Analyzing and Projection of Future Bangladesh Population Using Logistic Growth Model

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.


Introduction
Bangladesh is a small country with a huge (8 th [1]. The exponential model cannot define environmental limit point as it provides a stress-free calculation for future population and cannot catch the meticulous future population number. So the population will increase with the phase of time to infinity which is unrealistic.
Main limitation of Malthus model is that the population is increasing geometrically and twice in 25 or more years [2]. This model is appropriate for a very small time and a small amount of population as it doesn't reflect the environ- After reaching maximum level of population size, it decreases like S-shaped curve. In comparison, the exponential model doesn't provide estimation more accurate than a logistic one.
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 [3] [4]. Obaidullah (1976) presented an "Expo-linear Model" claiming that this model is superior to exponential or linear model in unfolding population growth over time [5]. However, there can be a difficulty in understanding its basic parameter unlike that of exponential or linear one. Mallik (1980) studies the motion of population in Bangladesh and suggested the prospect of a zero population growth rate in the ensuing hundred years (2080) [6]. Beekman (1981) had modified exponential growth model. That model had been applied a Markov chain to reproduce lower birth rate instigated by rural-urban development [7]. Applying this model, he estimated the future population of Bangladesh in 1998, 2018 and 2038 correspondingly. Kabir & Chowdhury (1982) scrutinized the affiliation between the population growth and food production of Bangladesh [8]. They suggested that the government should take necessary steps such as feeding policy etc. for declining the alarming growth of population in Bangladesh. Rabbani & Shadat Hossain (1981) studied the population trend of Bangladesh and forecasted the population of Bangladesh from 1975 to 2025 which is considered as the foundation work of forecasting population of Bangladesh [9].
The main limitation of their work is that they cannot indicate the year when Bangladesh will reach the apex of population. Though some researchers forecast Int. J. Modern Nonlinear Theory and Application on the population of Bangladesh, someone has used data of world meter or UN which is petite different from Bangladesh actual census data. But we carry on our work using actual census data of population of Bangladesh.
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 (Table 1) was found to work accurately in the estimation of population of Bangladesh and thus it is hoped that the findings will help in successful population projection, planning and development of the country, in next future.

Methodology
The method of arriving at reliant elucidations to the problems through the methodical assortment, analyzing and explanation of data was one of the bestunspoken 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.

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

Malthusian Exponential Growth Model
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 [2]. It reflects exponential growth of population and can be termed by the first order linear differential equation as the form: and evaluating the upper and lower limits yields Rearranging the equation, we get the exact solution where the initial condition

Logistic Model
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 [3]. The logistic model takes the shape of a sigmoid curve and describes the growth of population as exponential, followed by a reduction in growth, and bound by a carrying capacity due to ecological stresses which can be expressed first-order non-linear differential equation as the form: 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, Taking limit as t → ∞ (since 0 r > ), then from Equation (4) we get, max N K = (5) Suppose that at time 1 t = and 2 t = the values of N are 1 N and 2 N respectively.
Then from Equation (4) we get, Putting the value of e r − in first portion of Equation (6) and rearranging we get, which is the limiting value of N.
Let, at time t T = population be 1 N N = , where T is equally spaced years.
Then form Equation (4) we can write,

Determination of Carrying Capacity and Growth Rate
According to Pierre François Verhulst we calculate the parameters K and r from the population ( ) N t in three dissimilar consecutive years [3]. If 0 N is the population at time , then a monotonous calculation beginning from the exact solution of logistic model shows that Equation (8)  Proceeding this way, we get Table 2.

Estimation of Future Population of Bangladesh Using Logistic Model
From Equation (3) we can write, where K is assumed to be constant which is determined by the formula of Equa- future growth rate of Bangladesh population.
is determined by the given data in Table 1.
According to least square interpolation, Now applying Equations (4) & (12) with 6 th order RK method and MATLAB programming we estimate future Bangladesh population.

Result & Discussion
We  Figure 1 indicates that the graph of the forecasting population values is a sigmoid curve. Figure 2 states that the population of Bangladesh will gradually decrease in a parabolic manner i.e. sigmoid curve. This shows that the predicted values are fitted with the logistic model curve. Figure 1 indicates that Bangladesh to reach its carrying capacity would take more than 56 years i.e. the

Conclusion
Finally, it was obvious that the methodology of logistic model had amalgamated