Olagunju S. O.
89
Journal of Science and Science Education, Ondo Vol. 3(1), pp. 89 – 92, 19th November, 2012.
Available online at http://www.josseo.org
ISSN 0795135-3 ©2012
MATHEMATICAL MODEL FOR CONTROLLING POPULATION
EXPLOSION
Olagunju, S. O.1
1
Department of Mathematics, Adeyemi College of Education, Ondo
Abstract: It is very necessary to predict the future population of any community in
advance. This is for effective political and socio-economic planning. In this paper,
a new model to predict the future population of any community is developed,
putting population explosion into consideration bearing in mind that growth
restriction will limit the community’s population to its maximum expansion. Unlike
the previous population growth model which disregards effect of population
explosion, the result of this work leads to a model that gives regard to the effect of
explosion on the community. In conclusion, this model when properly implemented,
will give any community an insight to when its space and socio-economic plans
need a revisitation.
Keywords: Population, Socio-economic planning, Community, Population growth,
Population explosion.
Introduction
Population explosion, a sudden and rapid
increase in the number of individuals living in an
area or the rapid and dramatic rise in world
population, which may have occurred over a
certain number of years, may take after some
Mathematical ideology.
The world's population increased from 1.65
billion in 1900 to 3.02 billion in 1960, to about
6.74 billion in 2008 to about 6.82 billion in 2009
and recently on CCTV news to have hit 7 billion
in October 2011. Thus, the size of the population
had multiplied by about 425% in a span of about
100 years. This increase is historically
unprecedented.
Most of this growth is now taking place in the
world’s developing countries, where rates of
natural increase are much higher than they are in
industrialized countries. This is substantiated by
the analysis of the world’s population as outlined
from 1950 to 1980 on the appendix to this article.
Infact, recent records show that developing
countries rank first among nations when
population explosion is considered. The concern
that this might lead to overpopulation has led
some people, including the author of this paper to
look into the Mathematical implication(s) of such
imminent explosion. This will help any
community to prepare for such explosion, or cub it
before it explodes.
Among mankind, population explosion may
be as a result of an increased birth rate or a decline
in mortality, while in animals it may be because of
a lack of predators or altered environmental
conditions. This explains why many countries in
the world are facing serious problem of population
explosion, not because they do not have enough
resources to take care of their entire populace, but
because they do not have a means of checking
their population before it gets out of hand.
Though, population growth may assume
exponential nature as opined by some authors, it is
eventually limited by such factors as food and
space. This forces the growth of population in
such a community to assume a dangerous level as
it may have exploded beyond cope. These factors
90
Journal of Science and Science Education, Ondo
are taken into consideration in the development of
the model heralded by this paper.
REVIEW OF EXISTING MODEL
Considering the human population of a
community, let the number of individuals in that
community be x at a time t.
As t changes, x will change. (We note that
the change in x is not in the same continuous
manner as t)
The change in x occurs as it jumps once or
more at a time, as individuals produce or die.
Considering large population, not much error
is caused by the problem as though x changes
continuously.
Let the increase in x at a time interval t be
x .
For reasons of simplicity, let immigration and
emigration during time interval t be considered
negligible.
Since large population is being assumed,
Then, x  xt
 x  kxt (k being difference between birth
rate and death rate at time interval t .)
x

 kx
t
x
But as t  0, Lim
 Lim kx
t 0 t
t 0
Now, k  Z   population dying out ;
while k  Z   population increasing
Then,
dx
 kx
dt
dx
 kdt
x
So, loge x  kt  c

 x  kt c  c kt
Let population at time to be xo
i. e. xo  c
…
…
…
(i)
 xt  xo 
This (i) is the exponential model.
Consider the following cases:
Case 1: If k> 0  exponential growth model
without limit, leading to population explosion.
Case 2: If k< 0  exponential decay model
leading to zero.
kt
In practice, population growth may follow
exponential model for some time, but eventually
limited by such factors as food and space.
FORMULATION OF A NEW POPULATION
CONTROL MODEL
Since no research is an evolution, this work is
based on the above exponential model.
Now, considering space and socio-economic
facilities in a particular community,
Let the existing human population in the
community at time t = P(t)
Let maximum population that can be
accommodated in the community be M
Then, difference between maximum population
that be accommodated and the existing population
is given by M-P(t).
The rate of change of P(t) is proportional to MP(t).
The differential equation governing the situation
is:
dP
 ( M  P)
dt
dP
 k ( M  P)
dt
k>0 or k<0 (difference between birth and death
rates), being constant of proportionality.
Solving by the method of variable separation, we
have
dP
 k ( M  P) becoming
dt
dP
…
…
…
(ii)
 kdt …
M P
But we are interested in limiting population
explosion, we have
dP
So,
…
…
…
(iii)
 kdt
M P
Integrating, we have
loge (M  P)  kt  c
 M  P   kt c  c  kt
Given the initial population as Po  c
Then, M  P  Po  kt , t = 1, 2, 3, . . .
…
…
(iv)
 P(t )  M  Po  kt …
Equation (iv) is the Model for controlling
population explosion.
91
Olagunju S. O.
MODEL DISCUSSION AND ANALYSIS
Comparing this model with existing population
growth model, it is clear that this model sets limit
for population explosion.
P(t )  M  Po  kt , t = 1, 2, 3, . . .
Now, t = 1 , P1 = M - Po  -k
t = 2 , P2 = M - Po  -2k
t = 3 , P3 = M - Po  -3k
t = 4 , P4 = M - Po  -4k
.
.
.
.
.
.
.
.
.
-kt
As t   ,   0; and so, Po  -kt = 0.
Hence, P(t) = M .
This implies attainment of population limit; and so
population is checked to avoid explosion.
[To the best of the author’s knowledge, it is
noteworthy that previous model for population
growth prediction gives no limiting point].
Whereas, it can be observed that if any limiting
factor is not introduced to (ii) above,
dP
i.e.
…
…
(ii)
 kdt … …
M P
we would have it that
P(t )  M  Po kt , t = 1, 2, 3, . . .
This makes Po  kt become so large M becomes
negligible,
And so, if population at a time t = Xt , and M-P0 =
X0 ,
Then, Xt = Xo  kt ,t = 1, 2, 3, . . .
So that t= 1  X1 = Xo  k
t= 2  X2 = Xo  2k
t= 3  X3 = Xo  3k
t= 4  X4 = Xo  4k
.
.
.
.
.
.
.
.
.
This leads to t   ,  kt   , and so X t   .
Thus, the population explodes.
Thus what this model contributes to knowledge in
that population could be checked before
explosion.
Conclusion
In conclusion, the use of this model will make it
possible for any applying community to watch its
population growth keenly and put in necessary
measures of limitation before it gets out of hand.
Limitation
To every physical model, there is a limitation. It is
noteworthy that this model fails when error is
committed during implementation.
References
African Education Program Education Development
Centre, Massachusetts, U.S.A.Vol.2, Pp 678-683.
Benjamin Ginsberg, Microsoft ® Encarta ® 2007. ©
1993-2006 Microsoft Corporation.
CIA
World
Factbook(2012).
www.worldatlas.com/aatlas/populations/ctypopa.h
tm
Countries
Listed
by
Continent
(2012).
www.worldatlas.com
Mombasa Mathematics Workshop (1970): 1968
Workshop. Entebbe Mathematics Series.
UNESCO Institute for Statistics (2009) UNESCO
Institute for Statistics Data Centre. Montreal.
(http://stats.uis.unesco.org)
William, H. N. etal (1996): Power: The World of
Mathematics. World Book Inc, Vol 2, 2nd Ed. 525
W. Chicago. K 60661.
World Health Statistics (2011): Demographic and
Socioeconomic Statistics. Global HealthIndicators
II. 151-160.
www.english.cctv.com (2011): World Population.
APPENDIX
The Population of some countries of the world from 1950 to 1980 (x 1000)
#
Country or territory
1950
1955
1960
1965
1970
1975
1980
1
2
3
4
5
6
7
8
9
Afghanistan
Angola
Argentina
Armenia
Australia
Bangladesh
Bolivia
Bosnia and Herzegovina
Brazil
8,150
4,118
17,150
1,355
8,267
45,646
2,766
2,662
53,443
8,891
4,423
18,928
1,565
9,277
49,588
3,074
2,974
61,774
9,829
4,797
20,616
1,869
10,361
54,593
3,434
3,240
71,695
10,998
5,135
22,283
2,206
11,439
60,284
3,853
3,493
83,093
12,431
5,606
23,962
2,520
12,660
67,331
4,346
3,703
95,684
14,132
6,050
26,082
2,834
13,771
76,153
4,914
3,980
108,879
15,044
7,206
28,370
3,115
14,616
87,937
5,441
4,092
123,020
Journal of Science and Science Education, Ondo
92
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Burundi
Central African Republic
China
Congo (Kinshasa)
Costa Rica
Ivory Coast
Czech Republic
Egypt
El Salvador
Equatorial Guinea
Ethiopia
France
Georgia
Germany
Ghana
Haiti
Honduras
India
Iran
South Korea
Lebanon
Lesotho
Liberia
Mauritania
Morocco
Mozambique
Namibia
Nigeria
Pakistan
Philippines
Poland
Romania
Russia
Saudi Arabia
Senegal
Sierra Leone
Somalia
South Africa
Spain
Taiwan
Tanzania
Thailand
Turkey
Uganda
Ukraine
United Kingdom
United States
Venezuela
Yemen
Zambia
Zimbabwe
2,363
1,260
562,580
13,569
867
2,860
8,925
21,198
1,940
211
20,175
42,518
3,516
68,375
5,297
3,097
1,431
369,880
16,357
20,846
1,364
726
824
1,006
9,343
6,250
464
31,797
40,382
21,131
24,824
16,311
101,937
3,860
2,654
2,087
2,438
13,596
28,063
7,981
7,935
20,042
21,122
5,522
36,775
50,127
151,868
5,009
4,777
2,553
2,853
2,576
1,348
606,730
14,953
1,032
3,164
9,366
23,856
2,221
226
21,990
44,218
3,827
70,196
6,049
3,365
1,662
404,268
18,739
21,552
1,561
786
928
1,053
10,782
6,782
522
35,955
45,536
24,553
27,221
17,325
111,125
4,243
2,927
2,233
2,673
15,369
29,319
9,486
8,971
23,451
24,145
6,317
39,368
50,946
165,069
6,170
5,265
2,869
3,409
2,815
1,467
650,661
16,610
1,248
3,576
9,660
26,847
2,582
244
24,169
46,584
4,147
72,481
6,958
3,697
1,952
445,393
21,600
24,784
1,786
859
1,055
1,117
12,423
7,472
591
41,550
51,719
28,529
29,590
18,403
119,632
4,718
3,270
2,396
2,956
17,417
30,641
11,209
10,260
27,513
28,217
7,262
42,644
52,372
179,979
7,556
5,872
3,254
4,011
3,171
1,628
715,546
18,856
1,488
4,356
9,777
30,265
3,018
253
26,740
49,802
4,465
75,639
8,010
4,094
2,329
494,964
25,040
28,705
2,058
952
1,209
1,195
14,066
8,301
671
48,068
59,046
33,268
31,262
19,027
126,541
5,327
3,744
2,582
3,283
19,898
32,085
12,978
11,870
32,062
31,951
8,389
45,235
54,350
193,526
9,068
6,510
3,694
4,685
3,522
1,839
820,403
21,781
1,736
5,579
9,795
33,574
3,604
270
29,469
51,918
4,694
77,783
8,789
4,541
2,761
553,889
28,994
32,241
2,383
1,067
1,397
1,289
15,909
9,304
765
55,590
67,491
38,604
32,526
20,253
130,245
6,109
4,318
2,789
3,667
22,740
33,876
14,598
13,807
37,091
35,758
9,743
47,236
55,632
203,984
10,758
7,098
4,248
5,515
3,676
2,058
917,899
25,032
1,992
7,031
10,042
36,952
4,073
213
32,976
53,955
4,898
78,682
10,117
4,973
2,858
618,923
33,467
35,281
2,691
1,195
1,617
1,404
17,687
10,433
915
64,428
76,456
44,337
33,969
21,245
134,293
7,208
4,989
3,030
4,128
25,815
35,564
16,122
16,148
42,272
40,530
10,952
48,973
56,215
215,465
12,675
7,934
4,897
6,342
4,298
2,349
984,736
29,011
2,299
8,593
10,289
42,634
4,570
256
36,036
55,110
5,046
78,298
11,011
5,508
3,402
684,888
39,709
38,124
2,899
1,359
1,857
1,545
19,487
12,103
1,058
74,829
85,219
50,940
35,578
22,130
139,039
10,022
5,611
3,335
5,794
29,252
37,488
17,848
18,665
47,026
45,048
12,415
50,047
56,314
227,225
14,768
9,133
5,643
7,170
Source
United States Census Bureau - International Data Base (IDB), retrieved on 10 January 2012, including the minor
demographic corrections from the June 2011 release notes). The CIA World Factbook is sometimes attributed as a
source of these updated world population data; however, the CIA takes their demographic information from the
IDB. The estimates are all for 1 July of each year mentioned.