On the Three Properties of Stationary Populations and Knotting with Non-stationary Populations

Abstract

A population is considered stationary if the growth rate is zero and the age structure is constant. It thus follows that a population is considered non-stationary if either its growth rate is nonzero and/or its age structure is non-constant. We propose three properties that are related to the stationary population identity (SPI) of population biology by connecting it with stationary populations and non-stationary populations which are approaching stationarity. One of these important properties is that SPI can be applied to partition a population into stationary and non-stationary components. These properties provide deeper insights into cohort formation in real-world populations and the length of the duration for which stationary and non-stationary conditions hold. The new concepts are based on the time gap between the occurrence of stationary and non-stationary populations within the SPI framework that we refer to as Oscillatory SPI and the Amplitude of SPI.

Appendix

Let,

$$\begin{aligned} I= & {} [t_{0},t_{1})\bigcup \mathop {\mathop {\bigcup }\limits _{M,i=1}}\limits ^{k}[\delta _{i},t_{i+1}),\\ J= & {} \mathop {\mathop {\bigcup }\limits _{N,i=1}}\limits ^{k}[t_{i},\delta _{i}). \end{aligned}$$

and let $\dot{I}$ and $\dot{J}$ be the partitions of I and J, which are written as,

$$\begin{aligned} \dot{I}=\left\{ \left( I_{M}(t_{i})\right) :i=1,2,\dots ,k+1\right\} =\left\{ \left( I_{M}(t_{i})\right) \right\} _{i=1}^{k} \end{aligned}$$

and

$$\begin{aligned} \dot{J}=\left\{ \left( J_{N}(t_{i})\right) :i=1,2,\dots ,k\right\} =\left\{ \left( J_{N}(t_{i})\right) \right\} _{i=1}^{k}, \end{aligned}$$

where $I_{M}(t_{1})=[t_{0},t_{1}),$$I_{M}(t_{i})=[\delta _{i},t_{i+1})$ for $i=2,\dots ,k$ and $J_{N}(t_{i})=[t_{i},\delta _{i})$ for $i=1,2,\dots ,k.$ Since $I_{M}(t_{i})$ and $J_{N}(t_{i})$ are non-degenerate intervals, the lengths of $I_{M}(t_{i})$ and $J_{N}(t_{i})$ are always positive. Hence, $\text {max}I_{M}(t_{i})$, $\min I_{M}(t_{i})$, $\max J_{N}(t_{i})$, and $\min J_{N}(t_{i})$ exist. Let $f(a,t_{i})$ be the function specifying the proportion of individuals at age $a\in A$ during $I_{M}(t_{i})$ for $f(a,t_{i}):I_{M}(t_{i})\rightarrow \mathbb {R}^{+}$ and A be the set of all ages in the population. Since SPI holds in $I_{M}(t_{i})$, we have

$$\begin{aligned} Prob\left[ f(a,t_{i})=g(a,t_{i})\forall a,t_{i}\right]= & {} 1\,\, \text{ if } f(a,t_{i}):I_{M}(t_{i})\rightarrow \mathbb {R}^{+}\nonumber \\= & {} 0\,\, \text{ otherwise, } \end{aligned}$$

(15)

where $g(a,t_{i})$ is the function specifying remaining LR at age a during $I_{M}(t_{i}).$

Lemma 9

Suppose $\triangle f(a,t_{i})=\hat{f}(a,t_{i})-\check{f}(a,t_{i})$ for $i=1,2,3,\ldots ,k$, where $\hat{f}(a,t_{i})=\max _{a}f(a,t_{i})$ and $\check{f}(a,t_{i})=\min _{a}f(a,t_{i})$, then $\triangle f(a,t_{i})$ is bounded for each $I_{M}(t_{i}).$

Proof

If there are at least two age groups in A, then $\hat{f}(a,t_{i})$ and $\check{f}(a,t_{i})$ exists within $I_{M}(t_{i})$ and they are distinct. Suppose there are only two age groups in A, then (15) guarantees that there exist $\hat{g}(a,t_{i})$ and $\check{g}(a,t_{i})$ for $\hat{g}(a,t_{i})=\max _{a}g(a,t_{i})$ and $\check{g}(a,t_{i})=\min _{a}g(a,t_{i})$. This implies, $\triangle f(a,t_{i})<\hat{g}(a,t_{i})+\check{g}(a,t_{i}).$ This inequality follows even if there are more than two age groups in A, and hence, $\triangle f(a,t_{i})$ is bounded. $\square$

Theorem 10

$\frac{1}{\Sigma \triangle f(a,t_{i})}$ is bounded on $\left[ t_{0},t_{k+1}\right) .$

Proof

Since $\triangle f(a,t_{i})>0$ and $\triangle f(a,t_{i})$ are bounded on $I_{M}(t_{i})$ by Lemma (9), the result follows. $\square$

Suppose $\hat{f}(a,t_{i})$ is concentrated around mean age of the population and $\check{f}(a,t_{i})$ is concentrated around the very old age of the population, then $\triangle f(a,t_{i})$ is an increasing function indicates one or more of the following three situations; i) longevity of the population is increasing without much change in the mean age, ii) mean age is reducing without reducing in longevity, and iii) mean age is reducing and simultaneously longevity is increasing.

Theorem 11

Suppose the partitions $\dot{I}$ and $\dot{J}$ are given, then $1+\frac{k^{2}}{\hat{f}(a,t_{i})+\check{f}(a,t_{i})}>k\left( \frac{1}{\hat{f}(a,t_{i})}+\frac{1}{\check{f}(a,t_{i})}\right)$.

Proof

Consider the expression

$$\begin{aligned} \left( \check{f}(a,t_{i})-\sum _{i=1 \text{ for } t_{i}\in I}^{\infty }\int _{0}^{\infty }f(a,t_{i})da\right) \left( \hat{f}(a,t_{i})-\sum _{i=1 \text{ for } t_{i}\in I}^{\infty }\int _{0}^{\infty }f(a,t_{i})da\right) . \end{aligned}$$

(16)

Since $\sum _{i=1}^{\infty }\int _{0}^{\infty }f(a,t_{i})da=k$ and both the terms of the expression (16) are negative, (16) can be written as

$$\begin{aligned} \left( \check{f}(a,t_{i})-k\right) \left( \hat{f}(a,t_{i})-k\right) >0. \end{aligned}$$

(17)

Simplifying (17), we will obtain the desired result. $\square$

Remark 12

For each $t_{i}$ for $i=1,2,\dots ,k$, without taking the summations in (16), we have

$$\begin{aligned}&\left( \check{f}(a,t_{i})-f(a,t_{i})\right) \left( \hat{f}(a,t_{i})-f(a,t_{i})\right) \\&\quad {\left\{ \begin{array}{ll} \begin{array}{ll} =0 &{}\quad \text{ if } \hat{f}(a,t_{i})=f(a,t_{i}) \text{ or } \check{f}(a,t_{i})=f(a,t_{i})\\<0 &{}\quad \text{ if } \,\,\check{f}(a,t_{i})<f(a,t_{i})<\hat{f}(a,t_{i}) \end{array}\end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \left( \check{f}(a,t_{i})-\int _{0}^{\infty }f(a,t_{i})da\right) \left( \hat{f}(a,t_{i})-\int _{0}^{\infty }f(a,t_{i})da\right) >0. \end{aligned}$$

Let $\varphi (a,t_{i})$ be the function specifying the proportion of individuals at age $a\in B$ during $J_{N}(t_{i})$ for $\varphi (a,t_{i}):J_{N}(t_{i})\rightarrow \mathbb {R}^{+}$ and B be the set of all ages in the population when SPI does not hold. Suppose $\hat{\varphi }(a,t_{i})=\max _{a}\varphi (a,t_{i})$ and $\check{\varphi }(a,t_{i})=\min _{a}\varphi (a,t_{i})$. We note that equivalent versions of Theorem 11 and Remark 12 for the age functions $\varphi ,$$\hat{\varphi }(a,t_{i}),$$\check{\varphi }(a,t_{i})$ still hold. Under the continuous transition of decreasing population sizes over the interval $[t_{0},t_{k+1}),$ let us assume $\hat{f}(a,t_{1})>\hat{f}(a,t_{2})>\cdots >\hat{f}(a,t_{k+1})$ and $\hat{\varphi }(a,t_{1})>\hat{\varphi }(a,t_{2})>\cdots >\hat{\varphi }(a,t_{k}).$ This implies $\hat{f}(a,t_{1})>\hat{\varphi }(a,t_{1})>\cdots>\hat{\varphi }(a,t_{k})>\hat{f}(a,t_{k+1})$. Also, $\int _{0}^{\infty }f(a,t_{1})da-\hat{f}(a,t_{1})<\int _{0}^{\infty }\varphi (a,t_{1})da-\hat{\varphi }(a,t_{1})<\cdots<\int _{0}^{\infty }\varphi (a,t_{k})da-\hat{\varphi }(a,t_{k})<\int _{0}^{\infty }f(a,t_{k+1})da-\hat{f}(a,t_{k+1}),$ and this leads to $1-\hat{f}(a,t_{1})<1-\hat{\varphi }(a,t_{1})<\cdots<1-\hat{\varphi }(a,t_{k})<1-\hat{f}(a,t_{k+1}).$ We can model the dynamics of these maximum and minimum fractions over the time period using the following logistic growth models with certain limiting points of these fractions.

$$\begin{aligned} \frac{\hbox {d}\hat{f}(a,t)}{\hbox {d}t}= & {} r_{1}\hat{f}(a,t)\left( 1-\frac{\hat{f}(a,t)}{\left( \hat{f}(a,t)\right) _{e}}\right) \end{aligned}$$

(18)

$$\begin{aligned} \frac{\hbox {d}\hat{\varphi }(a,t)}{\hbox {d}t}= & {} r_{2}\hat{\varphi }(a,t)\left( 1-\frac{\hat{\varphi }(a,t)}{\left( \hat{\varphi }(a,t)\right) _{e}}\right) \end{aligned}$$

(19)

$$\begin{aligned} \frac{\hbox {d}\check{f}(a,t)}{\hbox {d}t}= & {} r_{3}\check{f}(a,t)\left( 1-\frac{\check{f}(a,t)}{\left( \check{f}(a,t)\right) _{e}}\right) \end{aligned}$$

(20)

$$\begin{aligned} \frac{\hbox {d}\check{\varphi }(a,t_{i})}{\hbox {d}t}= & {} r_{4}\check{\varphi }(a,t_{i})\left( 1-\frac{\check{\varphi }(a,t_{i})}{\left( \check{\varphi }(a,t_{i})\right) _{e}}\right) , \end{aligned}$$

(21)

where $r_{1},r_{2},r_{3}$, and $r_{4}$ are rates of declines in maximum and minimum fractions and $\left( \hat{f}(a,t)\right) _{e},$$\left( \hat{\varphi }(a,t)\right) _{e},$$\left( \check{f}(a,t)\right) _{e},$$\left( \check{\varphi }(a,t_{i})\right) _{e}$ are limiting points of the fractions $\hat{f}(a,t),$$\hat{\varphi }(a,t),$$\check{f}(a,t),$$\check{\varphi }(a,t_{i})$, respectively. Further, we provide partial differential equations models by treating $\hat{f}(a,t),$$\hat{\varphi }(a,t),$$\check{f}(a,t),$$\check{\varphi }(a,t_{i})$ as continuous variables. First, we consider two pairs of variables $\left\{ \hat{f}(a,t),\hat{\varphi }(a,t)\right\}$, $\left\{ \check{f}(a,t),\check{\varphi }(a,t_{i})\right\}$ and corresponding dependent variables $u_{1}\left( \hat{f}(a,t),\hat{\varphi }(a,t)\right)$, $u_{2}\left( \check{f}(a,t),\check{\varphi }(a,t_{i})\right)$ to build two models (22) and (23). These two models provide dynamics of simultaneous occurrences of stationary and non-stationary populations. If we want to follow dynamics of $\hat{f}$ and $\hat{\varphi }$ on the time interval $[t_{0},t_{\infty })$ by considering two pairs of independent variables $\left\{ t,\hat{f}(a,t)\right\}$, $\left\{ t,\hat{\varphi }(a,t)\right\}$ with corresponding dependent variables $v_{1}\left( t,\hat{f}(a,t)\right)$, $v_{1}\left( t,\hat{\varphi }(a,t)\right)$, then the PDE models we considered are given in (24) and (25). Here, $\tau _{1}$ and $\tau _{2}$ are constants, which could indicate speed of the dynamics of peaks of the maximum fractions. Similarly, dynamics of $\check{f}$ and $\check{\varphi }$ with dependent variables $w_{1}\left( t,\check{f}(a,t)\right)$ and $w_{2}\left( t,\check{\varphi }(a,t_{i})\right)$ are modeled as per equations given in (26) and (27), where $\tau _{3}$ and $\tau _{4}$ are constants indicate speed with which these variables move.

$$\begin{aligned} \frac{\partial u_{1}\left( \hat{f}(a,t),\hat{\varphi }(a,t)\right) }{\partial \hat{f}}= & {} -\hat{\varphi }\frac{\partial u_{1}\left( \hat{f}(a,t),\hat{\varphi }(a,t)\right) }{\partial \hat{\varphi }} \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial u_{2}\left( \check{f}(a,t),\check{\varphi }(a,t_{i})\right) }{\partial \check{f}}= & {} -\check{\varphi }\frac{\partial u_{2}\left( \check{f}(a,t),\check{\varphi }(a,t_{i})\right) }{\partial \check{\varphi }} \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial v_{1}\left( t,\hat{f}(a,t)\right) }{\partial t}= & {} -\tau _{1}\frac{\partial v_{1}\left( t,\hat{f}(a,t)\right) }{\partial \hat{f}}, \end{aligned}$$

(24)

$$\begin{aligned} \frac{\partial v_{2}\left( t,\hat{\varphi }(a,t)\right) }{\partial t}= & {} -\tau _{2}\frac{\partial v_{2}\left( t,\hat{\varphi }(a,t)\right) }{\partial \hat{\varphi }} \end{aligned}$$

(25)

$$\begin{aligned} \frac{\partial w_{1}\left( t,\check{f}(a,t)\right) }{\partial t}= & {} -\tau _{3}\frac{\partial w_{1}\left( t,\check{f}(a,t)\right) }{\partial \check{f}} \end{aligned}$$

(26)

$$\begin{aligned} \frac{\partial w_{2}\left( t,\check{\varphi }(a,t)\right) }{\partial t}= & {} -\tau _{4}\frac{\partial w_{4}\left( t,\check{\varphi }(a,t)\right) }{\partial \check{\varphi }}. \end{aligned}$$

(27)

Diffusion type of equations appears in several situations of modeling in biology, for example refer to the book (Perthame 2007). Further applications of diffusion type of equations appear in studying growth of cell populations, see Boulanouar (2001), Lebowitz and Rubinow (1974), and Rotenberg (1983).