Full Length Research Article

Science World Journal Vol. 15(No 4) 2020

www.scienceworldjournal.org
ISSN: 1597-6343 (Online), ISSN: 2756-391X (Print)
Published by Faculty of Science, Kaduna State University

ON LOGISTIC GROWTH MODEL FOR FORECASTING NIGERIA’S

POPULATION
Agog Nathan Samuel*, Bako Samuel Sunday, Bamanga Muhammad Ardo, Peter Magdaline, Byeli Usman

Department of Mathematical Sciences, Kaduna State University, P.M.B 2339, Tafawa Balewa Way, Kaduna.

*Corresponding Author’s Email Address: nathan.agog@kasu.edu.ng

ABSTRACT actual population of India and the estimated population using the
The population of Nigeria has been increasing steadily without a logistic model, where they deduced an error equation based on the
corresponding increase in resources required to cater for the trend line for the specified time period. They concluded that the
population explosion. When the growth in resources is not keeping data fits well into the logistic growth model. Also, they concluded
pace with the rapid growth in population, this poses a great threat that it is necessary to estimate the population of India yearly rather
to the nation in terms of social, environmental and economic than projecting it per census. The study also stated that it is
development. This paper focused on the application of logistic important to understand the changing population trend for planning
model to estimate the population growth of Nigeria. The carrying and implementation of policies related to population, environmental
capacity and vital coefficients of the population were estimated. and economy of India.
The study revealed that the population of Nigeria grows at the rate
of 3.0% per annum. The model was also employed to forecast for Ofori et al. (2013) used the exponential and the logistic growth
the Nigeria population. The study shows that the population of models to estimate the population growth of Ghana using data from
Nigeria will be approximately half of its carrying capacity of 1960 to 2011. The result of their findings reveals that the
485558039.4 in 58 years’ time which corresponds to the year 2065. exponential model predicted a growth rate of 3.15% per annum and
the population predicted was 114.8207 million by 2050. The
Keywords: Logistic Model, Forecasting, Carrying Capacity, Vital carrying capacity of Ghana was obtained using the logistic model
Coefficients, Population Growth Rate. and the vital coefficients a and b were 0.0523 and7.7468 × 10−5 ,
respectively. This indicated that the logistic model reveals a growth
1.0 INTRODUCTION rate of 5.23% and the population predicted was 341.2443 million
Population statistics of Nigeria in previous years reveals that it is by 2050.They conclude that the exponential model was the best for
increasing annually with about 2 to 3% growth rate. This is seen as projection due to small value of mean absolute percentage error as
one of the reasons for poor standard of living in Nigeria. Ashinze compared to logistic model.
(2015) indicated that among other proffered solutions, arise the
need to adequately model the Nigerian population that will Sintayehu et al. (2016) used the Malthus’s and the logistic growth
effectively checkmate the unprecedented growth of the population models to estimate the population growth of Ethiopia using data
with the resources available from 1980 to 2016. The result from their findings indicated that the
Kerry et al. (2017) discuss the last census exercise conducted by Malthus’s population model predicted a growth rate of 2.9% per
the National Population Commission (NPC) in 2006. Nigeria’s year while the logistic growth model predicted the carrying capacity
population was estimated to be at nearly one hundred and forty of 19767334 and growth rate of 2.9% per annum. In their result,
million. This is indeed an alarming figure for a country with both exponential and logistic model match well with the growth rate
distressed economy. The consequences of overpopulation is well estimated by International Data Base (IDB) for the past four years.
known, with characteristics of socio-economic problems such as The mean absolute percentage error was computed to be 0.64%
unemployment or underemployment, low level of per capita for Malthus’s growth model and 0.73% for the logistic growth
income, low standard of living, poverty, huge external debt burden model. From their research they conclude that the Malthus’
and many more social vices. Apart from information on the stock of population model seems to fit best to the original data since it has
the country’s population, it is essential to know the rate at which the the smallest value of mean absolute percentage error.
population is changing, structurally and in the aggregate. For a Augustus et al. (2012) applied logistic equation in modeling the
country to develop socially and economically, it needs to ensure population growth and carrying capacity of Uganda. The method of
that its population growth do not overstretch its available resources least squares was used to estimate the population growth rate of
to cater for the population. This is important not only for the present, Uganda. The values of the vital coefficients α and β were 0.0356
but also for future planning and implementation of policies that will and 1.2056 × 10−10 respectively. Using the logistic model, it
enhance economic growth and development. shows that the population growth rate of Uganda is 3.56% per
Ogbeide and Ikpotokin (2010) conducted a study using logistic annum and the population is expected to be 147,633,806 in the
growth model to estimate the population of the people of Esan West next 58 years from the year 2010, with a carrying capacity of
Local Government Area of Edo State, where the growth rate was 295,267,612. Minarul et al. (2012) used logistic population growth
estimated to be 3.5%. According to Shilpa et al. (2014), model for the projection of the population of Bangladesh for the
uncontrolled human population growth has been a threat to the year 2035. The results of their findings reveals that the carrying
earth’s resources and to the habitat themselves. Shilpa et al. capacity was K  176771641.8 , and the vital coefficients a
(2014) estimated the population growth of India from 2009 to 2012
10
using the logistic model approach. The study gave a comparison of and b were 0.05 and 2.84000 X 10 .

112

On Logistic Growth Model for Forecasting Nigeria’s Population

Science World Journal Vol. 15(No 4) 2020
www.scienceworldjournal.org
ISSN: 1597-6343 (Online), ISSN: 2756-391X (Print)
Published by Faculty of Science, Kaduna State University

2.0 MATERIALS AND METHOD a

In this study, secondary population data was taken from National Pmax  limt  K  (a 0) (7)
Bureau of Statistics. The Logistic growth mathematical model was b
used to compute the carrying capacity as well as the projected Putting t  1 and t  2 , the values of P1 and P2 respectively,
population values of Nigeria.
we obtain from (6) the following:
2.1 The Logistic Growth Model 1 e a
The logistic growth model was proposed by the Belgian
Mathematician Verhulst (1838), which incorporates the idea of the
b
a

1  e a   
P1 P0
(8)

carrying capacity. The population growth does not only depend on

1 e2 a
the population size but also how far this size is from its limit that is
(maximum supportable population). This model is a modification of
b
a

1  e2 a  
P2

P0
(9)
the Malthus model where population size is proportional to both the
previous population and a new term. Dividing (9) by (8) yields
dP a  bP(t )  1 e 2 a 
 (1)   
dt a
1  e a   P2 P0 
(10)
Where a and b are the vital coefficients of the population. This  1 e a 
term reveals how far the population is from its carrying capacity or   
maximum limit. Thus, as the population increases in size towards  P1 P0 
the carrying capacity, the value of the vital coefficients become very a
small and get to zero, providing the right information for the limiting Hence solving for e to obtain
P0  P2  P1 
value of the population growth. In this case, a modified equation
which tends to limit the population growth is used to model the e a  (11)
competition for available resources. The logistic growth model P2 ( P1  P0 )
follows the assumption that each individual reproduces at a rate
that deceases as a function of the population size. The modified Substituting equation (11) into equation (8) to obtain
equation is given as; b P12  P0 P2
dP (t ) aP (t )( a  bP (t ))  (12)
 , t0  t  t1 ; P(t0 )  P0 (2) a P1 ( P0 P1  2 P0 P1  P1P2 )
dt a
Therefore, the limiting value of P is given by;
Solving (2) and applying the initial conditions, a P1  P0 P1  2P0 P1  PP
1 2
dP K  Pmax  limt  P   (13)
 aP  bP 2 (3) b P12  P0 P2
dt
Applying variables separation in (3) and integrating to obtain 3.0 DISCUSSION OF RESULTS
1 1 b  This section discusses results obtained for the carrying capacity,
 a  P  a  bP  dP   dt vital coefficients of the population, estimated logistic model for the
population of Nigeria and finally the projected population of Nigeria
using the estimated logistic model.
1
a
 ln P  ln  a  bP    t  c at t  0 and 3.1 Estimating the Carrying Capacity of Nigeria
The carrying capacity is the maximum number of individuals that
P  P0 (4) can be supported in an environment without experiencing decline
in the capacity to support future generations within the area (Rees
1 1992).
c  (ln P0  ln(a  bP0 )) (5) Based on the population from the year 2006 in Table 1, let t=0, 1,
a 2 correspond to the years 2006, 2007 and 2008 respectively, where
Solving for P by substituting (5) into (4) yields
P0 , P1 , P2 are 144858000, 148604000 and 152429000
a
respectively. Then the carrying capacity is obtained as follows;
Pt  b (6) P1  P0 P1  2 P0 P2  PP
1 2
a  K
b  P12  P0 P2
1    1 e  at
2.5139612111021
 P0  K
  2.588734 1012
Now taking the limit as t   in (6) K  971116078.7
The above result for the carrying capacity is the maximum pressure

113

On Logistic Growth Model for Forecasting Nigeria’s Population

Science World Journal Vol. 15(No 4) 2020
www.scienceworldjournal.org
ISSN: 1597-6343 (Online), ISSN: 2756-391X (Print)
Published by Faculty of Science, Kaduna State University

or load that Nigeria can conveniently accommodate before

breaking down. A system breaks down when it can no longer Table 1: Actual and Projected Population of Nigeria using Logistic
accommodate the pressure of the population from the loads it is Growth Model
carrying.

3.2 Estimating the Vital Coefficients

The vital coefficients  a and b  reveal how far the population
is from its carrying capacity or maximum limit. As the population
increases in size towards the carrying capacity, the value of the
vital coefficients become very small and get to zero, providing the
right information for the limiting value of the population growth.
a
But, K  971116078.7 (14)
b
a
Using equation (11) we obtained e  0.970372657 and
by taking the natural logarithm of both sides, we obtained
a  0.030075098 .
This result indicated that the predicted growth rate of Nigeria’s
population is approximately 3% per annum. Also, using equation
(14) the value of the other vital coefficient b is obtained as;
0.030075098
b  3.096962213 1011
971116078.7
Lett  0 coincide correspond to the year 2006 where the initial
population P0  144858000 . The values of P0 , e and
a

a
are introduced into equation (6) to obtained;
b
971116078.7
Pt  (15)
1  (5.703917483)  0.970372657t
Equation (15) is the estimated logistic model for Nigeria. Table 1
shows the projected population values with the corresponding
actual population values. The projected population values were
computed using equation (15).
Hence the expected time for the population of Nigeria to be half of
 a 
its carrying capacity   485558039.35  can be
 2b 
obtained from equation (15), by substituting the value 48558039.35
for Pt ,
971116078.7
Pt 
1  (5.703917483)  0.970372657t
0.970372657t  0.1753181043
Taking logarithm of both side
log 0.1753181043
t
log 0.970372657
t  58 years
Therefore, it is expected that the population of Nigeria would be
half of its carrying capacity (48558039.35) in the year 2065.

4.0 Conclusion
In this study, we applied the logistic growth model in estimating the

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On Logistic Growth Model for Forecasting Nigeria’s Population

Science World Journal Vol. 15(No 4) 2020
www.scienceworldjournal.org
ISSN: 1597-6343 (Online), ISSN: 2756-391X (Print)
Published by Faculty of Science, Kaduna State University

population growth of Nigeria. The carrying capacity including the Kerry C.C., Subeno T., Ezeora, J.N. and Okafor J.I., (2017). A
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