Population growth with variable family size

"The Sharpe-Lotka continuous time deterministic model of population growth is developed to take account of some possible forms of mother-daughter fertility association....Model specific results relating the intergenerational fertility effect to the long term population growth rate and magnitude are established. The quantitative implications of the theory are illustrated by a consideration of a general bilinear form of A and in this context numerical results illustrating the finite time growth and also the long term distribution of fertility levels in the stable female population are obtained. In particular, it is shown that different fertility specific subpopulations can coexist indefinitely."     

Statistical modeling of selected aspects of the childbearing process with application to world fertility survey countries∗

The childbearing process should be monitored in developing countries experiencing high population growth rates and high levels of maternal and infant mortality. A mathematical model for estimation of certain aspects of the childbearing process, which requires only data on age‐specific fertility rates, is developed. Synthetic maternal childbearing indices, namely, mean ages at first and last birth, length of reproductive life span, inter‐birth spacing, and proportion of childless women, in addition to the well‐known mean age at childbearing, for the WFS countries are obtained using the proposed model. The indices are free from age truncation effects, and, under certain assumptions, provide information about a cohort's completed fertility before the women stop reproducing. The effects of women's residence and education on fertility are also examined.     

Macro‐demographic effects of the transition to adulthood: Multistate stable population theory and an application to Italy

We exploit a multistate generalisation of a classical, one‐sex, stable population model to evaluate structural and long‐term effects of changes in the attainment of adulthood. The demographic framework that inspired this paper is provided by Italy, where a strong delay in the transition to adulthood and union formation has been observed over the last several decades. Italy has also experienced very low fertility levels, and the subsequent ageing problems have become of primary concern. We first discuss a theoretical framework based on the model developed by Inaba (1995) and then include the process of transition to adulthood. We consider explicitly some specifications of the general model, and we present two distinct empirical applications, one using macrosimulation and the other one using a linear approximation. Our principal aim is to evaluate the impact of the delay in the attainment of adulthood on reproduction and on the age structure of the population.     

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 multistate stable population model with immigration

This paper outlines the discrete‐time and continuous‐time formulations of the stable population model with immigration, showing their commonality. It then illustrates how the model can be extended to include multiple interacting populations, and goes on to consider a multistate version of reproductive value that further illuminates the evolutionary dynamics of an “open”; model of multistate population growth and redistribution. Attention is restricted to results arising from a fertility regime that is below replacement level.     

Estimating the goodman,keyfitz, pullum kinship equations: An alternative procedure∗

As is often the case in demography, Goodman, Keyfitz and Pullum (1974) developed their theory of the interrelationships of fertility, mortality and kinship numbers by means of continuous mathematics [integrals], but resorted to ad hoc finite approximations for calculating results in concrete empirical cases. Their reason: ‘Ordinarily, we cannot evaluate the l(x) and m(x) functions for arbitrary values of x, since the data are usually collected for five‐year age intervals’ [p. 24]. Recent developments in computer software now provide an alternative, two‐step procedure that avoids extensive programming of finite approximation algorithms: 1) using a popular scientific curve‐fitting package, functions are found to represent particular sets of discrete data on fertility and mortality, 2) the resulting functions and parameter estimates are then inserted directly into the kinship equations, and the integrals evaluated numerically using readily available mathematics software. This procedure has potentially wide application in other areas of population mathematics where theory is given by integrals and other continuous expressions, but data are for discrete age groups.     

Populations with constant immigration

We consider a Leslie‐type model of a one‐sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie‐type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigratioa     

Human capital,technological progress and the demographic transition*

We emphasize the importance to consider components of population growth — fertility and mortality ‐ separately, when modeling the mutual interaction between population and economic growth. Our model implies that two countries with the same population growth will not converge towards the same level of per capita income. The country with the lower level of birth and death rates will be better off in the long run. Introducing a spill over effect of average human capital on total productivity our model implies multiple equilibria as illustrated in Becker el al. (1990) and Strulik (1999). Besides the existence of a low and high level equilibrium ‐ as characterized by low and high levels of per capita output respectively ‐ we show the existence of multiple low level (Malthusian) equilibria. Initial conditions and parameters of technological progress and human capital investment determine whether an economy is capable to escape the low level equilibrium trap and to enjoy sustained economic growth.     

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.     

A stochastic version of the malthusian trap model: Consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross‐sectional correlation between these two growth rates under two additional assumptions: i) the relations in the model at national levels include country‐specific and time‐invariant random components, and ii) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these two assumptions can explain near‐zero correlations between the two growth rates even if there exist a strongly negative effect of population growth on economic growth. However it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

Turbulent dynamics in a Xviith century population

A reconstruction of the population of the Pays de Caux (1589–1700) yields the time series of a fertility behavior indicator, the overall Coale index If. In spite of the noisy appearance of its evolution, the trajectory of If looks ordered, as if it were confined alternatively to two given zones, looping in each of them for a while, then suddenly jumping from the low one to the higher one, or slowly whirling down from the high to the low one.

An attempt is made to explain this general temporal structure by using a simulation model based on the autoregulation model (the so‐called European Marriage Pattern), putting into play a choice of the spouse function, a fertility function, modalities of marriage and remarriage, under the environmental forcing of the reconstructed mortality conditions.

The correspondence between reconstruction and simulation turns out to be quite good, not only for the population size or the Coale index, but also for the marriage series, quite independently of the reconstructioa

A second simulation with simulated mortality conditions shows a bifurcation point: as the mean frequency of crisis increases, the state of the system leaves the lower level and concentrates more and more in the higher level.

Thus, not only does the autoregulator model appear validated by empirical data, but its bi‐modal structure is revealed, depicting the dynamic response of a traditional community both to the environment and to the endogeneous demographic process.     

Population momentum: A wider definition

Summary

Keyfitz has derived an elegant formula for estimating the ultimate size of an initially stable, growing population that abruptly reduces its fertility to replacement level. Reduction of fertility is achieved by the rather unrealistic device of dividing the original age schedule nffertility rates by the net reproduction rate. Only the inertia of the age distribution is thus accounted for, but not that of the fertility schedule. The key idea of an abrupt imposition of a fixed regimen capable in the long run of generating zero population growth may be retained, but the regimen made more realistic. By elaborating the population setting, such disparate ZPG regimens as reduction of marital fertility by contraception, delayed and/or less universal marriage, raised mortality risks, or permanent net out-migration may be formulated. Convergence of the populaton to stationarity becomes a two-phase process: a primary adjustment period of changing fertility rates followed by a period of age adjustment.

The present paper treats what happens when a fixed ZPG sterilization regimen, defined by a minimum age of sterilization γ and constant continuous risk φ of sterilization among unsterilized wives aged γ to β, is imposed abruptly (or else progressively over an interval T) upon an initially stable, growing population. Additional sources of residual growth are: (1) the nine-month lag in sterilization effect owing to pregnancy: (2) the more youthful pattern of child-bearing under sterilization: (3) the extra adjustment period (of length β-γ-0.75) of changing fertility rates; and (4) any delays in exposing elements of the population to the sterilization regimen.

Two questions are pursued. First, how important are the additional sources of residual growth? Secondly, how do their relative sizes vary as a function of the characteristics of the initial population?     

Age patterns of mortality for older women: An analysis using the age‐specific rate of mortality change with age

The age‐specific rate of mortality change with age, defined by k(x) = d Inμ(x)/dx, where μ(x) is the age‐specific death rate at exact age x, is estimated for middle and old ages in ten selected populations that are considered to have relatively accurate age data. For females in each of the study populations, k(x) follows a bell‐shaped curve that usually peaks around age 75. In some of the populations, the age pattern of k(x) for males is confounded with substantial cohort variations, which seem to reflect long‐term impacts of their World War I experiences.

Among the mathematical models proposed by Gompertz, Makeham, Perks and Beard, only the Perks model is consistent with the bell‐shaped pattern of k(x). It is shown that, if the risk of death for every individual follows the Makeham equation and if the individual frailty is gamma‐distributed, then the age‐specific death rate follows the Perks equation.     

The Census Bureau on Prospects for US Population Growth in the Twenty‐First Century

The U.S. Census Bureau periodically releases projections of the US resident population, detailed by age, sex, race, and Hispanic origin. The most recent of these, issued 13 January 2000, for the first time extend to the year 2100 and also include information on the foreign‐bom population. (Earlier projections were carried up to 2080.) The extensive tabulations presenting the new set, and detailed explanation of the methodology and the assumptions underlying the projections, are accessible at the Census Bureau's web site: http://www.census.gov . A brief summary of some of the main results of these projections is reproduced below from United States Department of Commerce News, Washington, DC 20230. (The Census Bureau is an agency of the Department of Commerce.) Uncertainties as to future trends in fertility, mortality, and net migration over a period of some 100 years are very great, as is illustrated by the massive difference in the projected size of the population for 2100 in the three variants produced. The “middle” projected population figure of 571 million (which represents a growth of some 109 percent over its current level) is bracketed by “lowest” and “highest” alternative projections of 283 million and 1.18 billion, respectively. With somewhat lesser force, the point also applies to the 50‐year time span considered in the well‐known country‐by‐country projections of the United Nations. These projections are also detailed in three variants: low, middle, and high. The UN projections (last revised in 1998) envisage less rapid growth in the United States during the first part of the twenty‐first century than do the Census Bureau's. The projected population figures for 2050 in the three variants (low, middle, and high) are as follows (in millions):

Cyclically stable populations∗

The cyclically stable population relaxes the stable population assumption of fixed vital rates and replaces it with the assumption of a recurring sequence of schedules of vital rates. From any point (or stage) in one cycle of the sequence to the same stage in the next cycle, the cyclically stable population grows at a constant rate (λ). While the age composition of the cyclically stable population is different at different stages of the same cycle, it always has the same age composition at the same stage of every cycle. The essential dynamics of the cyclically stable model are captured by its birth projection matrix (BPM). The dominant eigenvalue of the BPM is growth rate A, and the right eigenvector associated with λ gives the within cycle‐birth sequence.

An important special case occurs when λ = 1, and a cyclically stationary population arises. Such populations challenge simplistic ideas about “Zero Population Growth.”; A population projection based on the sets of rates observed in the United States, 1970–90, shows a cyclically stationary population arising in less than 100 years. While it experiences no long term growth, that cyclically stationary population exhibits fluctuations in total size and considerable variability in age structure.     

A model of numbers of births in three countries,with persistent forty‐year cycles

A non‐linear model of fertility is described, which was derived from data for more than a century from England and Wales, New Zealand, and the U.S.A. The demographic transition is modelled with a logistic function, and age‐specific fertility rates are estimated using lognormal distributions. The stepwise inclusion of a partner availability estimate in the model showed that it accounts for twenty‐nine percent of otherwise unexplained variance. Projected future births stabilise in sustained or limit cycles with periods a little longer than 40 years, and amplitudes at least 7% of the mean. The necessary conditions for cycle persistence are outlined on a graph of maximum and minimum fertility parameters.     

Posterity and population growth: fertility intention among a cohort of Nigerian adolescents

Population growth is probably the greatest global challenge of the twenty-first century and fertility is a central element of this growth. Fertility is a human attribute which depends almost entirely on social, economic, political, cultural and psychological frameworks, making fertility intention an element of what individuals learn from a very young age as part of their socialisation into society. The fundamental significance of socio-psychological, environmental and cultural factors in what adolescents are assimilating on fertility cannot be exaggerated, yet, relevant information is limited. Eight factors deduced from ecological model and theory of planned behaviour were used to predict fertility intention among a cohort of Nigerian adolescents, using cross-sectional design. Mean fertility intention was 4.06 ± 1.34. Age and religion had no effect, but gender did. Self esteem, perceived parental expectation of fertility, attitude towards fertility and peer-related subjective norm are significant predictors. Media and ethnic attitude are insignificant predictors of, but are significantly related to fertility intention. Attitude towards a four-child family and perceived behavioural control yielded insignificant relationships with, and also failed to predict fertility intention. Perceived parental expectation of fertility, an interpersonal factor of the ecological model is the single most important predictor (β = 0.707, R² = 0.506, r = 0.711, and partial r = 0.710). Fertility intention points towards fertility decline, though sluggish and diminutive, thereby failing to reflect the need of Nigeria’s population pyramid to thin out from the base.

     

Age structure,growth, attrition and accession: A new synthesis

This paper shows that each equation describing relationships among demographic parameters in a stable population is a special case of a similar and equally simple equation that applies to any closed population and demonstrates some implications of these new equations for demographic theory and practice. Much of formal demography deals with functions that pertain to individuals passing through life, or to a stationary population in which births of individuals are evenly distributed over time. These functions include life expectancy, probabilities of survival, net and gross reproduction rates, expected years spent in various states and the probability that certain events will occur in the course of life. The stable population model permits the translation of population structure or processes in a more general type of population, with constant growth rates, back into equivalent populations for a stationary population. The method for translation developed in this paper, requiring only a set of age-specific growth rates is even more general, applying to any population. Age specific growth rates may also be useful for performing reverse translations, between a population's life table and its birth rate or its age distribution. Tables of numbers of females by single years of age in Sweden are used to illustrate applications. Tables summarize the basic relations among certain functions in a stationary population, a stable population and any population. Applications of new equations, particularly to demographic estimation of mortality, fertility and migration, from incomplete data, are described. Some other applications include; the 2 sex problem, increment decrement tables, convergence of population to its stable form, and cyclical changes in vital rates. Stable population models will continue to demonstrate long term implications of changes in mortality and fertility. However, in demographic estimation and measurement, new procedures will support most of those based on stable assumptions.     

Long‐Term Contribution of Immigration to Population Renewal in Canada: A Simulation

We analyze the direct and indirect demographic contribution of immigration to the foreign‐origin composition of the Canadian population according to various projection scenarios over a century, from 2006 to 2106. More specifically, we use Statistics Canada's Demosim microsimulation model to assess the long‐term sensitivity to immigration levels and the frequency of mixed unions of the share of immigrants in Canada and of persons who have at least one ancestor who arrived after 2006. The results of the simulations show that the population renewal process through immigration happens at a fast pace in a high immigration and low fertility country such as Canada. Under the scenarios developed, immigrants who entered after 2006 and their descendants could form the majority of the population by 2058 at the earliest and by 2079 at the latest and could represent between 62 percent and 88 percent in 2106. They also show that mixed unions are a key element of the speed at which the changes are likely to occur in the long run.     

How responsive is U.S. population growth to immigration? A situational sensitivity analysis

There has been public concern about the effect of immigration on population growth in the U.S. But how responsive is population growth to immigration? This paper examines the sensitivity of intrinsic population growth to immigration and situates such sensitivity in fertility and survival changes. The application of second derivatives on a modified Leslie matrix facilitates the analysis of situational sensitivity of U.S. population growth to immigration. The results show that the sensitivity to immigration is not as influential as the sensitivity to fertility, and that the sensitivity to immigration further depends on changes in fertility and survival.