Ultimate Population Size: Some Investigations Under Stable Population Theory

Some models have been proposed for the projection of future size of population for short and long terms under the stability conditions with changed regime of fertility schedule. The main aim of this paper is to see the size of population if fertility is curtailed up to the level of replacement, especially in developing countries. Models have been illustrated taking a set of real and hypothetical data consistent with the current demographic scenario of India. It was found that the proposed models are the extended forms of the models developed by the previous researchers and the projected population are more or less consistent with them.


Introduction
In a stable population theory when population growth rate becomes zero, the population size becomes fixed and such population is known as stationary population or a population at the replacement level or an ultimate population. Several ways of reduction in fertility schedule have been proposed to obtain population size under stable and stationary conditions (Frauenthal [3]; Keyfitz [4]; Kulkarni [7]; Mitra [9]; Pathak and Ram [10]; Ryder [11]; Singh et.al. [12] and [13]; Yadava [14]; Yadava et.al. [15] and [16]). Keyfitz [4] considered an abrupt change in age specific fertility rate say in m(x,t) at time t as m(x)/Ro, where Ro is the net reproduction rate. The state of stationary is supposed to yield a new NRR as unity (see Figure 1), but it is not feasible due to population momentum built up in the past. That is the size of ultimate population would be larger than the expected one due to the existence of a high fertility rate built up by the past age structure (Keyfitz [4] and Kulkarni [7] (see Figure 2). Nevertheless, the size of population does not depend only on the way of reduction in fertility but also on the several other existing parameters of the population.  Keyfitz [4] has shown that when the age specific fertility rate m(x) is replaced by m(x)/Ro, such that new NRR, say Ro*, becomes unity, the ultimate population size is given by where, b: the present initial annual number of births; e 0 0 : expectation of life at birth; µ : mean age of child bearing in stationary population; Ro : the present net reproduction rate and r :the present annual rate of natural increase However, as mentioned above, an abrupt change in fertility schedule is not realistic even not feasible. Moreover, due to momentum, the population tends to increase for some time as shown in Figure 2 Accounting such type of decay if it for A years ( A is less than or equal to the lower limit of the reproductive period), Ryder [11] made an adjustment to Keyfitz's model (1) and according to him the ultimate population size becomes,.
Further if fertility schedule m(x) declines exponentially during the transition period of A years, an approximate formula for ultimate population size given by Coale [1] is: Where ≅ ln 0 is the mean length of generation.
Usually varies from country to country ranging between 27 to 29 years. On an average, if = 28 years, then by the Coale's model one can project population for a period more than the lower limit of the reproductive value. Kulkarni [7] has proposed two types of reduction pattern in fertility schedule, namely exponential and linear if time to reach the replacement level is less than or equal to the lower limit of the reproductive period. According to him, if m(x) declines exponentially as : Where = − ln / , so that = 1/ and α is the lower limit of the reproductive period. Under such reduction, the ultimate population size becomes (Kulkarni [7]), and when ( ) declines linearly as : then the ultimate population size becomes (Kulkarni [7]), It should, however, be noted that for a general value of A, it is very difficult to find an exact formula for the ultimate population size. All the above formulae have been derived under the assumption that a constant proportion of decline occurs in the age specific fertility schedule ( , ) at time t at all the ages of the reproductive period which may not be reasonable as well as practicable for a society. An age dependent reduction in fertility schedule would be more reasonable and accordingly Mitra [9] suggested a reduction in fertility to replacement level that would be more for older women than the younger ones i.e. age specific fertility rate ( )be replaced by * ( ) = − ( )rather than Taking such a change in fertility schedule, Kumar [6] has derived expressions for the ultimate population size, allowing exponential and linear decay as proposed by Kulkarni [7] for A years ( ≤ α) (see equations (4) and (6)). Allowing a decay of fertility to replacement level as a mixture of exponential for the first 1 years and linear between 1 to 2 years ( 1 < 2 ≤ ≤ α), Yadava et.al. [17] has also derived some formulae for the size of ultimate population under * ( ) = ( ) − . However, it is expected that at the start of any family welfare activity, programme goes slowly and picked up its speed at some later stage. Thus, a linear type of reduction in fertility schedule would be more useful at the beginning and then exponential decline at the later stage. This approach has also been taken into account in this study and models are derived accordingly. For simplicity, formulae developed in this paper dealt only with one sex i.e. the female sex. The male population can be obtained as Male population = (Female population) * [(Life expectancy for male) / (Life expectancy for female)] * sex-ratio at birth

Derivation of Models for Ultimate Population
Lotka [8] has given a renewal equation for female births ( ), at time t for a population which is closed to migration, as follows: where ( ) is the age distribution of females at time = 0; ( ) is the survival function of a female up to age ; ( ) is the probability that a female of age will bear a female child during dx period of her life. where ( ) is the trajectory of births to women born before time zero (i.e. before start of the reduction in fertility) and the integral term gives births to women born after time zero i.e. during the transition period of fertility reduction.
If the base line population is stable then, we have where is the instantaneous birth rate at = 0. Thus from equation (8) and (9), we get For > 0 For ≤ , equation (10) reduces to Because ∫ − ( ) ∞ ( , ) = 1 for ≤ is the fundamental equation for the intrinsic rate of increase r. The equation (11) gives the birth trajectory at any time ( ≤ ) obtained by Frauenthal [3].
If all the changes in fertility schedule takes during ( α) and after α, fertility remains constant i.e. ( , ) = ( ) for > α, thenthe renewal equation reduces to and the population ultimately reaches the stationary state with size (Keyfitz [4]) where G(t) is defined as, For an initially stable population, Further, if change in fertility is completed by A years ( ≤ α), then ( ) can be written as Now taking the different way of reduction in fertility schedule m(x) and hence evaluating the value of ( ), ultimate population size are projected by putting its values in equation (13).

Mitra Form of Reduction in Fertility Schedule:
As mentioned in the introductory section, if we take Mitra's form of reduction in ( ) i.e. ( , ) = ( ) − (Mitra [9]), then an age dependent equation (6) becomes where is a small positive integer Now the value of ( ) is, It is very difficult to evaluate the integrals like unless some functional form of the maternity function say, is assumed. Mitra [9] assumed the nature of the maternity function as normal distribution and accordingly Therefore, and the size of population is, The model M12 is same as Kumar [6] under a linear change in fertility schedule.
(iii) if 1 = 0 and 2 = then, the size of ultimate population is This model M13 is same as Kumar [6] under an exponential change in fertility schedule.
(iv) Further, for small , 2 2 ≅ 1 and then models M12 and M13 reduce to Kulkarni's [7] under exponential and linear change in fertility schedule respectively.

A Mixture of Abrupt and Gradual Change in Fertility Schedule:
If we fix a reduction in fertility schedule in between Keyfitz's abrupt change and Kulkarni's gradual change, then a formula for ultimate population size is derived as follows. In such a situation Kulkarni's [7] model under a linear change in fertility schedule.

An Uniform Reduction in Fertility Schedule:
If we assume an uniform decline in ( ) in ( 0, ), such that

A Linear Reduction in Fertility Schedule:
Kulkarni [7] proposed a linear change in m(x) (see equation (24)). Under such a linear change in fertility schedule, the value of G(t) is and hence the ultimate size of population as,

A Simplified Model:
It should, however, be noted that all the above models require data for many parameters to predict the size of the population. Some very specific models may be approximated as follows: Let us take the example of model M13 from equation (23), where the ultimate population size is (since 2 2 ≅ 1, for a small value of ) Thus, the size of an ultimate population is be 0 0 times the initial population. That is if only two parameters viz., birth rate and expectation of life at birth are available for a country/area, ultimate population can be projected for that country. By projecting population from equation (38), an extent of bias involved there can also be illustrated.

Ultimate Population Size : Application of the Reproductive Value
Keyfitz [5] has outlined that if current age specific fertility rate ( ) is replaced by * ( ) such that 1 < 0 * < 0 as A more simplified case of model M6 may be by putting and Into (42) and hence an ultimate population size as Further, if k is small enough to put it zero, then the ultimate population is just 0 1/2 times the initial population, which is given by Singh et.al. [13]. This shows that if only the net reproduction rate of a country/nation is known its ultimate population size can be worked out.
Thus several models proposed under the alternative fertility reduction are the extended form of the previous researchers like Keyfitz [4], Kumar [6], Kulkarni [7], Mitra [9] and many others.

Data and Methodologies
To obtain the size of ultimate population by the various models have developed here, we need the values of the demographic parameters b (birth rate of the initial stable population), (growth rate) and 0 0 (expectation of life at birth for females). For illustration, the values of these parameters are taken from the "Regional Model Life Tables and Stable  Population" Table 1 presents the ultimate size of the population under various models for India upto the lower limit of the reproductive period. Moreover, the ultimate population size estimated under various proposed models has been compared with the ultimate population size obtained by the other researchers including the model of Keyfitz [4]. Table 2 gives the projected population for different values of 1 and 2 ( 1 < 2 ≤ ) by model M1. For example, if 1 = 5 years and 2 = 10 years, the ultimate population size by the Model came out to be 1 .134 times the initial population (see also Figure 3) . This figure shows that a linear change in fertility for a longer time yield lesser size of ultimate population than exponential change.  Similarly if we take the value of k=0.001, the ultimate population size from model M6 was found to be 1.069 times the initial population. The ultimate size of population by models M5 and M7 was 1.414 and 1 .083 times the initial population respectively (see also Figure 4). This figure 4 also shows that the ultimate population size obtained by the proposed models provide more or less similar results provided by the models developed by the previous researchers. An upward bias of about 27 and 4 per-cent was found involved in the size of the population under the simplest models M5 and M7 respectively. Thus the models M5 and M7 would be useful for the projection of population size where reliable and accurate data on demographic parameters are not available. However, the validity of a population model depends on the way of increase or decrease of the net maternity function ( ( ) ( )). Nevertheless, the reduction in fertility schedule from ( ) to − ( ) seems to be more reasonable as it depends on the age of females in the reproductive period in comparison to reduction like ( ) = ( ) 0 (Keyfitz, 1971). Further, an abrupt change in fertility (as shown by Figure 2) may not yield the expected size due to momentum created by a high force of fertility build up in the past (Keyfitz [5]).

Conclusions
To obtain the size of an ultimate population under the stability conditions, a number of models have been developed under the various paths of fertility reduction. A model for the ultimate population size is also developed making use of the reproductive value as well as taking an increase in the survival function. Taking some hypothetical data consistent with the Indian demographic scenario, the models have been illustrated where possible explanations are given regarding merits and demerits of a model. Some models have also been approximated to extent of minimum number of demographic parameters to predict the size of ultimate population. These models are useful for many underdeveloped and developing countries where demographic data are found unreliable or inaccurate.
It has also been found that models derived in this paper are the generalization of all most all the previous models.

Computation of m(x)
If the age specific fertility rate (ASFR) for female births only is known then it is needed to convert it into age specific marital fertility rate (ASMFR).Since ASFR includes all the females and this does not take into account whether a female is eligible for conception or not. Thus the values of ASFR are divided by the proportion of eligible couples of a particular group for finding the values of ASMFR. Once the values of ASMFR are known, m(x) can be calculated as: m(x)=ASMFRxQx5 where Q is the sex ratio at birth. Multiplication by 5 comes due to 5 yearly group of the females (couples).