Study of Mathematical Models for Population Projection
Meng Yi
Wang Zizheng
Sia Wai Leng (supervisor)
Raffles Girls' School (Secondary)
20 Anderson Road, Singapore 259 978
mengyi_22@yahoo.com.cn,
wzz321@hotmail.com, wailing@rgs.edu.sg
Tel: 6838 7811, Fax: 6235 3731
Abstract: Three main mathematical models for the prediction of population growth,
Thomas R. Malthus’s model, logistic model and coalition model, are examined in this
project. First of all, the explicit solutions for each model are exactly reached via using
mathematical techniques of differentiation and integration. Then the tests of practical
data and analysis for these models are done based on the explicit solutions; the
comparisons of predicted data and actual data of population growth for the models are
given by chart, tables and figures. They prove the efficiency of these models.
Furthermore, the advantages and shortcomings of these models are summarized. Finally
two new probably improved mathematical models for the predication of population
growth are proposed. Here, these models have been applied to the prediction of China’s
population growth. The results from the predictions show the importance of adopting the
family planning program in China. Moreover, describing and analyzing impacts on
population growth, such as economy, education, unemployment and poverty in social,
and environment, reveal the family planning cannot fail to become a national policy in
China.
Keywords: population, population size, population growth model, logistic model,
coalition model, Thomas R. Malthus’s model, family planning program.
1. Introduction
The population projection has become one of the most important problems in the world.
Population sizes and growth in a country directly influence the situation of economy,
policy, culture, education and environment etc of this country, and determine exploring
and cost of natural sources. No one wants to wait until those resources are exhausted
because of population explosion. Therefore the study of population projection has been
started earlier.
In 1798 the Englishman Thomas R. Malthus proposed a mathematical model of
population growth. His model, though simple, has become a basis for the most future
modeling of biological populations. He successfully discussed the caveats of
mathematical modeling through his paper, “An Essay on the Principle of Population".
Furthermore, he gave a pessimistic view over the dangers of over-population and
described the dangers of over-population would eventually lead to a shortage of food on
a global scale, poverty, hunger and disease. He believed that human population increases
geometrically (i.e. 2, 4, 8, 16, etc.) whereas food supplies can only grow arithmetically
(i.e. 2, 4, 6, 8, etc.) as it is limited by available land and technology. The geometric
population growth outruns an arithmetic increase in food supply. He stated that the ‘laws
of nature’ dictate that a population can never increase beyond the food supplies necessary
to support it.
However, most populations are constrained by limitation on resources -- even in the
short run -- and none is unconstrained forever. In 1840 a Belgian Mathematician Verhulst
modified Malthus’s Model, he thought population growth not only depends on the
population size but also on how far this size is from its upper limit.
Then in 1960 Heinz von Foerster, Patricia Mora, and Larry Amiot published a
now-famous paper in Science, “Population Density and Growth”. The authors argued
that the population growth pattern in the historical data can be explained by
improvements in technology and communication that have molded the human population
into an effective coalition in a vast game against Nature—reducing the effect of
environmental hazards, improving living conditions, and extending the average life span.
They proposed a coalition growth model for which the productivity rate is not constant,
but rather is an increasing function of P, namely, a function of the form P  , where the
power  is positive and presumably small.
Here, we have also done some research, trying to portrait a different picture of China
when there is no population control. We calculated some numbers on several important
development indicators and compared the needs of different situation. We found that
economy, social and environment have mutual influence upon each other. The conflicts
would aggravate if overpopulation occurs. More people need to live and eat, exhausting
natural resources. They also cause the shortage of public facilities. The government has
to invest more on housing, education and medical supply to satisfy the increased demand.
Less money therefore will be put into economic development. On another hand, economy
needs to grow rapidly to improve the living standard of population; a larger scale of
industries will definitely cause more pollution.
In the project, the construction is that in section 2 three main mathematical
techniques for the prediction of population growth, Thomas R. Malthus’s model, the
logistic model and the coalition model are examined in this project. First of all, the
explicit solutions for each model are exactly reached via using mathematical techniques
of differentiation and integration. Then the tests of practical data and analysis for the
models are done based on the explicit solutions, and these models are applied to the
predictions of China’s population growth. They prove the efficiency of these models.
Furthermore, the comparisons of predicted and actual data of population growth for the
models are given by chart, tables and figures. Finally, the advantages and shortcomings
of these models are summarized. In section 3, two new probably improved mathematical
models for the predication of population growth are proposed. In section 4, the
predictions of China’s population growth based on these models show the importance of
adopting the family planning program in China. Moreover, describing and analyzing
impacts on population growth, such as economy, education, unemployment and poverty
in social, and environment etc reveal the family planning cannot fail to become a national
policy in China. The conclusion is in section 5.
2. The Techniques for Predicting Population Sizes
2.1 Malthus’s Model
In 1798 the Englishman Thomas R. Malthus proposed a mathematical model of
population growth.
 dP
 P,   0

 dt

 P(t 0 )  P0  0
(2.1.1)
(2.1.2)
Where  in equation (2.1.1) expresses population growth rate and it reflects a strong
impact on how fast the population will grow. (2.1.1) indicates
dP
increases to infinity as
dt
t increases, i.e. the model population increases to infinity as time goes to infinity. (2.1.2)
expresses the population size when t  t 0 . Malthus’s model voices such principles:
(1) Food is necessary for human existence;
(2) Human population tends to grow faster than the power in the earth to produce
subsistence and that
(3) The effects of these two unequal powers must be kept equal.
(4) Since humans tend not to limit their population size voluntarily, population reduction
tends to be accomplished through the "positive" checks of famine, disease, poverty and
war.
2.1.1 The Solution of Malthus’s Model
We give the mathematical resolution of equations (2.1.1)-(2.1.2) here. Based on equation
(2.1.1), it is obtained that
P 1 dP  dt
t
P
t0
1
(2.1.3)
t
dP   dt
(integrating for two sides of (2.1.3))
t0
(2.1.4)
Thus
P  ( P0 e  t0 )e t  ce t
(2.1.5)
Where c  P0 e  t0 is a constant for some fixed  .
The resolution (2.1.5) of equations (2.1.1)-(2.1.2) clearly indicates P is an
exponential function of e. Its physical meaning expresses that population sizes grow
exponentially as time t .
2.1.2 The Practical Data Test of Malthus’s Model
We calculate and test the model (2.1.1)-(2.1.2) here through the following example. Start
with an initial value for the population size, P0  5 at t 0  0 and one of  , say  = 3.
Substitute the numbers into (2.1.1)-(2.1.2), it is obtained that P1  P(t  1)  100 .4 ,
P4  P(t  4)  813774 .0 and
then P2  P(t  2)  2017 .1 , P3  P(t  3)  40515 .4
P5  P(t  5)  16345086 .9 etc.
Notice how quickly the population can grow. The following figure 2-1 shows the
population growths as time, all of which start with an initial population size of 5, i.e.
P0  5
Fig.2-1
Population
20000000
15000000
10000000
5000000
0
t1
t2
t3
t4
t5
Year
As you can see, all the different curves have a similar shape. They are referred to as
J-shaped curves. The mathematical name for this type of growth is exponential growth.
This growth equation can be used in cases where there is truly this type of growth.
For example, when a new species arrives to an island where there is plentiful of food,
perfect conditions for reproduction, and no predators, or when a scientist starts growing
bacteria under perfect conditions, one can certainly observe this (almost perfect) type of
growth; although, not forever. That is why other mathematical models were developed.
Let’s investigate how good of a model this is. To do so, we need to estimate the
constant  . This constant is the continuous relative growth rate, i.e.
1 dP
. Let’s look at
P dt
some actual data. This chart shows the China population in the years 1949 and 1959,
measured in millions.
Table 2-1
Year
Population
1949
54 1. 67
1959
672. 07
Using this information with
t  0 corresponding to the year 1949, we
have P0  541.67 . We can solve for  using the fact that P  672.07 when t  10 . By
using (2.1.5), that is
672.07  541.67e10   
1  672.07 
ln 
  0.0215
10  541.67 
(to 3 s.f.) (2.1.6)
This gives us the general solution
P(t )  541.67e 0.0215t
(2.1.7)
Let’s compute the population at later years and compare it with the actual data. We
have done this and put it in the following chart.
Table 2-2
Actual
Predicted
1960
662. 07
686. 19
1961
65 8. 59
701. 10
1962
672. 95
716. 34
1963
661. 72
731. 90
1964
704. 99
747. 81
1965
725. 38
764. 06
1966
745. 72
780. 67
1967
763. 68
797. 63
1968
785. 34
814. 97
1969
806. 71
832. 86
1970
829. 92
850. 78
Predicted
Actual
950
900
Population sizes(million)
Year
850
800
750
700
650
600
550
500
1971
852. 29
869. 27
1972
871. 77
888. 16
1960
1962
1964
1966
1968
Fig. 2-2 Population vs time
1970
1972
Year
Based on fig.2-2, we see when   0.0215 , how approximate of the prediction value
and actual value of population sizes as time increases except of 1963. It showed the
efficiency of the model of Malthus under supposes of the Malthus’s model.
If we used the above model to predict the China population in 2000 and 2049(which
is after China was founded 100 years), we get,
P(2000)  1621.58 million, P(2049)  4650.16 million.
2.2 Logistic Model
Malthus’s model is unconstrained growth, i.e. model in which the population increases in
size without bound. It is an exponential growth model governed by a differential equation
of the form
dP
1 dP
 P 
  (Constant)
dt
P dt
As we have seen, the equation solution is equation (2.1.5).
P  ( P0 e  t0 )e t  ce t
Where c  P0 e  t0 is a constant for the constant  . Therefore, the population number P
increases to infinity as time t goes to infinity.
Nevertheless, most populations are constrained by limitations on resources -- even in
the short run -- and none is unconstrained forever. The following figure 2-3 shows three
possible courses for growth of the population. The red curve expresses super-exponential
growth and approaching a vertical asymptote (the dashed line), the green curve follows
an exponential growth pattern, and the blue curve is constrained so that the population is
always less than some number K . When the population is small related to K , the patterns
are virtually identical -- in particular, the constraint doesn't make much difference. But as
P becomes a significant fraction of K , the curves begin to diverge, and, in the constrained
case, as P gets close to K , the growth rate drops to 0. (Here K   )
Fig. 2-3 Population growths as time
In 1840 a Belgian Mathematician Verhulst modified Malthus’s Model. He thought
population growth not only depends on the population size but also on how far this size
is from its upper limit. He proposed a new model which is,
dP
P
 P (1  )
dt
M
(2.2.1)
Where   0 expresses population growth rate, and M  0 is called the carrying capacity
or the maximum supportable population. This equation is also known as a logistic
difference equation.
2.2.1 The Solution of the Logistic Model
We may account for the growth rate declining to 0 by including in the exponential model
a factor of  - P -- which is close to 1 (i.e., has no effect) when P is much smaller than
 , and which is close to 0 when P is close to  .
The constant solutions are P =0 and P =M. The non-constant solutions may obtained
by separating the variables.
dP
P
dP(1  )
M
  dt
(2.2.2)
Taking indefinite integration for the two sides of equation (2.2.2)

dP
P(1 
P
)
M

 dt
(2.2.3)
The partial fraction techniques give
1
dP
1
  (  M )dP

P
P
P
P(1  )
1
M
M
(2.2.4)
Which gives
ln P  ln 1 
P
 t  c
M
(2.2.5)
Easy algebraic manipulations give
P
P
1
M
 Ce t
(2.2.6)
Where C is constant. Solving for P, we get
MCe t
P
M  Ce t
(2.2.7)
If we consider the initial condition P(0)  P0 (assuming that P0 is not equal to both 0
or M), we get
C
P0 M
M  P0
(2.2.8)
Which, once substituted into the expression for P (t ) and simplified, we find
P(t ) 
MP0
P0  ( M  P0 )e t
(2.2.9)
It is easy to see that
lim P (t )  M
t  
(2.2.10)
Fig. 2-4 Population sizes vs. time based on logistic growth model
The curve produced by the logistic equation resembles an S. That is why it is called an
S-shaped curve or a Sigmoid. As you can see, when the population starts to grow, it does
go through an exponential growth phase, but as it gets closer to the carrying capacity, the
growth slows down and it reaches a stable level. This slow down to a carrying capacity is
perhaps the result of war, pestilence, and starvation as more and more people contend for
the resources that are now at their upper bound. There are many examples in nature that
show that when the environment is stable the maximum number of individuals in a
population fluctuates near the carrying capacity of the environment. However, if the
environment becomes unstable, the population size can have dramatic changes.
2.2.2 The Practical Data Test of the Logistic Model
If all we know about P is its values at certain times t, then we have to approximate the rate of
change,
dP
P
1 dP
by the fraction
. The relative rate of growth is
. Using population data
dt
P dt
t
from 1842~1970 we can estimate the relative growth rate at the years 1895, 1900, 1905, etc, with
the formula
1 dP 1 P P(t  5)  P(t  5)


P dt P t
10 P
This table has a few of these estimates.
(2.2.11)
Table 2-3
Y
ear
Popul
ation
1
895
495.
0.0018083
509.
0.0018070
523.
0.0018084
537.
0.0018066
614.
0.020409
825.
0.018625
50
1
910
12
1
925
11
1
940
48
1
955
48
1
970
P(t  5)  P(t  5)
10 P
81
We can find a better model by using this data to estimate how the relative growth
rate is changing with respect to population. That is, instead of assuming that the relative
growth rate is constant, let’s try to find a function which estimates this rate as a function
of the population P .
Here we used nonlinear least-squares regression to plot the estimates for
P along with the line which best fits the points.
Fig. 2-5 Population Growth rate vs. population
1 dP
versus
P dt
By using Origin 5.0, we get, the equation of this line is
1 dP
 0.0 2 6 625.8 7 6 121 05  P
P dt
(2.2.12)
Assuming the carrying capacity M is 14.8 billion (which is predicted by Census of
China). Then, the growth rate is
1 dP
 0.02662  5.87612  10 5  1 4 8 0
1480 dt
(2.2.13)
1 dP
 0.0 6 0 3 4 7
1 4 8 0dt
(2.2.14)
We can see that the growth rate increases as population size increases.
The following graph is a work which shows the population growth from 0 to 1972 in
China.
600
500
Population
Population
400
300
200
100
0
12
80
14
10
14
40
14
70
15
00
15
30
15
60
15
90
16
20
16
50
16
80
17
10
17
40
17
70
18
00
18
30
18
60
18
90
19
13
19
25
19
33
19
40
0
Year
Fig.2-6 Population growth
From the graph, we assume P0 is 40 million.
Then,
P(t ) 
1480P0
P0  (1480  P0 )  e 0.060347t
(2.2.15)
P(t ) 
73600
40  1440e 0.060347t
(2.2.16)
2.3 Coalition Model
In 1960 Heinz von Foerster, Patricia Mora, and Larry Amiot published a now-famous
paper in Science, “Population Density and Growth”. The authors argued that the growth
pattern in the historical data can be explained by improvements in technology and
communication that have molded the human population into an effective coalition in a
vast game against Nature -- reducing the effect of environmental hazards, improving
living conditions, and extending the average life span. They proposed a coalition growth
model for which the productivity rate is not constant, but rather is an increasing function
of P, namely, a function of the form P  , where the power  is positive and presumably
small. (If  were 0, this would reduce to the natural model -- which we now know it
does not fit.)
The differential equation for this model is
dP
 P 1 ,
dt
,  0
1 dP
 P 
P dt
(2.3.1)
(2.3.1)’
Where  growth is rate of population, and  is productivity rate.
The model asserts that the derivative of P should be proportional to a power of P,
that is, the rate of change should be a power function of P. If that is the case, then the
logarithm of the derivative should be a linear function of the logarithm of the population,
i.e.
ln
dP
 ln   (1   ) ln P
dt
(2.3.2)
Where 1   is the power.
2.3.1 The Solutions of Coalition Model
We now use the separation-of-variables technique to obtain a symbolic representation of
the solutions of this differential equation. Then we will consider the implications of
faster-than-exponential growth.
Separate the variables in the differential equation
dP
 P 1
dt
(2.3.3)
And write it in the form
P  (1 ) dP  dt
(2.3.4)
Taking indefinite integration for the two sides of equation (2.3.4), there is the following
P
(1 )
dP   dt

1
dP (1 )1   dt

  dP (1 )1    dt
(2.3.5)
That is
P   t  c
P
1
1
,
i.e.
(2.3.6)
(2.3.7)
[t  c] 
Where c is constant. Finally
P
1
(T  t )
1
(2.3.8)
This model only makes sense if t is less than T. This calculation shows that there is a
finite time T at which the population P becomes infinite -- or would if the growth pattern
continues to follow the coalition model.
2.4 Results from the models
Notice that Malthus’s model works fairly well for a short period that is the predicted
population sizes are quite similar to those of the actual sizes. It starts to fall apart after
that. However, this model assumes that the relative growth rate is constant. In fact, even
if we ignore natural disasters, wars, and changes in social behavior, the growth rate
would change as the population increased due to crowding, disease, and lack of natural
resources. The model predicts that the population would grow without bound, but this
cannot possibly happen indefinitely. One of the biggest failures in the Malthusian theory
was that Malthus failed to foresee the immense technological innovation that was to
occur, which increased crop yields and discovered new resources. Malthus interprets his
mathematical conclusions in terms of the real world and compares the real world to the
model. Well, ideally, he would do that. But, he doesn't live in an information-rich age,
and he's dealing with lengthy time spans, so he can't make such comparisons very easily.
The logistic growth equation is a useful model for demonstrating the effects of
density-dependent mechanisms: discrete-time model, in population growth. Under such
model, it is possible for the population to overshoot its carrying capacity. There is no
instantaneous adjustment of the population growth rate. The discrete-time model tells us
something about what happens when the effects of density-dependence aren't
instantaneous, but lag behind the population's growth in time. However, the logistic
growth equation utility in real populations is limited because the dynamics of populations
are complex and because it is difficult to come up with the real value for M in a given
habitat. In addition, M is not a fixed number over time; it is always changing depending
on many conditions. It is often limited by the current level of technology, which is
subject to change. More generally, species can sometimes alter and expand their niche. If
the carrying capacity of a system changes during a period of logistic growth, a second
period of logistic growth with a different carrying capacity can superimpose on the first
growth pulse. For example, cars first replaced the population of horses but then took on a
further growth trajectory of their own.
Comparatively speaking, the coalition growth model is the most comprehensive
among the models. However, this model only makes sense if t is less than T. The
calculation shows that there is a finite time T at which the population P becomes infinite
-- or would if the growth pattern continues to follow the coalition model.
All in all, there are quite a lot of potentially valuable factors which affect population
growth have been left out in all three models. Therefore, we cannot say which model is
the better one.
3. Modification of the Logistic Model
There are many shortcomings in the three models. However, here, we only use the
logistic model as an example to do the modification.
3.1 Adding health-condition factor
It is important to include the effect of health conditions (not sure about the term) in the
model. Having a model of this sort can help population researchers (same as the above
term) set scientifically sound population dying limits? If H is the number of people taken
by, for example, disease each time period (usually years) then the logistic model can be
modified to account for this by subtracting H from the population each time period. The
resulting model is:
P(t )  P(1 
P
)H
M
3.2 Adding minimum viable population factor
There is a second modification to the logistic model that further increases the model's
realism and usefulness. This is the addition of the ecological idea of a minimum viable
population. The idea here is that a population of any species has a minimum level at
which the population can thrive. If the population drops below this minimum level
various environmental and genetic factors lead to the elimination of the population. The
relevant factors might include inability to find mates, loss of genetic diversity and
increased vulnerability to short and long term environmental changes and disease events.
The logistic model can be modified to account for the existence of a minimum viable
population. One way of doing this is to use the difference equation:
P(t )  P(
P
P
 1)(1  )  H
Min.
M
4. Impacts on Population Growth
4.1 Economy Impacts
China’s national economy has developed at an accelerated speed, especially since the
economic reform to market-oriented system in 1978. With the gross national product
(GNP) quadrupled over that of 1980 ahead of schedule, China stands as the second
largest economy in the world after the US. However, when this significant achievement
is divided by China’s enormous population, each individual gets only 4,300 dollars a year,
accounting for 17.3% of the US per capita GNP in 1995.
Fig.4-1 Predicted population and GDP
2500
2000
1500
1000
500
0
1984
1985
1986
1987
1988
1989
1990
1991
Year
Predicted Population
GDP
According to the calculation of science institution of China ；The gross domestic product
(GDP) per capita increases by 0.36~0.59% while population growth rate decreases by
0.1% correspondingly. China’s population growth rate has dropped from 2.60% to 0.88%
since 1970. That means the family planning program helped to increase GDP per
capita by 6.192~10.148%.
GDP per capita under different situations
(RMB)
With family planning (million)
1
979
1
998
4
6
17.7
Without family planning (million)
490.2
3
63.0
4
099.5
By using formula (2.1.7), we have such chart of virtual population without control.
Year
Predicted
Year
Predicted
Year
Predicted
Year
Predicted
1973
907.5
1980
1054.9
1987
1226.2
1994
1425.3
1974
927.2
1981
1077.8
1988
1252.8
1995
1456.3
1975
947.3
1982
1101.2
1989
1280.1
1996
1488.0
1976
967.9
1983
1125.1
1990
1307.9
1997
1520.3
1977
989.0
1984
1149.6
1991
1336.3
1998
1553.3
1978
1010.5
1985
1174.6
1992
1365.3
1999
1587.1
1979
1032.4
1986
1200.1
1993
1395.0
2000
1621.6
Then, we can draw graph of the chart.
1800
9000
1600
8000
1400
7000
1200
6000
1000
5000
800
4000
600
3000
400
2000
200
1000
0
GDP
Population (million)
Fig. 4-2 Real and Predicted Population&GDP
0
1984
1987
1990
1993
1996
1999
years
population (predicted)
population (real)
GDP
Now we see the blue line indicating the predicted population growth without control in
China, the highest growth rate is 2%. The GDP growth would be lower than it and thus
the GDP per capita would only increase from RMB 363 to RMB 4099.5 through the
twenty years of 1979~1998 compared to the actual increase from RMB 417.7 to RMB
6490.21.
We cannot repudiate the idea of a huge population size is resource itself. But in
China, overpopulation will but only decrease welfare quality of each individual. The
births of 338 million avoided by family planning policy would require a relevant cost of
43.4 billion US dollars for bringing up, excluding the costs for more public facilities such
as schools, hospitals and household. This may result in lack of capital for economic
growth. Moreover, human resources depend on the quality of the population. In
developed countries, the contribution of technologies progress due to the improvement of
human quality is accounting for 60~80% of its economic growth. In China, 72% of the
economic growth relies upon the investment of natural resources and very little on
technologies. This deepens the conflicts between economic development and
environment degradation.
4.2 Social Impacts
4.2.1 Education and Medical Supply
Education and medical treatment play a significant role in the improvement of population
quality. Only population of a high quality could be conversed into plentiful resources.
Through years’ efforts of the government, the nine-year compulsory education [4] is
implemented national wide, which ensures 80.5% literacy. The number of health
institutions also increased from 3670 to 324771, barely satisfying people’s needs.
China had to expend much more on these public facilities if family planning policies
did not launch, as more children would ask for education and more people had to get
medical supply. Shortage of those public facilities would cause various social problems.
Fig.4-3 Comparison betw een real and required number of schools and
health institutions
1600000
1400000
1200000
1000000
800000
600000
400000
200000
0
number of primary and junior middle
schools
number of health institutions
predicted
required
Fig.4-4 Public expenditure on education as percentage of GNP
United States
United Kingdom
Brazil
year 1995
Japan
India
China
Nigeria
0
1
2
3
4
5
6
%
We can see from the graph that even India invested a larger percentage of its GNP on
education than China. Nevertheless, there is still more than 50% of Indian who can
neither read nor write. Indian’s huge efforts to promote education have been offset by its
rapid population growth (You know every year India adds at least 12 million people each
year).
4.2.2 Unemployment
What if Chinese government could not afford to pay for the cost of so many people’s
education fee? That would definitely lead to more illiterate people who were not able to
adapt to a modern and open China, which requires but well-educated and skilful labor.
Unemployment would undoubtedly increase massively in spite of more jobs created by
economic development.
In rural areas, as the cropland inevitably shrinks by years, the surplus of agricultural
labor force (already 120 million) is expanding substantially. Many people living in big
cities lost their jobs now due to the changing system of many national industries. The
situation of too many people competing for comparatively scarce job opportunities would
be unimaginably terrible if 300 million people were to join in.
Fig.4-5 Unemployment rate
4
3.5
3
2.5
2
1.5
1
0.5
0
1984
1986
1988
1990
1992
1994
1996
1998
2000
years
unemployment rate
Unemployment will cause a lot of problems when a country lacks a strong and mature
welfare system. For example, a jobless person may commit thefts or robberies when he
could not find another way to earn livings. Crime rate therefore goes up, affecting the
stability of society.
4.2.3 Poverty
In recent 20 years of economy reform, China has made tremendous achievement to
reduce poverty. Nearly 220 million Chinese have shaken off poverty during the past 20
years. In late 70s, China had 1/4 of people below poverty line in the world. However, this
ratio has decreased to 10%. A range of preferential policies have been mapped out to
help poor rural families have more access to education, wealth and basic services to
improve their lot.
China would have an entirely different picture without family planning policy, the
birth of 300 million people could not be avoided and the cost of bringing up the children
were to mount up to 360 billion dollars (about 43.4 billion US dollars), accounting for
0.08% of its GDP.
If China chose not to enlarge the scale education and medical supply, the population
quality would definitely decrease drastically. Literacy rate would drop and
unemployment rate would mount up. The increased unemployment leads to high crime
rate and more people below poverty line.
4.3 Environment
China, the third largest country in the world with its rich natural resources, is identified
very poor when those resources are divided by its 13-billion-strong population. Moreover,
the country now faces multitudes of environmental threats of including air pollution and
acid rain, water shortages and pollution, desertification and soil erosion, the destruction
of ecosystems and severe deforestation which are the consequences of ineffective
utilization of natural resources when developing economy.
According to China’s own statistics, the amount of land, farmland, and grassland
water resources owned by each individual Chinese is less than one-third of the world’s
average figure. Forest and oil resources per capita are just one-tenth of the world’s
average.
Merely 7% of vast land of China is arable, on which has to feed one-fifth of the
world population. However, cropland is still shrinking from 103.3 million ha in 1965 to
95.0 million ha in 1995, or a decline of 8% during the last three decades (Table 1). The
modernization process in China will take away more fertile farmland from agriculture.
Table 4-1 Agricultural Areas in China, million ha
Year Cultivated Areas Paddy Fields Irrigated Sown areas Total Grain Areas Cropping Index, %
1965 103.6
25.0
33.1
143.3
119.6
138
1975 99.7
25.5
43.3
149.5
121.1
150
1985 96.8
25.0
44.0
143.6
108.8
148
1995 95.0
24.9
49.3
149.9
110.1
158
The cropland per capita of China is currently about 43% of world average, about
0.101 hectares. The cropland per capita is less than 0.8 ha, the alarm line set by United
Nation in more than 600 counties. Well, if an extra 338 million people were added to
share the arable land, the cropland per capita would be less than 0.068 hectares (not
including the loss area for increasing residential and industrial use).
Green Revolution and new technologies nearly tripled the yield of China’s
agriculture production. However, the current technologies obviously afford another 355.8
million people’s stomachs, which increase the requirement for food production. China
could choose either to reclaim 64.4 hectares of pastures and forests for agricultural use or
buy food from other countries. The first way would only form a vicious circle of
destruction of environment (the forests were reduced, more flooding occurred and thus
more arable land lost by natural disasters); the second way might lead a increase in prices
of global agricultural production, affecting the world food security.
Fig.4-6 comparison betw een real agriculture production and agriculture production
required by 1.6 billion people
700
600
million tons
500
agriculture
production (real)
agriculture
production (virtual)
400
300
200
100
0
1984
1986
1988
1990
1992
1994
1996
1998
2000
years
Water scarcity is none the better in China. China has a total of 2,800 billion cubic meters
of annually renewed fresh water; the world's most populous country is fourth in the
world in terms of total water resources. Considering per capita water resources, China
has the second lowest per capita water resources in the world, less than one third of the
world average. Northern China is especially water-poor, with only 750 cubic meters per
capita; this geographic region has one-fifth the per capita water resources of southern
China and just 10 percent of the world average. 70 million people still have not the
access to drinking water.
8000
7000
World Average
China
Southern China
Northern China
6000
5000
4000
3000
2000
1000
0
1
The total supply of water resources is estimated to mount up from over 500 to 800
billion cubic meters if the population size mounted up to 1.6 billion due to the lack of
family planning，i.e. increased by 60%. It is 28% of available water resources in China.
According to global experiences, once the amount of water use exceeds 20% of the
county’s total water resources, water crisis will mostly occur.
The forest resource also decline through years due to the growth of population and
massive deforestation. Forests in China cover 133.7 million-hectares, not more than 14%
of China’s total area. Forests per head equals only to 18% of world average. People will
destroy forests to reclaim new farmland in order to survive. If an extra 355.8 million of
people lived on this land, 46.3 hectares of forests would disappear for residence. That
would break the biological balance and would result in severe soil erosion, contributing
to floods of major rivers in China.
Overpopulation not only exhausted natural resources but also contributed heavily to
environmental pollution. Water pollution and air pollution in China, already rank among
the top of the world, would certainly worsen if another 355.8 million would live on the
land. The air, water and land pollution were the cause of many diseases such as Diarrheal
disease and viral hepatitis. The continuing environment pollution and the degradation of
people’s living condition form a vicious circle.
4.3 Disadvantages on Family Planning Program
Although family planning policies have many positive impacts, we cannot deny the
shortcomings it brought as well. Firstly, the sex selection due to strict one child policy
has made sex ratio skewed. The sex ratio is 117 boys to 100 girls according to 2000
census.
Female infant abandonment is still common in rural areas of China. Secondly, China
may have to face a large group of aging people, which will account for 10% of its total
population. The productive, employable population sectors will shrink to less than
two-thirds of their population, meaning that even if all people of working age were to be
gainfully employed, they would have to support at least a third (and growing) of their
population that is not yet or no longer capable of working.
Fig.4-7
Fig. 4-8
Fig. 4-9
5. Conclusion
In the project, we examine Malthus’s model, logistic model and coalition model. Using
mathematical techniques of differentiation and integration, we exactly reach the explicit
solutions for each model. They are greatly clear and simple for tests of practical data and
analyses for each model. Furthermore, with chart, tables and figures, we compare
predicted data and actual data of population growth for the models. They prove the
efficiency of the models. Then the advantages and shortcomings of these models are
summarized. Finally two new probably improved mathematical models for the
predication of population growth are proposed
Experiences of a considerable number of countries, especially those, which are less
developed, can bespeak that overpopulation will but lead to severe problems such as
slowing development of economy, instability or even collapse of social systems, vicious
circle of poverty and environmental degradation and pollution.
Using mathematics techniques, Malthus’s population model, the logistic model, as
well as the coalition model, we predicted the virtual population size without family
planning policies. We tried to analyze the influences of family planning polices in China,
the most populous country, and evaluate its merits and shortcomings. Our study, though
very limited and sallow, showed that family planning programs have benefited the whole
country drastically and even avoided some terrible social or environmental disasters.
The success of family planning programs in China deserves attention from other
developing countries that also face the problem of massive population and its rapid
growth and the negative impacts have already affected the whole world. So we suggest a
global population program should be planned and launched as an important part of
sustainable development which emphasis an ideal relationship among population growth,
economic development and environmental protection. Our project is merely a puny effort
to study scientifically the measures to manage population that make it go along with the
other two factors.
Reference and Notes
1.
http://episte.math.ntu.edu.tw/applications/ap_population/
2.
http://www.ento.vt.edu/~sharov/PopEcol/popecol.html#mark5
3.
http://www.econlib.org/library/Malthus/malPlong.html
4.
http://www.ento.vt.edu/~sharov/PopEcol/lec5/explog.html
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http://www.co.rug.nl/~Maddison/China.html
6.
http://www.stats.gov.cn/index.htm
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http://www.adb.org/Documents/Books/Key_Indicators/2002/default.asp
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http://www.undp.org/hdr2001/pr2.pdf
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http://www.cia.gov/cia/publications/factbook/
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