Net Maternity Function Research Articles

Population Projection under Changing Fertility Conditions

This paper attempts to derive model for population projection under gradual change in fertility schedule on stability conditions using birth trajectory at time t for 0 < t ? a . The projected population can be obtained when the data of the rate of natural increase, net reproduction rate, age variance of the net maternity function, birth rate of the initial stable population along with its age distribution are available of the population.Journal of Institute of Science and Technology, 2014, 19(2): 86-88

Generalized Lotka stability

The recently developed macroscopic approach to demography describes the age distribution of mothers and the net maternity function for a given human population entirely in terms of five parameters. Tracking of these parameters provides a number of new tools for analyzing populations and predicting their future states. Within the macroscopic approach, the new concept of generalized Lotka stability is presented in this paper, as an extension of a strong version of classic Lotka stability. The two leading parameters of the macroscopic approach, the Malthusian parameter r and the perturbation s, are computed from population data and plotted in two-dimensional parameter space. Generalized Lotka stability is then defined in terms of the movement of the (r,s)-vector over time. It may be observed in a number of human populations at specific periods of their history.

Parametrizing the Net Maternity Function

AbstractThe net maternity function represents the chance of a newborn female surviving to a certain age class, multiplied by the average number of female babies she is likely to bear while sojourning in that age class. Although the net maternity function does not lend itself well to modeling in terms of conventional time, it may be parametrized effectively as a function of the logarithm of a mother’s age. Choice of this independent variable leads to absolute dimensionless quantities that do not have to be rescaled when the units of time are changed. The parametrized model provides a close fit to the real net maternity function for stable populations, while divergences between the model and actual function may serve as a diagnostic for new features within a population. Reduction of the specification of the net maternity function to a small number of parameters enables those parameters to be tracked in long-term historical studies, and also serves as the basis of techniques for predicting future states of a...

Parametrizing the Net Maternity Function

The net maternity function represents the chance of a newborn female surviving to a certain age class, multiplied by the average number of female babies she is likely to bear while sojourning in that age class. Although the net maternity function does not lend itself well to modeling in terms of conventional time, it may be parametrized effectively as a function of the logarithm of a mother's age. Choice of this independent variable leads to absolute dimensionless quantities that do not have to be rescaled when the units of time are changed. The parametrized model provides a close fit to the real net maternity function for stable populations, while divergences between the model and actual function may serve as a diagnostic for new features within a population. Reduction of the specification of the net maternity function to a small number of parameters enables those parameters to be tracked in long-term historical studies, and also serves as the basis of techniques for predicting future states of a population. Among the parameters, a new logarithmic generation time emerges as an important population variable.

A macroscopic approach to demography.

A canonical/lognormal model for human demography is established, specifying the net maternity function and the age distribution for mothers of new-borns using a single macroscopic parameter vector of dimension five. The age distribution of mothers is canonical, while the net maternity function normalizes to a lognormal density. Comparison of an actual population with the model serves to identify anomalies in the population which may be indicative of phase transitions or influences from levels outside the demographic. Tracking the time development of the parameter vector may be used to predict the future state of a population, or to interpolate for data missing from the record. In accordance with classical theoretical considerations of Backman, Prigogine, et al., it emerges that the logarithm of a mother's age is the most fundamental time variable for demographic purposes.

Exploring cyclic net reproduction

This paper advances a new approach that provides closed form expressions for the birth trajectory produced by a regime of changing vital rates. An exponentiated sinusoidal net maternity function is considered in detail, as populations with cyclically varying net maternity are of particular interest because of their connection to the Easterlin hypothesis. The dynamics of the model are largely determined by the ratio of the population's generation length (A) to the period of cyclicity (T), and relatively simple expressions are found for the phase difference and relative amplification of the birth and net reproduction functions. More generally, an analytical expression for a population's birth trajectory is derived that applies whenever net reproductivity can be written as an exponentiated Fourier series. In the cyclic model, Easterlin's inverse relationship between cohort size and cohort fertility holds whenever the phase difference is zero. At other phase differences, the birth‐reproduction equations have the form of predator‐prey equations. The present analytical approach may thus be relevant to analyses of interacting populations.

On the intrinsic force of convergence to stability.

"Observed populations differ greatly in the speed with which they approach the stable form, but what determines rates of convergence is not fully understood.... Here we examine the trajectory to stability, derive a mathematical expression for the force of convergence, and provide an approximate relationship in terms of the mean and variance of the stable net maternity function." Trajectories for Japan, Togo, and the United States are used as illustrations. (SUMMARY IN FRE)

On the process of stabilization in the renewal model: approximations for the time to convergence.

The transient behaviour of the renewal model leading to the stable age distribution is studied for weakly skewed net maternity functions (found in human as well as in some animal populations). The study, which is partly based on heuristic arguments, first provides approximate expressions for the damping constant and the circular frequency (in terms of the moments of the net maternity function) belonging to the principal oscillatory component of the birth trajectory. The time to stability (defined as the time interval after which the principal oscillatory component has become less than a certain fraction of the stable solution) is then determined in two cases: For the genesis model and for a stable population in which the net reproduction rate is reduced to one. The results are applied to a problem which arises in the mass rearing of pest insects.

Some Reducible Models of Age Dependent Dynamics

Three models of age dependent dynamics are developed which can be reduced to systems of ordinary differential equations. These include the cohort model, which is based on the Easterlin hypothesis, the welfare-cohort model and the weighted cohort model. It is shown that the behavior of solutions of the cohort model approaches that of the first order difference equation modeling populations with discrete, nonoverlapping, generations as the net maternity function of the cohort model approaches the Dirac delta function. It is shown for relatively flat maternity functions that the weighted cohort model more readily produces persistent oscillations than does the cohort model.

Lotka's roots under rescalings.

In the mathematical theory of stable populations, when the net maternity function is scaled by a constant divisor , changing its level without changing its shape, the rates of attrition of transient waves in the age structure of the population as it converges toward stability are altered. The attrition rates are specified by the real parts of the complex roots of Lotka 's equation. Conditions are given for the falsity of the longstanding claim that there always exists some rescaling that brings to zero the real part of the complex root governing the lowest frequency wave. A general account of scalable and unscalable roots follows for the discrete-age, Leslie formulation, elucidating and setting limits to the standard account of approach to stability.

The long-term effects of time-dependent maternity behavior

Using the integral population model of Sharpe and Lotka, it is demonstrated that if the time variation of the maternity function is assumed to only affect the parent population, then standard methods of obtaining the long-term behavior may still be used. Further, if the net maternity function has explicit time dependence, in contrast to age dependence, only for time less than the minimum age of childbearing, the standard techniques still may be used. It is shown that the recent extensions of Cerone and Keane to include exponential time dependence may be applied, together with the above observations, to define piecewise time-dependent net maternity functions and to determine the resulting long-term asymptotic behavior of the population. The population management problem of determining the time path of change needed to approach a given population also is considered in some detail.

The asymptotic behaviour due to a piecewise time dependent net maternity function

AbstractThe long term asymptotic behaviour of a population is evaluated where the age specific fertility behaviour is allowed to change with time. Thus in this article the behaviour of a population is determined with a time dependent net maternity function. It is shown that methods used when the net maternity function was independent of time are still applicable if the change with time is explicit only for the initial population. Further, using the fact that for realistic situations the net maternity function is non-zero over a finite interval α < x < β, it is shown that traditional methods can again be used if the time dependence is associated with ages less than α, the minimum age of childbearing. Recent extensions of Cerone and Keane to include exponential time dependence are utilized and models are presented which are piecewise defined, allowing general and exponential time dependence for the parent and new-born populations respectively. The Sharpe-Lotka single sex determinstic population model is used as the basis for the analysis.

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.

Influence of instantaneous fertility decline to replacement level on population growth: An alternative model

If age-specific birth rate mx of a stable population drop abruptly to mx/Ro, where Ro is the net reproduction rate, then, according to Keyfitz, the size of the ultimate stationary population relative to that at the beginning of the process is given by I = beoo(Ro - 1)/(rmuRo), where b and r are the birth rate and the rate of growth, respectively, of the stable population, eoo the life expectancy at birth, and mu the average age at childbirth in resulting stationary population. Noting that the decline in mx need not necessarily be uniform, investigation has veen carried out to examine the effect on I when fertility decline is more rapid at higher ages. In particular, the effect of the reduced age-specific rates such as mxe-rx (which also produces a stationary population) has been analyzed, and simplifications of the results carried out separately for three different models of the net maternity function. It has also been shown the when mx drops abruptly to some m* x, where the form of m*x need not be specified except for the restriction that the resulting population will be stationary, the value of the index can be approximately obtained from I* = beoo (1 - rmu/2), where mu is the average age at childbearing of theinitial stable population.

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.

A systems model for the population renewal process

Abstract A synthetic cohort of women is followed from birth through childbearing age. The transition from parity i to parity i + 1 is quantitatively described utilizing systems analysis theory. This description results in a parametrization of the renewal process in which the net maternity function is completely determined by five parameters: maturation time, 0th parity time constant, survival fraction to 0th parity, average child spacing time, and parity progression ratio. To detect trends in these parameters, estimates are made of them for the U. S. A. population from 1950 to 1965.

General population theory in the age-time continuum

A linear first-order partial differential equation is derived for the age density k(x,t) of a biological population. It contains factors which are time-dependent functions defining attrition and immigration. Emigration may be included in the attrition function. The general solution k(x,t) of the differential equation contains an arbitrary function that is identified as the birth rate B(t) of the community. A Volterra integral equation for B(t) arises if the total population N of the community is planned to be a prescribed function of time. Also, projection of the population on the basis of a given net maternity function and a given immigration function requires the solution B(t) of a linear integral equation. This equation can be transformed to a Volterra type, but another form in which the limits of integration are constant bounds of the fertility period has some advantages. Some properties of solutions of the integral equation are discussed, and effects of stepwise discontinuities in the kernel are considered.

A Taylor series expansion for Lotka’s r

Abstract 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.

Asymptotic properties of a human age distribution under a continuous net maternity function

A useful and intuitively appealing proposition in theoretical demography asserts that the age distribution of a closed human populationis asymptotically independent of this shapein thedistant past, and is therefore exclusively determined by the historyof fertility and mortalitythat has prevailed during a reasonably long period of time. The mathematical foundations of this ergodic principle arelaid out in this article and thedetailsof its proofareworked out afteremphasizing an intuitive understanding of the process through which an age distribution tends to "forget" its past. The tendency for an unchanging schedule of vital ratesto produce a fixed agestructure in a closed population, is presented as a corollary of the main proposition dealt with in this article.