Derivations of complex roots of a stable model for a special distribution of net maternity rates: an alternative method

"Several mathematical models representing the age distribution of net maternity rate p(x) have...tried in the past [to] generate estimates of the parameters of a stable population model.... In an earlier paper, Mitra and Levin (1990) experimented with a generalized model.... Among other things, that model generated a pair of nonlinear equations from which the intrinsic roots, real as well as complex, had to be solved. The purpose of this paper is to present an alternative method of solving the same pair of equations which reveals another set of interesting features characteristic of this different approach."

Growth rate, population entropy, and perturbation theory

This paper is concerned with the connection between two classes of population variables: measures of population growth rate--the Malthusian parameter, the net reproduction rate, the gross reproduction rate, and the mean life expectancy; and measures of demographic heterogeneity--population entropy. It is shown that the entropy functions predict the response of the growth rate parameters to perturbations in the age-specific fecundity and mortality schedule. These results are invoked to introduce the notion of environmental intensity. The intensity function, expressed in terms of the entropy parameters, is applied to give a comparative study of the effect of environmental factors on the dynamics of Swedish and French populations.

Conditions determining the local stability of populations

The conditions that determine the local stability classification of an equilibrium population configuration are analyzed. The population investigated is age-structured and density-dependent, where density is determined by an age-weighted population size. 2 demographic parameters are introduced: the marginal birth rate and the marginal death rate, which describe the marginal density-dependence of the birth and death rates of the equilibrium population. Certain necessary and/or sufficient conditions determining stability are developed, most of them involving the net reproduction rate of the population, and examples illustrating these conditions are presented.

Migration and stability

"In this endeavor to incorporate net migration in a model of population growth, we have found that when net maternity rates are held constant, birth trajectories can be traced in several examples. We have basically experimented with (a) unchanging age-specific net migration rates and also with (b) constant number of net migrants with unchanging age composition. In most cases, the birth trajectory approaches an exponential pattern of growth with two interesting exceptions. When the intrinsic rate of growth is negative, the population approaches stationarity in the second example. Also a linear trend is found in the same example, when the net reproduction rate is equal to one." (SUMMARY IN FRE AND ITA)

Recent fertility trends in Suriname

A rapid decline in fertility took place in Suriname between 1962 and 1974, and then stopped. While this sudden stabilization is surprising, it is not unusual. Similar trends have occurred in some Caribbean and Latin-American countries. The 1962–74 fall in fertility in Suriname seems to have been due to a combination of socioeconomic factors and the activities of the Suriname Family Planning Association, founded in 1968.

[Tempo and parity structure of Japanese fertility]

The present study is part of a larger project on the formal demographic analysis of Japanese fertility. It deals with the most elusive aspect of fertility, that is the tempo of fertility. The temporal aspect of fertility, particularly in terms of cohort fertility framework with speical reference to parity of women, has been emphasized by Whelpton and Ryder, but in the Japanese demography there have scarcely been studies of this aspect, probably due to lack of interest. In very recent years, however, interest has grown, reflecting the need for more accurate fertility projections in the future in connection with population projections. The present study is an application of life table techniques to explore the temporal aspect of fertility in Japan. 2 methods have been used. One is the age-marriage-parity specific net reproduction rate tables, which were first constructed by the senior author of this paper. The other is by Professor Chin Long Chiang at the University of California, Berkely. In the application of the former type, net reproduction rate tables were constructed for 3 dates -- 1970-71, 1975-76, and 1980-81. Through this method, it was demonstrated that recent fertility for the period 1980-81 was reduced, partly because of the decline in fertility in live-birth order of 1 and 2 among the relatively young ages up to 25 years old. On the other hand, for the ages up to 30 years old, some catching-up mechanisms took place. The 2nd method is by Chiang's new approach to the fertility table. According to his method, which was included in his new book entitled Life Tables and Its Applications, parity-oriented fertility tables have been constructed for the Japanese females for every year from 1971 through 1983, both inclusive. In these cases, denominator is for year z while numerator is concerned with yea z + 1. This method is particularly interesting since it can calculate waiting time for childbearing until the completion of family formation. Some of the findings indicated: 1) recent increases in both parity progession ratio and parity-specific fertility rate in low parity augur a future increase in fertility in the period measure in the near future; and 2) a shortening of the waiting time until the completion of family formation around 1981 and then the recovery of its lengthening in the years 1982 and 1983 suggest a recent trend of lengthening actual reproductive period which definitely contributes to an increase in fertility.

Reproductive life of Bhoksa women

The paper discusses the reproductive life of 111 ever-married Bhoksa women. The mean age at marriage for women of all ages among Bhoksas, like other tribal populations, is high, unlike the caste populations. The mean ages at first birth of the pooled sample and of the completed fertility cases suggest late and early marriages of the older and younger generations. The maximum number of marriages occur between 15 and 19 years and of first births between 16 and 20 years. Percentage of reproductive wastage is high in both the lower and higher age groups. Young mothers with low birth orders and older mothers with high birth orders display a high frequency of reproductive wastage. Evidently, both birth order and the age of the mother have effects on reproductive wastage. Average number of children ever born (including stillbirth but not abortion or miscarriage) per mother of all ages is the highest among Bhoksas of all the studied ethnic groups of India. The Bhoksa, like caste populations, show a high number of children ever born per mother of completed fertility. Quite a high masculinity in the secondary sex ratio, like other mongoloid population is noticed. The contribution of mortality component to the Total Index of Opportunity for Selection is more than that of the fertility component. Bhoksas conform to the general low range of net reproductive index, which is however greater than unity, suggesting that they are in a growth stage.

Effects of fertility and mortality on extinction of family and number of living children

A model is developed to study the effects of fertility and mortality on number of living children and on extinction of family. Numerical illustrations are provided by applying the model to three different populations differing in fertility and mortality. It is found that a combination of a high birth rate and a low death rate is required to assure a reasonable probability that the family will not ultimately become extinct. A moderate to high birth rate and a low death rate are necessary to assure that there will be at least one surviving son in the parents' old age. This finding implies that zero population growth is impossible as long as parents depend on children for old age security. Thus, alternate old age security measures must be adopted by societies attempting to reduce their birth rate to the replacement level.