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Bangladesh is a densely populated country than many other countries of the world. The population growth is termed as alarming, however, knowledge of growth in the years to come would be useful in planning for the development of the country. This article is based on the projection of future population growth of the country. The available actual population census data during 1991-2011 of Bangladesh was applied to the application of a non-linear, non-autonomous ordinary differential equation familiar as Verhulst logistic population model with the maximum environmental capability of Bangladesh. Bangladesh will reach its carrying capacity of 245.09 million population in the next 56 years i.e. the year 2067 and then it decreases as S-shaped curve. The article has provided a focus on the changing trends of the growth of the population of Bangladesh.

Bangladesh is a small country with a huge (8^{th} most in the world) population in South Asia. Overpopulation is the main cause of the poverty, illiteracy, unemployment, and malnutrition of the country. So the aggregative tendency in population growth is a great alarming to the people of Bangladesh. Therefore population control should be given top most priority in planning national policy. Application and use of a feasible technique of estimation and projection of the future population of the country can find importance at this juncture. Population dynamics uses population model, basically leading an individual form of a mathematical model. There are a lot of techniques which assist to develop a sound population model. Among these, Malthusian exponential model by Thomas Robert Malthus (1766-1834) and the Verhulst (1838) logistic differential equation model are well known [

Population census data of Bangladesh from 1991 to 2011 were used to arrive at an accurate prediction of the population. The growth rate in the population of Bangladesh has been strictly decreasing, its overall population has been strictly increasing and it would continue to increase until a zero growth rate is reached. The country will have the largest population when the growth rate will be zero. The logistic model is found to be used in census data of different country populations [

At the backdrop of stringent economy and environment of the country, it is crucially important to comprehend the fluctuating population movement for better planning in the country, in the days to come. The principal aim of the exercise was to prepare an estimation of growth of Bangladesh by using logistic model in predate actual census data from the year 2012 onward, up to 1991 to 2011. The Logistic model used and applied in population data (

The method of arriving at reliant elucidations to the problems through the methodical assortment, analyzing and explanation of data was one of the best-unspoken tools in any kind of research work. For this research work, secondary classified census data of Bangladesh (1991-2011) were collected from the Bangladesh Bureau of Statistics, the government of Bangladesh. 6^{th} order RK method (6^{th} order) in logistic model through MATLAB programming was used for the direct projection of future population of Bangladesh. Verhulst deduced formula was used to find out the carrying capacity and growth rate of Bangladesh population. Least square interpolation method was used to enumerate the growth rate of the future population of Bangladesh as a function of time.

A mathematical model is a benign picture of the whole structure using mathematical thoughts and verbal.

Malthusian Growth Model, sometimes entitled a simple exponential growth model, is essentially exponential growth based on a constant rate. The model is named after Thomas Robert Malthus, who wrote “An Essay on the Principle of Population” (1798), one of the most primitive and prominent books on population dynamics [

Year | Bangladesh Census Data (in thousand) |
---|---|

1991 | 106,313 |

2001 | 124,355 |

2011 | 142,319 |

d N d t = r N (1)

where r is the growth rate (Malthusian Parameter).

Equation (1) is habitually mentioned as the exponential law. It is usually considered in the arena of population ecology as the leading principle of population dynamics where Malthus as the initiator. Also occasionally it mentioned as the Malthusian Law.

The solution to Equation (1) is easily achieved by separation of variables and integrating both sides of the equation, supposing that N ( 0 ) = N ( t ) = 0 , to yield

∫ N ( 0 ) N ( T ) d N N = ∫ t = 0 t = T r d t ⇒ ln | N | | N ( 0 ) N ( T ) = r t | 0 T + C

and evaluating the upper and lower limits yields

ln ( N ( T ) ) − ln ( N ( 0 ) ) = ( r T + C ) − ( r × 0 + C )

⇒ N ( T ) N ( 0 ) = e r T

where N ( T ) and N ( 0 ) are both positive.

Rearranging the equation, we get the exact solution

N ( t ) = N 0 e r t (2)

where the initial condition N ( 0 ) = N 0 .

One of the utmost elementary and revolutionary models of population growth was the logistic model of population growth articulated by Pierre François Verhulst in 1838 [

d N d t = r N ( 1 − N K ) (3)

where parameter r is the growth rate and K is environmental maximum support i.e. limiting population as t → ∞ and N is the population size.

Solving Equation (3) for N, talking integration on both sides we get,

ln | K N − 1 | = − r t + c ⇒ K N − 1 = e − r t + c ⇒ K N − 1 = A e − r t

where A = e c = c o n s t a n t .

Assuming N = N 0 (initial population) at t = 0 then we get, A = K N 0 − 1

∴ N = K 1 + A e − r t = K 1 + ( K N 0 − 1 ) e − r t (4)

Taking limit as t → ∞ (since r > 0 ), then from Equation (4) we get,

N max = K (5)

Suppose that at time t = 1 and t = 2 the values of N are N 1 and N 2 respectively.

Then from Equation (4) we get,

1 − e − r K = 1 N 1 − e − r N 0 , 1 − e − 2 r K = 1 N 2 − e − 2 r N 0 (6)

Dividing 2^{nd} portion of Equation (6) by 1^{st} portion of Equation (6) we get,

1 + e − r = 1 N 2 − e − 2 r N 0 1 N 1 − e − r N 0

⇒ e − r = N 0 ( N 2 − N 1 ) N 2 ( N 1 − N 0 ) (7)

Putting the value of e − r in first portion of Equation (6) and rearranging we get,

K = N 0 N 1 2 + N 1 2 N 2 − 2 N 0 N 1 N 2 N 1 2 − N 0 N 2 (8)

which is the limiting value of N.

Let, at time t = T population be N = N 1 , where T is equally spaced years.

Then form Equation (4) we can write,

N 1 = K 1 + ( K N 0 − 1 ) e − r T

⇒ 1 + ( K N 0 − 1 ) e − r T = K N 1

⇒ ( K N 0 − 1 ) e − r T = K N 1 − 1

⇒ e − r T = K N 1 − 1 K N 0 − 1

⇒ e r T = K N 0 − 1 K N 1 − 1

⇒ r = 1 T ln ( K N 0 − 1 K N 1 − 1 )

⇒ r = 1 T ln ( 1 N 0 − 1 K 1 N 1 − 1 K ) (9)

According to Pierre François Verhulst we calculate the parameters K and r from the population N ( t ) in three dissimilar consecutive years [

Proceeding this way, we get

Percentage of relative error of exponential model in 1991 is 0%, in 2001 is 11.394% and 2011 is 35.033% while Percentage of relative error of logistic model in 1991 is 0%, in 2001 is 0.002% and 2011 is 0.004%. The error term of logistic has little difference to the census data of the population of Bangladesh. From

From Equation (3) we can write,

d N d t = r ( t ) N ( t ) ( 1 − N ( t ) K ) (10)

where K is assumed to be constant which is determined by the formula of Equation (8)

∴ (10) ⇒ 1 N ( t ) d N d t = r ( t ) ( 1 − N ( t ) K ) = F (t)

future growth rate of Bangladesh population.

⇒ r ( t ) = F ( t ) 1 − N ( t ) K (11)

where N ( t ) < K (carrying capacity)

Estimating r ( t ) , we model F ( t ) with a linear equation in time.

So,

F ( t ) = F ( 0 ) + m t (12)

where m = a ( 2011 ) − a ( 2001 ) 10 is determined by the given data in

According to least square interpolation, F ( t ) = 0.26554935849191336 − 0.029621318688351928 t [

Now applying Equations (4) & (12) with 6^{th} order RK method and MATLAB programming we estimate future Bangladesh population.

We predicted the population of Bangladesh applying Verhulst logistic population model. Firstly we have analyzed the census data of population of Bangladesh during 1991-2011 compared with the logistic model that matched as congenial harmony with the census data of population of Bangladesh. To make the non-linear differential equation as linear we used least square interpolation method and expressed growth rate F ( t ) as a function of t. Mathematically, the tactic of logistic model is a congenial apparatus to forecast the population of any country like Bangladesh. Higher order Runge-Kutta method is more accurate than lower order because of less error. Using higher order RK method (6^{th} order) in logistic model through MATLAB programming we may assume the future population of Bangladesh. For prediction, we use a generalized logistic population model.

portrait of the population which is well-disciplined by reducing population growth with different likely measures.

Finally, it was obvious that the methodology of logistic model had amalgamated with 6^{th} order RK method which was a congenial apparatus in population estimation. The principal aim of the exercise was to prepare an estimation of growth of Bangladesh by using logistic model in predate data from the year 2012 onwards, up to 1991 to 2011. The realistic mathematical logistic model is convenient in many disciplines from experimental utensils to arduous mathematical study and methods. Hence, the findings have been taken from this research work which will assist the government of Bangladesh in successful population planning and sustainable development in the country.

The authors declare no conflicts of interest regarding the publication of this paper.

UllaH, M.S., Mostafa, G., Jahan, N. and Khan, Md.A.H. (2019) Analyzing and Projection of Future Bangladesh Population Using Logistic Growth Model. International Journal of Modern Nonlinear Theory and Application, 8, 53-61. https://doi.org/10.4236/ijmnta.2019.83004

t: Time t in years

N t : Population at time t

N 0 : Population at reference time t = 0 i.e. year 1991

K: Human population capacity i.e. carrying capacity

r: Growth rate

d N d t : Rate of change of population with respect to time t

F ( t ) : Future growth rate of Bangladesh Population