# The PAT model of population dynamics

Z. C. Feng  Department of Mechanical and Aerospace Engineering, University of
Missouri, Columbia, MO 65211<EMAIL_ADDRESS>and  Y. Charles Li  Y.
Charles Li, Department of Mathematics, University of Missouri, Columbia, MO
65211, USA<EMAIL_ADDRESS>https://liyan.mufaculty.umsystem.edu

###### Abstract.

We introduce a population-age-time (PAT) model which describes the temporal
evolution of the population distribution in age. The surprising result is that
the qualitative nature of the population distribution dynamics is robust with
respect to the birth rate and death rate distributions in age, and initial
conditions. When the number of children born per woman is $2$, the population
distribution approaches an asymptotically steady state of a kink shape; thus
the total population approaches a constant. When the number of children born
per woman is greater than $2$, the total population increases without bound;
and when the number of children born per woman is less than $2$, the total
population decreases to zero.

##  1\. Introduction

In human history, technological advances are the main factors for human
population increase, such as tool-making revolution, agricultural revolution
and industrial revolution. Technological advances provide human with more food
supply and medicine. More food supply is the main driver of population
increase. Medicine prolongs human life span. Diseases such as plagues could
cause human population to temporarily decrease. But since 1700, human
population has been monotonically increasing due to technological advances.
Since 1960s, due to the introduction of high yield grains, agricultural
machineries, fertilizers, chemical pesticides, and better irrigation systems,
human population has been increasing by 1 billion every 12 years, from 3
billion to 8 billion by 2023. Thus enrichment of food has increased human
population dramatically. It seems that both the Cornucopian and the Malthusian
views were realized [5] [7]. Human indeed dramatically advanced technology to
provide abundant food supply to meet the demand of population growth according
to Cornucopian view. Human population also dramatically increased with the
abundant food supply according to Malthusian view. The question is whether or
not we are heading to a new Malthusian catastrophe, i.e. some people are going
to starve. Technologies may be advanced further to support more humans. But
the earth resource is limited, and the human population cannot increase
without bound on earth.

Human overpopulation not only can cause huge damage to earth resource and
environment, but also has serious sustainability consequence. If there is a
global food scarcity, huge famine can cause major population loss. According
to World Wide Fund for Nature [9], the current human population is already
exceeding its earth carrying capacity. On the other hand, estimating earth’s
carrying capacity for human is more difficult than for other animals due to
the fact that human choices may play an important role [1]. In the long run,
human population cannot continue to grow; there are clear human resource
limits of food, energy and territory (individual human space) as discussed by
von Hoerner [4]. The key moment is when human population reaches its maximum.
The crucial question is: How will the human population change afterward? Will
human population more or less stay at a stagnation population or decrease
substantially? If human population decreases, is the decrease due to birth
control, normal death or abnormal death? Birth control and normal death are
hopeful for reducing human population from the example of China. Abnormal
death corresponds to various kinds of disasters such as diseases, wars etc..
Von Hoerner also proposed the possibility of moving humans out of earth, i.e.
stellar expansion [4]. But wars and diseases are more probable.

There have been a lot of effort in fitting human population historical data
with a function such as the nice fitting by $c(t_{*}-t)^{-\alpha}$ for
positive parameters $c$, $t_{*}$ and $\alpha$ [6]. There were also studies on
human population dynamics from logistic point of view [8] and ecological
perspectives [3].

Here we are focusing on the temporal evolution of the population distribution
in age, and introduce the population-age-time model.

##  2\. The population-age-time (PAT) model

Let $p(t,a)$ be the population of age $a$ (in year) at time $t$ (in year). One
can think $p(t,a)$ as the spectrum of population in age, $a=0,1,2,\cdots,A$;
$t=0,1,2\cdots$. We introduce the following population-age-time (PAT) model,

(2.1) $\displaystyle p(t+1,a)$ $\displaystyle=$ $\displaystyle
p(t,a-1)-\omega(t,a-1)p(t,a-1),\quad a=1,2,\cdots,A;$ (2.2) $\displaystyle
p(t+1,0)$ $\displaystyle=$
$\displaystyle\sum_{b=B_{1}}^{B_{2}}\alpha(t,b)p(t,b);$

where $\omega(t,a-1)$ is the death rate of the population $p(t,a-1)$,
$\alpha(t,b)$ is the birth rate of the population $p(t,b)$, $p(t,A)$ is set to
zero, and $A$, $B_{1}$ and $B_{2}$ are e.g.

(2.3) $A=120,\ B_{1}=18,\ B_{2}=40.$

Here the effects of immigration/emigration and migration are not included. The
total birth rate

(2.4) $\alpha=\sum_{b=B_{1}}^{B_{2}}\alpha(t,b),$

roughly represents half of the number of children born per woman (under the
assumption that male and female populations are roughly the same). For Europe,
$\alpha$ is about $0.7$; for Africa, $\alpha$ is $2$ to $2.5$; and for China,
$\alpha$ is about $0.5$. Factors that can affect birth rate include food
supply, medical technology, social norm, and affordability (expense). Factors
that can affect death rate include age, disease (medical technology), food
supply, war, accident, disaster.

The evolution of the total population

(2.5) $P(t)=\sum_{a=0}^{A}p(t,a),$

satisfies

$P(t+1)=P(t)+\sum_{b=B_{1}}^{B_{2}}\alpha(t,b)p(t,b)-\sum_{a=1}^{A}\omega(t,a-1)p(t,a-1),$

where the second term is the total birth and the last term is the total death
in the year $t$.

The steady state population distribution $p(t,a)=p(a)$, ($a=0,1,2,\cdots,A$)
satisfies

(2.6) $\displaystyle p(a)$ $\displaystyle=$ $\displaystyle
p(a-1)-\omega(a-1)p(a-1),\quad a=1,2,\cdots,A;$ (2.7) $\displaystyle p(0)$
$\displaystyle=$ $\displaystyle\sum_{b=B_{1}}^{B_{2}}\alpha(b)p(b);$

thus

$\displaystyle p(a)$ $\displaystyle=$
$\displaystyle[1-\omega(a-1)]\cdots[1-\omega(0)]p(0),\quad a=1,2,\cdots,A;$
$\displaystyle p(0)$ $\displaystyle=$
$\displaystyle\sum_{b=B_{1}}^{B_{2}}\alpha(b)[1-\omega(b-1)]\cdots[1-\omega(0)]p(0);$

and we arrive at the following constraint on $\alpha(b)$ and $\omega(b)$:

$1=\sum_{b=B_{1}}^{B_{2}}\alpha(b)[1-\omega(b-1)]\cdots[1-\omega(0)].$

We are interested in the dynamics of the population distribution and the total
population, and its dependence upon birth/death rate distribution and initial
conditions.

##  3\. Robustness of the population distribution dynamics

With the choice of parameters (2.3), we start with the following piecewise
linear birth and death rates:

(3.1)
$\alpha(t,b)=\left\\{\begin{array}[]{lr}\frac{\epsilon}{12}(b-18),&18\leq
b\leq 30,\cr\frac{\epsilon}{10}(40-b),&30<b\leq 40,\end{array}\right.$ (3.2)
$\omega(t,a)=\left\\{\begin{array}[]{lr}0,&0\leq a\leq
n,\cr\frac{1}{119-n}(a-n),&n<b\leq 119.\end{array}\right.$

See Figure 1 for a graphical illustration. These are the simplest birth rate
and death rate that we can think of. The key point is that we are going to
show that the qualitative nature of the population distribution dynamics is
robust with respect to different forms of birth rate and death rate
distributions in age.

The total birth rate is given by

$\alpha=\sum_{b=18}^{40}\alpha(t,b)=\sum_{b=18}^{30}\frac{\epsilon}{12}(b-18)+\sum_{b=31}^{40}\frac{\epsilon}{10}(40-b).$

Choosing $n=50$, the temporal dynamics of the population is shown in Figures 2
\- 4. The most interesting feature is the development of a sharp transition
region right after the death rate starts to be nonzero. That is, old age
population sharply declines with aging due to death. The sharp transition
region makes the population distribution bearing a kink shape. We will show
later that the kink shape is universal with respect to different forms of
birth and death rates. When $\alpha=1$ (roughly two children born per woman),
the population distribution $p(t,a)$ approaches an asymptotically steady state
(see Figure 2). The steady state bears the typical kink shape with the age
region before the sharp transition being constant. When $\alpha<1$, the total
population decreases to zero (see Figure 3). The left portion of the kink
shape population distribution is an increasing function in age. When
$\alpha>1$, the total population increases without bound (see Figure 4). The
left portion of the kink shape population distribution is a decreasing
function in age.

Now we are going to show the robustness of the qualitative nature of the
population distribution dynamics with respect to initial conditions, birth
rate, and death rate. When perturbing the initial condition with zero mean
random perturbation (with Gaussian distribution), the random perturbation is
quickly washed away in time; with $\alpha=1$, the population distribution
$p(t,a)$ approaches the same asymptotically steady state as in Figure 2, see
Figure 5. When $\alpha=1$, the birth dynamics (2.2) amounts to a weighted
averaging of the fertile population. Such an averaging washes away the random
perturbation.

Replacing the linear birth or death rates with sublinear or superlinear rates
does not change the nature of the dynamics (see Figure 6). Since the birth
dynamics amounts to a weighted averaging of the fertile population when
$\alpha=1$. Such an averaging does not change substantially when we replace
the linear birth rates with sublinear or superlinear rates. The death rate
always creates a sharp transition region in the population distribution.

Replacing the initial condition with general initial conditions does not
change the nature of the dynamics either (see Figure 7). When $\alpha=1$, the
weighted averaging in the birth dynamics keeps reducing the maximum and
increasing the minimum of the population distribution in the region $a\leq
40$. This leads to the constant asymptotics of the population distribution
before the sharp transition.

Replacing the birth rates $\alpha(t,b)$ in (2.2) with

(3.3) $\alpha(t,b)\frac{P(0)}{P(t)},$

we have the nonlinear birth rates that we are interested in. The idea is that
when the total population $P(t)$ increases, the birth rate decreases. The
total birth rate is replaced by $\alpha\frac{P(0)}{P(t)}$. When
$\alpha\frac{P(0)}{P(t)}>1$, the total population increases (as in Figure 4),
thus $\alpha\frac{P(0)}{P(t)}$ will decrease. When
$\alpha\frac{P(0)}{P(t)}<1$, the total population decreases (as in Figure 3),
thus $\alpha\frac{P(0)}{P(t)}$ will increase. This causes that
$\alpha\frac{P(0)}{P(t)}$ approaches $1$ as $t\rightarrow+\infty$. Thus the
population distribution $p(t,a)$ always approaches an asymptotically steady
state with the total population

(3.4) $P(t)=\alpha P(0),$

given by $\alpha P(0)/P(t)=1$ (see Figures 8 \- 10).

##  4\. Conclusion

A population-age-time (PAT) model is introduced that models the temporal
evolution of population distribution in age. The model is focused on the
effects of birth rate and death rate distribution in age, ignoring the effects
of immigration/emigration and migration. To our surprise, the qualitative
nature of the population distribution is very robust with respect to different
forms of initial conditions, birth rate distributions, and death rate
distributions. This indicates that the population distributions often take
certain universal asymptotic shape - a kink, which indeed agrees with typical
population distribution in reality (such as the population distribution in
US). When the number of children born per woman is $2$, the population
distribution approaches an asymptotically steady state of a kink shape; thus
the total population approaches a constant. When the number of children born
per woman is greater than $2$, the total population increases without bound;
and When the number of children born per woman is less than $2$, the total
population decreases to zero.

(a) linear birth rate (b) linear death rate

Figure 1. (a). The graphical illustration of the linear birth rate (3.1). (b).
The graphical illustration of the linear death rate (3.2).

Figure 2. (a). From the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=1$ for $n=50$. (b). The temporal evolution
of the total population $P(t)$. (c). From the top to the bottom is the
temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 3. (a). From the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=0.77$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 4. (a). From the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=1.32$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 5. (a). With zero mean random perturbation on the initial condition of
Figure 2, from the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=1$ for $n=50$. (b). The temporal evolution
of the total population $P(t)$ with zero mean random perturbation on the
initial condition of Figure 2. (c). With zero mean random perturbation on the
initial condition of Figure 2, from the top to the bottom is the temporal
evolution of the population distribution difference: $p(20,a)-p(0,a)$;
$p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 6. (a). The graphical illustration of a mixed superlinear and sublinear
birth rates. (b). With the mixed birth rates in (a) and zero mean random
perturbation on the initial condition of Figure 2, from the top to the bottom
is the temporal evolution of the population distribution: $p(0,a)$; $p(20,a)$;
$p(100,a)$; $p(200,a)$. The total birth rate $\alpha=\alpha_{n}=1.0001$ for
$n=50$. (c). The temporal evolution of the total population $P(t)$. (d). Ffrom
the top to the bottom is the temporal evolution of the population distribution
difference: $p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$;
$p(220,a)-p(200,a)$.

Figure 7. (a). With the mixed birth rates in Figure 6 and a general initial
condition, from the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=1.0001$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 8. (a). With the nonlinear birth rate (3.3) where $\alpha(t,b)$ is
given in Figure 6, from the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=1.0001$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 9. (a). With the nonlinear birth rate (3.3) where $\alpha(t,b)$ is
given in Figure 6, from the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=0.65999$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

Figure 10. (a). With the nonlinear birth rate (3.3) where $\alpha(t,b)$ is
given in Figure 6, from the top to the bottom is the temporal evolution of the
population distribution: $p(0,a)$; $p(20,a)$; $p(100,a)$; $p(200,a)$. The
total birth rate $\alpha=\alpha_{n}=2.2672$ for $n=50$. (b). The temporal
evolution of the total population $P(t)$. (c). From the top to the bottom is
the temporal evolution of the population distribution difference:
$p(20,a)-p(0,a)$; $p(40,a)-p(20,a)$; $p(120,a)-p(100,a)$; $p(220,a)-p(200,a)$.

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