1 Stationary Population

Stationary population assumptions and related mathematical formulations were of interest to Edmond Halley, an astronomer who also published in demography (in 1693, Halley 1693), to influential mathematician Leonard Euler in the eighteenth century (Euler 1760, 1923, 1922), as well as to twentieth-century famous theoretical population biologist (Lotka 1925, 1939). A population is said to be stationary if it has a zero growth rate and a constant population age-structure, and if not the population is said to be non-stationary. Lotka associated his theory of stationary populations with the average rate at which a woman in her lifetime will be replaced by a girl (which we call the net reproduction rate). In fact, Lotka (1925, 1939) logically argued that the rate of natural growth, r,  of a population will be zero in (1) when the net reproduction rate of the population, \(R_{0}\), will be equal to one (the notation \(R_{0}\) was introduced by Lotka). The relation between r and \(R_{0}\) is given by

$$\begin{aligned} r=\frac{\ln (R_{0})}{L}, \end{aligned}$$
(1)

where L is the length of the generation of a population, which is expressed as,

$$\begin{aligned} L=\frac{1}{R_{0}}\intop _{0}^{\infty }xa(x)s(x)dx, \end{aligned}$$
(2)

where a(x) is the age-specific fertility rate of women of age x to give births to girl babies, and s(x) is the survival probability function for women to live up to the age x. Although \(R_{0}\) is a popular demographic measure, its practical applicability in timely measuring the stationary status of a population is questioned. See, for example, Rao (2021).

We define below the stationary population identity (SPI) or the Life Table Identity for a discrete population as given in Rao and Carey (2019).

Definition 1

Stationary Population identity (SPI) or the Life Table Identity: Let, for each age between 0 and a maximum age present in a discrete stationary population or discrete stationary life table population, X denote the set of elements representing the proportions of individuals present in the population for these ages. Furthermore, let Y denote the set of elements representing the proportions of individuals in the population whose remaining times to live are equal to these ages. We say that the SPI holds if \(X=Y\).

Let, for a discrete stationary population with maximum age \(\omega \), A denote the set of all integers from 0 to (and including) \(\omega \): \(A=\{0,a_{1},a_{2},\ldots ,\omega \}\). So, A equals the set of discrete ages in a discrete stationary population or discrete stationary life table. The statement that the SPI holds for a discrete stationary population or discrete stationary life table means that the proportion \(f_{1}(x)\) of individuals with age \(x\in A\) in the population is equal to the proportion \(f_{2}(x)\) of individuals in the population who will life for x years. i.e., \(f_{1}(x)=f_{2}(x)\) \(\,\forall \,x\in A.\)

This identity was theoretically demonstrated by means of partitioning large populations (Rao and Carey 2019). This is frequently referred to also as the life table identity in mathematical population biology and demography literature. Such identities are found in various published literature, for example, see Muller et al. (2004), Carey et al. (2012), Rao and Carey (2015), Brouard (1986), Carey et al. (2018), and Swanson and Tedrow (2021). There are other related equalities in stationary populations which suggest that the average age in a stationary population is equal to the average expectation of remaining life time (for continuous versions) (Kim and Aron 1989; Cox 1962). Although Lotka and Cox in their respective works have chosen continuous frameworks, in this article the population ages are treated in a discrete framework.

In this article, we state and prove a novel theorem that states a criterion to partition a randomly selected large discrete population into stationary and non-stationary components. The procedure consists of comparing \(f_{1}(x)\) and \(f_{2}(x)\) and see if they are identical. Instead of the life table population, the theorem suggests comparing the fractions of the population for each age of the actual population at a time point with the fractions of the population that have remaining ages in the corresponding life table population constructed at the same time.

In general, we do not often see stationary populations, except in life tables. But a careful investigation of any large population data suggests that a sub-population of a population could obey the properties of a stationary population. Now, through the partition theorem that is proved in this article, it becomes easier to understand the stationary component of large populations instantly, without being dependent on measures like NRR (net reproduction rate). Moreover, the theorem guarantees with certainty if a component of a large population is stationary.

2 Partition Theorem and Proof

Theorem 2

Stationary Population Identity (SPI) partitions a randomly selected large population into stationary and non-stationary components (if such components are non-empty).

Proof

Let a randomly selected large population be given, and let the maximum age present in the population be denoted by \(\omega \). Let P(t) denote the size of the population at time t, and for \(x\in A\), where \(A=\{0,a_{1},a_{2},\ldots ,\omega \}.\) For \(x\in A\), let \(P_{x}(t)\) denote the number of individuals with age x in the population at time t. In the current proof, we considere P as a summation over discrete ages. Then

$$\begin{aligned} P(t)=\Sigma _{x=0}^{\omega }P_{x}(t). \end{aligned}$$
(3)

Let \(g_{1}(x,t)=\frac{P_{x}(t)}{P(t)}\) for all \(x\in A\). We define the stationary component of P(t), denoted by M(t), as the size of the sub-population of all (variously aged) individuals for which there exists a value \(y\in A\) such that \(g_{1}(y,t)=f_{2}(y,t)\) holds, where \(f_{2}(y,t)\) denotes the proportion of individuals in the population at time t who will have y remaining years to live. The non-stationary component of P(t),  denoted by N(t), is defined as the size of the sub-collection of individuals for which \(g_{1}(z,t)\ne f_{2}(z,t)\) for any \(z\in \) \(A\). It will be clear that at any t the equality

$$\begin{aligned} P(t)=M(t)+N(t) \end{aligned}$$
(4)

holds. Let us choose all individuals at age y in P(t) and consider

$$\begin{aligned} g_{1}(y,t)=\frac{P_{y}(t)}{P(t)}. \end{aligned}$$
(5)

From the life table constructed for the population P(t) (assumably for the single ages of the set A), we can obtain the proportion of the population who have expected remaining years within A,  and compare this with the proportion of the population at each age \(x\in A\) in P(t). Note that a life table is a mathematical model to synthetically demonstrate the age-specific death rates of a population and to compute remaining average years to live at each age in A. There are several books available to understand details of life table constructions, for example, see (Wachter 2014; Misra 1998; Preston et al. 2000; Carey and Roach 2019; Lahiri 2018). We will compare \(g_{1}(y,t)\) with another proportion \(f_{2}(x,t)\) \(\forall x\in A,\) where

$$\begin{aligned} f_{2}(x,t)=\frac{L_{x}(t)}{L(t)}. \end{aligned}$$
(6)

Here \(L_{x}(t)\) is the life table population at time t whose remaining years to live is x and L(t) is total life table population at t. First, we match \(\frac{P_{y_{1}}(t)}{P(t)}\) for a given \(y_{1}\in A\) in (5) with \(\frac{L_{x}(t)}{L(t)}\) for each \(x\in A.\) If \(\frac{P_{y_{1}}(t)}{P(t)}\) is equal to \(\frac{L_{x}(t)}{L(t)}\) for some x,  then we call the corresponding life table proportion of the population as \(\frac{L_{y_{1}}(t)}{L(t)}\) \((say,\) \(f_{2}(y_{1},t)\)). That is, \(\frac{L_{y_{1}}(t)}{L(t)}\) is the proportion of life table sub-population who have \(y_{1}\) years to live. This follows,

$$\begin{aligned} g_{1}(y_{1},t)=\frac{P_{y_{1}}(t)}{P(t)}=\frac{L_{y_{1}}(t)}{L(t)}=f_{2}(y_{1},t). \end{aligned}$$
(7)

Suppose \(\frac{P_{y_{1}}(t)}{P(t)}\) does not equal to any of the proportions \(\frac{L_{x}(t)}{L(t)}\) for \(x\in [0,\omega ),\) then we denote \(y_{1}\) in (7) by \(z_{1}\) and write this situation as \(g_{1}(z_{1},t)\ne f_{2}(z_{1},t).\) That is, for any of the proportions \(\frac{L_{x}(t)}{L(t)}\) for \(x\in [0,\omega ),\) the remaining years to live is not equal to \(y_{1}.\)

We will continue matching \(\frac{P_{y_{2}}(t)}{P(t)}\) for some \(y_{2}\ne y_{1}\) and \(y_{2}\in [0,\omega )\) with \(\frac{L_{x}(t)}{L(t)}\) for each \(x\in A\) except for \(x=y_{1}.\) If there is a value of \(\frac{L_{x}(t)}{L(t)}\) that equals \(\frac{P_{y_{2}}(t)}{P(t)},\) we call the corresponding proportion in the life table population as \(\frac{L_{y_{2}}(t)}{L(t)}\) (say, \(f_{2}(y,t)\)). \(\frac{L_{y_{2}}(t)}{L(t)}\) is the proportion of the life table sub-population who have \(y_{2}\) years to live. That is,

$$\begin{aligned} g_{1}(y_{2},t)=\frac{P_{y_{2}}(t)}{P(t)}=\frac{L_{y_{2}}(t)}{L(t)}=f_{2}(y_{2},t). \end{aligned}$$
(8)

Suppose \(\frac{P_{y_{2}}(t)}{P(t)}\) is not equal to any of the proportions \(\frac{L_{x}(t)}{L(t)}\) for \(x\in A\), then we denote \(y_{2}\) in (8) by \(z_{2}\) (if \(z_{1}\) already arises in an earlier situation such that \(g_{1}(z_{1},t)\ne f_{2}(z_{1},t))\), and we write \(g_{1}(z_{2},t)\ne f_{2}(z_{2},t).\) However, if \(g_{1}(y_{1},t)=f_{2}(y_{1},t)\) exists but \(\frac{P_{y_{2}}(t)}{P(t)}\ne \frac{L_{x}(t)}{L(t)}\) \(\forall x\in A\), then we denote \(y_{2}\) as \(z_{1}\) and write \(g_{1}(z_{1},t)\ne f_{2}(z_{1},t).\)

Fig. 1
figure 1

Proportions of sub-population in an actual population and in a life table. \(P_{y_{k_{j}}}(t)\) denotes the size of sub-population whose individuals are of age \(y_{k_{j}}\) at time t, and P(t) is the total actual population at time t. \(L_{y_{k_{j}}}(t)\) is the life table population at time t of these individuals whose remaining years to live at time t is equals \(y_{k_{j}}\) and L(t) is the total life table population at t. Here time is not to be measured on a continuous scale

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Fig. 2
figure 2

Stationary component of a randomly selected large population. Proportional sub-populations at ages \(y_{k_{1}},y_{k_{2}},\ldots \) in the actual total population are identical with the proportional life table sub-populations whose individuals have remaining years \(y_{k_{1}},y_{k_{2}},\ldots \) to live

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Fig. 3
figure 3

Non-stationary component of a randomly selected large population. Proportional sub-populations at ages \(z_{k_{1}},z_{k_{2}},\ldots \) in the actual total population are not identical with any of the life table proportional sub-populations whose individuals have remaining years to live equal to \(z_{k_{1}},z_{k_{2}},\ldots \)

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Similarly, for a randomly selected age \(y_{i}\in A\), the proportion \(\frac{P_{y_{i}}(t)}{P(t)}\) (i.e. \(g_{1}(y_{i},t)\)) is matched with the proportions \(\frac{L_{x}(t)}{L(t)}\) \(\forall x\in A.\) If there is a value of \(\frac{P_{y_{i}}(t)}{P(t)}\) that matches with \(\frac{L_{x}(t)}{L(t)}\) then we denote it by \(f_{2}(y_{i},t),\) otherwise we denote it by \(f_{2}(z_{i},t)\) (if the previous z value in the order of unmatched proportions was denoted as \(z_{i-1}\) for \(i=2,3,\ldots \)). Through this procedure we will match for all the values of \(y_{i}\in A,\) and decide whether or not \(g_{1}(y_{i},t)\) is equal to the \(f_{2}(y_{i},t).\) The criteria is: of for any age \(y_{i}\) in a randomly selected large population (actual population) the, if the value of the proportion \(\frac{P_{y_{i}}(t)}{P(t)}\) is equal to any of the life table sub-population proportions \(\frac{L_{x}(t)}{L(t)}\) \(\forall x\in A\) whose remaining years to live is exactly \(y_{i},\) then \(g_{1}(y_{i},t)=f_{2}(y_{i},t).\)

Existence of two sets \(\{y_{k_{j}}\}\) and \(\{z_{k_{j}}\}\):

In general, at each age \(x\in A\) we can check whether

$$\begin{aligned} \frac{P_{x}(t)}{P(t)}=\frac{L_{x}(t)}{L(t)} \end{aligned}$$
(9)

holds, or

$$\begin{aligned} \frac{P_{x}(t)}{P(t)}\ne \frac{L_{x}(t)}{L(t)} \end{aligned}$$
(10)

holds. See Fig. 1 for application of (9), Fig. 2 for the stationary component, and Fig. 3 for the non-stationary component.

Suppose we start this procedure from age 0. At \(x=0\), either (9) holds or (10) holds, but not both. If (9) holds, then let us denote \(y_{k_{1}}\) for 0, or, if (10) holds, then let us denote \(z_{k_{1}}\) for 0. Hence, at age 0 one of the \(y_{k_{1}}\) or \(z_{k_{1}}\) exists, and

$$\begin{aligned} \left\{ y_{k_{1}}\right\} \cup \left\{ z_{k_{1}}\right\} =\left\{ 0\right\} \end{aligned}$$
(11)

Now, let us consider age \(a_{1}>0\) for \(a_{1}\in A\). At age \(x=a_{1}\), either (9) holds or (10) holds, but not both. If (9) holds at \(x=a_{1}\), and \(y_{k_{1}}\) exists, then let us denote \(y_{k_{2}}\) for \(a_{1}\). If (9) holds at \(x=a_{1}\) and \(z_{k_{1}}\) exists, then let us denote \(y_{k_{1}}\) for \(a_{1}\). If (10) holds at \(x=a_{1}\) and \(y_{k_{1}}\) exist, then let us denote \(z_{k_{1}}\) for \(a_{1}\). If (10) is true at \(x=a_{1}\) and \(z_{k_{1}}\) exists, then let us denote \(z_{k_{2}}\) for \(a_{1}\). From the arguments constructed so far, we have shown the existence of one of the following sets:

$$\begin{aligned} \left\{ \{y_{k_{1}},y_{k_{2}}\},\{z_{k_{1}},y_{k_{1}}\},\{y_{k_{1}},z_{k_{1}}\},\{z_{k_{1}},z_{k_{2}}\}\right\} \end{aligned}$$
(12)

The union of all the elements of the set (12) is

$$\begin{aligned} \left\{ \{y_{k_{1}},y_{k_{2}}\}\cup \{z_{k_{1}},y_{k_{1}}\}\cup \{y_{k_{1}},z_{k_{1}}\}\cup \{z_{k_{1}},z_{k_{2}}\}\right\} =\left\{ 0,a_{1}\right\} . \end{aligned}$$
(13)

Now, let us consider age \(a_{2}\) for \(a_{2}>a_{1}>0\) and \(a_{2}\in A\). At age \(x=a_{2}\), either (9) holds or (10) holds, but not both.

Suppose (9) holds at \(x=a_{2}\). For \(x=0\) and \(x=a_{1}\), there is a possibility of occurrence of one of the four sets of (12) . For \(x=a_{2}\) exactly one of the following four possibilities then will occur:

(i):

(9) holds at \(x=a_{2}\) and \(\left\{ y_{k_{1}},y_{k_{2}}\right\} \) exists; then let us denote \(y_{k_{3}}\) for \(a_{2}\).

(ii):

(9) holds at \(x=a_{2}\) and \(\{z_{k_{1}},y_{k_{1}}\}\) exists; then let us denote \(y_{k_{2}}\) for \(a_{2}\).

(iii):

(9) holds at \(x=a_{2}\) and \(\{y_{k_{1}},z_{k_{1}}\}\) exists; then let us denote \(y_{k_{2}}\) for \(a_{2}\).

(iv):

(9) holds at \(x=a_{2}\) and \(\{z_{k_{1}},z_{k_{2}}\}\) exists; then let us denote \(y_{k_{1}}\) for \(a_{2}\).

Suppose (10) holds at \(x=a_{2}\). For \(x=0\) and \(x=a_{1}\), there is a possibility of occurrence of one of the four sets of (12). Also here one of the following possibilities will occur:

(v):

(10) holds at \(x=a_{2}\) and \(\left\{ y_{k_{1}},y_{k_{2}}\right\} \) exists; then let us denote \(z_{k_{1}}\) for \(a_{2}\).

(vi):

(10) holds at \(x=a_{2}\) and \(\{z_{k_{1}},y_{k_{1}}\}\) exists; then let us denote \(z_{k_{2}}\) for \(a_{2}\).

(vii):

(10) holds at \(x=a_{2}\) and \(\{y_{k_{1}},z_{k_{1}}\}\) exists; then let us denote \(z_{k_{2}}\) for \(a_{2}\).

(viii):

(10) holds at \(x=a_{2}\) and \(\{z_{k_{1}},z_{k_{2}}\}\) exists; then let us denote \(z_{k_{3}}\) for \(a_{2}\).

Through (i) to (viii) we have shown the existence of one of the following sets:

$$\begin{aligned} \left\{ \begin{array}{c} \{y_{k_{1}},y_{k_{2}},y_{k_{3}}\},\{z_{k_{1}},y_{k_{1}},y_{k_{2}}\},\{y_{k_{1}},z_{k_{1}},y_{k_{2}}\},\{z_{k_{1}},z_{k_{2}},y_{k_{1}}\}\\ \{y_{k_{1}},y_{k_{2}},z_{k_{1}}\},\{z_{k_{1}},y_{k_{1}},z_{k_{2}}\},\{y_{k_{1}},z_{k_{1}},z_{k_{2}}\},\{z_{k_{1}},z_{k_{2}},z_{k_{3}}\} \end{array}\right\} \end{aligned}$$
(14)

The union of \(2^{3}\) number of sets is

$$\begin{aligned} \left\{ 0,a_{1},a_{2}\right\} . \end{aligned}$$
(15)

Continuing in a similar manner with \(x=a_{3}\) we construct the following \(2^{4}\) number of sets:

$$\begin{aligned} \left\{ \begin{array}{c} \{y_{k_{1}},y_{k_{2}},y_{k_{3}},y_{k_{4}}\},\{z_{k_{1}},y_{k_{1}},y_{k_{2}},y_{k_{3}}\},\{y_{k_{1}},z_{k_{1}},y_{k_{2}},y_{k_{3}}\},\{z_{k_{1}},z_{k_{2}},y_{k_{1}},y_{k_{2}}\}\\ \{y_{k_{1}},y_{k_{2}},z_{k_{1}},y_{k_{3}}\},\{z_{k_{1}},y_{k_{1}},z_{k_{2}},y_{k_{2}}\},\{y_{k_{1}},z_{k_{1}},z_{k_{2}},y_{k_{2}}\},\{z_{k_{1}},z_{k_{2}},z_{k_{3}},y_{k_{1}}\}\\ \{y_{k_{1}},y_{k_{2}},y_{k_{3}},z_{k_{1}}\},\{z_{k_{1}},y_{k_{1}},y_{k_{2}},z_{k_{2}}\},\{y_{k_{1}},z_{k_{1}},y_{k_{2}},z_{k_{2}}\},\{z_{k_{1}},z_{k_{2}},y_{k_{1}},z_{k_{3}}\}\\ \{y_{k_{1}},y_{k_{2}},z_{k_{1}},z_{k_{2}}\},\{z_{k_{1}},y_{k_{1}},z_{k_{2}},z_{k_{3}}\},\{y_{k_{1}},z_{k_{1}},z_{k_{2}},z_{k_{3}}\},\{z_{k_{1}},z_{k_{2}},z_{k_{3}},z_{k_{4}}\} \end{array}\right\} \end{aligned}$$
(16)

The union of the \(2^{4}\) sets in (16) is

$$\begin{aligned} \left\{ 0,a_{1},a_{2},a_{3}\right\} . \end{aligned}$$

Note that the elements of each of the sets in (16) are drawn from the set

$$\begin{aligned} \{y_{k_{1}},y_{k_{2}},y_{k_{3}},y_{k_{4}},z_{k_{1}},z_{k_{2}},z_{k_{3}},z_{k_{4}}\} \end{aligned}$$

which is of size \(2.4=8.\)

In a similar way, by induction, the number of possible sets at age \(x=a_{j}\) for \(a_{j}>a_{j-1}>\cdots>a_{1}>a_{0}\) would be \(2^{j+1}\). Each of these \(2^{j+1}\) sets will be of size \(j+1,\) and the \(j+1\) elements for each set are drawn from the following unique combinations of set of \(2j+2\) elements:

$$\begin{aligned} \left\{ y_{k_{1}},y_{k_{2}},\ldots ,y_{k_{j+1}},z_{k_{1}},z_{k_{2}},\ldots ,z_{k_{j+1}}\right\} . \end{aligned}$$
(17)

The union of the \(2^{j+1}\) sets would be

$$\begin{aligned} \left\{ 0,a_{1},a_{2},\ldots ,a_{j}\right\} . \end{aligned}$$
(18)

When \(a_{j}=\omega \) in (18), this union becomes the set A. The elements of A are drawn from

$$\begin{aligned} \left\{ y_{k_{1}},y_{k_{2}},\ldots ,\omega ,z_{k_{1}},z_{k_{2}},\ldots ,\omega \right\} . \end{aligned}$$

Here,

$$\begin{aligned} \bigcup _{p=1}^{j}\left\{ y_{k_{p}}\right\} \cup \left\{ z_{k_{p}}\right\} =A, \end{aligned}$$
(19)

and the corresponding sub-populations’ totals for the individuals who are all at ages \(\left\{ y_{k_{j}}\right\} \) and \(\left\{ z_{k_{l}}\right\} \) are \(P_{y_{k_{j}}}(t)\) and \(P_{z_{k_{l}}(t)},\) respectively. See Fig. 4 for partitioned components. Due to (19), we can write,

$$\begin{aligned} \Sigma _{j=1}^{\omega }P_{y_{k_{j}}}(t)+\Sigma _{l=1}^{\omega }P_{z_{k_{l}}(t)}=P(t), \end{aligned}$$
(20)

where \(\Sigma _{j=1}^{\infty }P_{y_{k_{j}}}(t)\) is M(t) formed by satisfying the SPI at the ages \(\left\{ y_{k_{j}}\right\} \), and \(\Sigma _{l=1}^{\infty }P_{z_{k_{l}}(t)}\) is N(t) formed by not satisfying the SPI at the ages \(\left\{ z_{k_{l}}\right\} \). Hence, the proof. □

Fig. 4
figure 4

Partitions into stationary and non-stationary components of a randomly selected large population. All the populations within the dotted lines on the left hand side form the stationary component and others make the non-stationary component of the total population, as shown in the right side of the Figure. Note that partitioning is not about geographical partitioning

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Example 3

Suppose for an arbitrary age \(a_{m}\in A\), we have

$$\begin{aligned} \frac{P_{a_{m}}(t)}{P(t)}=\frac{L_{a_{m}}(t)}{L(t)}, \end{aligned}$$

then the population at \(a_{m}\) belongs to the stationary component, and for an arbitrary age \(a_{n}\in A,\) suppose

$$\begin{aligned} \frac{P_{a_{n}}(t)}{P(t)}\ne \frac{L_{a_{n}}(t)}{L(t)}, \end{aligned}$$

then the population at the age \(a_{n}\) belongs to the non-stationary component.

Remark 4

If (9) is satisfied for all the ages in A then the entire randomly selected population is stationary. If (9) is not satisfied for all the ages in A then the entire randomly selected population is non-stationary.

Remark 5

Since we are calculating the proportions of sub-populations at each age of the actual population, we might come across values of these proportions calculated at two or more ages that could be identical. All the ages with identical proportions would fall within the same component of the population, i.e. stationary or non-stationary by the construction explained in the proof of the Theorem 2.

Remark 6

The partition theorem can be extended to more than two partitions if there are multiple decrement life tables available for a randomly selected large population.

3 Conclusions

The main theorem stated and proved is the first such observation in the literature. Moreover, we have not come across in the literature where the life table identity was being used to relate to actual populations and applied to decide on stationary and non-stationary components of a large randomly selected actual population. The partitioning procedure described in this work can be used to decide what proportion of the population is stationary and what proportion is not by considering the world population as a whole as one unit, individual countries, continents and groups of countries, etc, We can also apply this procedure to test the stationary and non-stationary status of all sub-regions of a large country.

The partition theorem stated and proved is not an improvement of any previously shown results in stationary populations. The statement of the partition theorem in the article is original and the method demonstrates whether a component of a large population is stationary or non-stationary did not exist previously. Cases of a given large population or subpopulation showing oscillatory behaviour of transitioning from a stationary to a non-stationary state or vice versa were earlier given in Rao and Carey (2019). The theory and partition methods described could be used in insect demography (Unnikrishnan and Gadagkar 2020; Chi et al. 2019; Tasnin et al. 2021), plant sciences (Mohammadi-Tabar et al. 2015; Chalmandrier et al. 2021), or other research areas where age- and aging-based stratification is done, for example Mondal et al. (2016), Ning et al. (2017) and Srivastava et al. (2021).

With many countries in the world approaching or at replacement levels but often with widely varying if not unstable age distributions, the partitioning theory outlined here has the potential to provide new metrics on age structure in particular, and on overall population dynamics in general. Metrics to measure population stability status between two populations were proposed in Rao (2014). Both of these can be used for both between-country comparisons as well as for projections into the future at both regional, national and global levels.