a variety of models of care including midwife led models. Childbearing women, however, need to understand the differences between these models if they are to make an informed decision about their choice of care. Decision Aids (DA) help people decide when there is not a single best option and the best decision will be based upon the values of the decision maker. There is no current tool that focuses on the choice of midwife led vs other models of maternity care.AimThis research aimed to develop, and pilot test a Decision Aid focusing on the choice between midwife led and standard models of maternity care.MethodsThe DA was developed using the International Patient Decision Aid Standards and pilot tested for acceptability with a group of clinicians who provide antenatal care in one jurisdiction in Australia. A posttest only study was conducted assessing knowledge, acceptability and decisional conflict, with a group of women of childbearing age living in the jurisdiction.FindingsA DA was developed and pilot acceptability testing with 14 women and 13 clinicians of Australian Capital Territory (ACT) health demonstrated its acceptability and highlighting areas for further development.DiscussionSome revisions may be needed to address issues of balance and bias toward midwife-led care identified by some recipients.ConclusionPilot acceptability testing with women and staff of ACT health provides a steppingstone to further research, development and evaluation of this DA.     

Age composition in the absence of mortality and in other odd circumstances

The age composition of populations experiencing no mortality and various kinds of fertility is derived. It is then shown that, if successive mortality schedules can be expressed as the sum of a component that varies with age and a component that varies with time, only the former has any effect on the age structure of the population. One implication is that a population in which mortality varies with time but not with age has the same composition as a population with no mortality at all.     

The use of fourier analysis to express the relation between time variations in fertility and the time sequence of births in a closed human population

A fundamental theorem of mathematical demography states that two closed populations sharing the same extended history of net fertility, no matter how variable, have the same age composition. In consequence, when in any population net fertility has long followed a purely repetitive cycle, age composition must also be repetitive with the same cycle length. Under these conditions annual births follow a path that is an exponential multiplied by a periodic time-function. If the time variation of fertility is a small amplitude sinusoidal oscillation, the periodic component of the birth sequence is also sinusoidal. The relative amplitude and timing of fertility oscillations and the consequent birth fluctuations are functions of the duration (or frequency) of the fertility cycle; the nature of the functions is determined by the age structure of net fertility.Any time sequence of fertility variation over an extended period of time can be treated as if part of a repetitious cycle of very long duration. By Fourier analysis the fertility sequence can be expressed as the sum of a series of sinusoidal terms; and finally the sinusoidal components of the resultant birth sequence can be derived. The calculated birth sequence closely approximates the actual.     

D. V. Glass on the Problems of a Declining Population

During the past quarter century fertility has dropped below replacement levels in many parts of the world. According to United Nations estimates, in 2005 this was the case in 65 countries, comprising 43 percent of the world's population. In many cases, most notably in Europe and East Asia, the shortfall of fertility from the level that would be necessary in the long run to sustain a stationary population is substantial. In Europe, for example, the average total fertility rate for the period 2000–2005 was 1.4. Indefinite maintenance of such a level implies a shrinkage of the total population by one‐third over a generation–roughly every 30 years. Accompanying that rapid decline of total numbers would be an age structure containing a preponderance of the elderly, posing extreme adjustment difficulties for the economic and social system. Societies that wish to avoid radical depopulation would have to engineer a substantial rise infertility–if not to full replacement level (slightly more than two children per woman), then at least to a level that would moderate the tempo of population decline and make population aging easier to cope with. An additional counter to declining numbers, if not significantly to population aging, could come from net immigration. This is the demographic future assumed in the UN medium‐variant projections for countries and regions currently of very low fertility. Thus, for example, in Europe over the period up to 2050 fertility is assumed to rise to 1.85 and net immigration to amount to some 32 million persons. The UN projections also anticipate further improvement in average life expectancy–from its current level of 74 years to 81 years. This factor slows the decline in population size but accelerates population aging. Under these assumptions, Europe's population would decline from its present 728 million to 653 million by 2050. At that time the proportion of the population over age 65 would be 27.6 percent, nearly double its present share. Demographic change of this nature is not a novel prospect. It was envisioned in a number of European countries and in North America, Australia, and New Zealand in the late 1920s and early 1930s. Concern with the possible economic and social consequences generated much discussion at that time among demographers and social scientists at large and also attracted public attention. Possible policy measures that might reverse the downward trend of fertility were also debated, although resulting in only hesitant and largely inconsequential action. The article by D. V. Glass reproduced below is an especially lucid and concise treatment of demographic changes under conditions of low fertility and their economic and social implications. It appeared in Eugenics Review (vol. 29, no. 1, pp. 39–47) in 1937 when the author was 26 years old. Glass's line of argument is broadly representative of the main focus of demographic analysis in the mid‐1930s on aspects of population dynamics, applying the then still novel analytical tool of the stable population model. It also echoes the work of economists then witnessing the great difficulties capitalist economies faced in adjusting to structural changes in consumer demand and labor supply. While Glass addresses these issues primarily with reference to England and Wales, he sees the issues as affecting all industrialized countries. The Malthusian problem of relentless population growth he persuasively declares to be irrelevant for these countries. The Western world faces the opposite problem: population decline, a trend only temporarily masked by the effects of an age distribution that still has a relatively high proportion of women in the child‐bearing ages, reflecting the higher fertility level of the past. A stationary population, Glass cogently argues, is to be welcomed, and he considers the absolute size at which zero growth would be achieved relatively unimportant. In contrast, a continuous population decline would have “thoroughly disastrous” results in an individualist civilization and in “an unplanned economic system.” And, he concedes, somewhat quaintly, that sustained below‐replacement fertility would pose a great problem “even in a country in which the means of production were owned communally.” Glass's conclusions about the reversibility of low fertility are as pessimistic as those of most informed observers today. Still, he sees hope in a future “rationally planned civilization” that would “produce an environment in which high fertility and a high standard of life will both be possible.” In this context, high fertility means the level necessary to sustain the population in a stationary state. By present‐day standards the level Glass calculates as needed for long‐term zero growth is indeed fairly high: 2.87 children per woman. But that figure reflects the fact that, when he wrote, mortality up to age 50 was still fairly high and fertility occurred almost wholly within marriage; it also assumes zero net immigration. In the last 70 years much has changed in each of these three components of population dynamics, both in England and Wales and in the rest of Europe. Still, Glass's commentary remains highly relevant to the discussion of the problems of low fertility today. David Victor Glass (1911–78) was associated with the London School of Economics throughout much of his scientific career. He followed R. R. Kuczynski as reader in demography in 1945 and became professor of sociology in 1948. His work on demography, population history, and population policy had already made him one of the most influential demographers in pre‐World War II Britain. After the war he rose to international prominence through pioneering work on the Royal Commission of Population; through his research on historical demography, the history of demographic thought, and social mobility; and through founding, in 1947, the journal Population Studies, which he edited until his death.     

Immigration and the stable population model

This paper reports on work aimed at extending stable population theory to include immigration. Its central finding is that, as long as fertility is below replacement, a constant number and age distribution of immigrants (with fixed fertility and mortality schedules) lead to a stationary population. Neither the level of the net reproduction rate nor the size of the annual immigration affects this conclusion; a stationary population eventually emerges. How this stationary population is created is studied, as is the generational distribution of the constant annual stream of births and of the total population. It is also shown that immigrants and their early descendants may have fertility well above replacement (as long as later generations adopt and maintain fertility below replacement), and the outcome will still be a long-run stationary population.     

Pro-natalist population policies in Czechoslovakia

Summary Given the high rate of women's employment and the lack of labour reserves, other than the natural replacement of the population, pro-natalist population policy in Czechoslovakia should be seen as a response to an anticipated shortage of labour. The rapid post-war decline in the birth rate has been caused by the greatly increased opportunity structure for women in education and employment, and by other policies favouring lower natality - rapid urbanization, inadequate provision of housing, insufficient investment in consumers' goods and services, low wages and relatively free availability of abortion. To reverse this undesirable population trend, the Czechoslovak government has adopted a more restrictive attitude towards abortion, lengthened paid maternity leave, increased family allowances and single grants given at childbirth and introduced the so-called maternity allowance, which is a direct monthly payment given by the state to mothers who wish to stay at home to raise a second or subsequent child, until the child is two years old. The time so spent counts towards the mother's retirement pension and other kinds of seniority, and her job is held open for her. These measures have contributed to the recent increase in the Czechoslovak birth rate, but more time is needed for the assessment of the long-term effectiveness of these measures.