Thomas Robert Malthus published his famous book “An Essay on the Principle of Population” in 1798. In the basic argument, Malthus suggests that there are diminishing returns to land, that is if we add more workers to the land, the land produces little amounts of additional value. This means that if the population doubles and we have twice the number of workers on the land, we don’t produce double the amount of food. So, doubling the population means that each person actually has less food than that before. One of the earliest social theorists who connected the difference between modern Western cultures and non-Western and non-modern Western countries’ opulence to human behavioral patterns was Thomas Malthus. His finding that inconsistent population growth contributes to modern fortune has had a significant impact on contemporary social theorists as well as Western social theorists. Malthus believed that family planning, as it is known now, needed the typical Western capacity to evaluate the advantages and disadvantages of having children and make informed decisions about whether to get married now or later [14] - [16] . Malthus gives the following population model:

$\frac{\text{d}N\left(t\right)}{\text{d}t}=rN\left(t\right)$, (1)

where $r$ is the population growth rate which is a positive constant and also a constant of proportionality, and $N\left(t\right)$ is the number of individuals of the considered population at time t. The positive value of r indicates the population is increasing and the negative value of r indicates the population is decreasing. Since the solution of Equation (1) exponentially increases, the model can be called an unlimited population growth model. In real-world problems, population growth is restricted by food, space, and other necessities requirements for survival [17] .

We will solve the Malthusian model that is Equation (1) by variable separation method of ODE as follows:

$\frac{\text{d}N\left(t\right)}{\text{d}t}=rN\left(t\right)$

$\Rightarrow \frac{\text{d}N\left(t\right)}{N\left(t\right)}=r\text{d}t$

$\Rightarrow {\displaystyle \int \frac{\text{d}N\left(t\right)}{N\left(t\right)}}={\displaystyle \int r\text{d}t}$

$\Rightarrow \ln \left(N\left(t\right)\right)=rt+C_1$

$\Rightarrow {\text{e}}^{\ln \left(N\left(t\right)\right)}={\text{e}}^{rt+C_1}$

$\Rightarrow N\left(t\right)=C{\text{e}}^{rt}$, (2)

where $C$ is an integrating constant. Using initial condition, say at the starting time, that is when time $t=0$, the initial population is $N\left(0\right)=N_0$. Putting $t=0$ in the (2) equation we get

$N\left(0\right)=C{\text{e}}^0$

$\Rightarrow C=N_0$

Then out Equation (2) becomes

$\Rightarrow N\left(t\right)=N_0{\text{e}}^{rt}$

which is the solution of the Malthusian population model [18] - [20] . Solution curve is given below:

In Figure 1 , we consider artificial values $N_0=1.0$ (initial value), and for positive growth rate $r=0.1$ and $r=-0.1$ for negative growth rate in order to find a solution curve of the Malthusian population model. For negative growth rate population is decreasing, whereas for the positive growth rate population is increasing as time passes.

A self-limiting component was added to logistic equations to represent population increase, acting as a withstand on the unbounded expansion of the Malthusian model b Belgian mathematician Pierre François Verhulst in 1838. It commonly used to forecast technological and economic progress as well as populations of humans, plants, animals, and microbes. The logistic equation is a non-linear equation of the form:

$\frac{\text{d}N\left(t\right)}{\text{d}t}=rN\left(t\right)\left(1-\frac{N\left(t\right)}{K}\right),N\left(0\right)=N_0$(3)

where $r$ is the intrinsic growth rate, $N\left(t\right)$ denotes the population density at time $t$, $K$ is the environmental carrying capacity and N₀ is the population density at time $t=0$.

The solution of the logistic Equation (3) is

$N\left(t\right)=\frac{N_{0}K}{K{\text{e}}^{-rt}+N_0\left(1-{\text{e}}^{-rt}\right)}$(4)

where the carrying capacity, constant $K$ is not realistic. Its value may vary if the environment changes. When the carrying capacity $K$ and growth rate a is constant, we say Equation (3) is an autonomous system. For example, after the change of environment food availability can be changed. This change may create a positive or negative effect on the population of the species [21] - [23] .

Instead of constant carrying capacity, time-dependent carrying capacity $K\left(t\right)$ is used in many studies for various applications. Using a time-varying transport capacity makes the logistic equation non-autonomous due to the explicit time dependence of $K\left(t\right)$. Then our above (3) logistic equation changed to

$\frac{\text{d}N\left(t\right)}{\text{d}t}=rN\left(t\right)\left(1-\frac{N\left(t\right)}{K\left(t\right)}\right)$, (5)

where the carrying capacity

$K\left(t\right)=K_s\left(1-{\text{e}}^{-ct}\right)$

Here $K_s$ is the bacterial fertilization level, $c$ is the fertilization constant, $b=\left(1-\frac{K_0}{K_s}\right)$, with $K\left(0\right)=K_0$ and $0<b<1$ [24] - [26] .

Equation (3) is an ordinary differential Equation (ODE). To solve this equation, we will use the variable separation method. In this method, we first separate two variables on the right and left sides and then we integrate both sides. For finding the value of the integrating constant we will consider an initial guess.

The solution to Equation (3) is given below:

$\frac{\text{d}N\left(t\right)}{\text{d}t}=rN\left(t\right)\left(1-\frac{N\left(t\right)}{K}\right)$

$\Rightarrow \frac{\text{d}N\left(t\right)}{N\left(t\right)\left(1-\frac{N\left(t\right)}{K}\right)}=r\text{d}t$

$\Rightarrow {\displaystyle \int \frac{\text{d}N\left(t\right)}{N\left(t\right)\left(1-\frac{N\left(t\right)}{K}\right)}}={\displaystyle \int r\text{d}t}$

$\Rightarrow {\displaystyle \int \left(\frac{1}{N\left(t\right)}+\frac{1}{K-N\left(t\right)}\right)}={\displaystyle \int r\text{d}t}$

$\Rightarrow \ln \left(N\left(t\right)\right)-\ln \left(K-N\left(t\right)\right)=rt+c$

$\Rightarrow \ln \left(\frac{\left(K-N\left(t\right)\right)}{N\left(t\right)}\right)=-rt-c$

$\Rightarrow \ln \left(\frac{\left(K-N\left(t\right)\right)}{N\left(t\right)}\right)=-rt-c$

$\Rightarrow \frac{K-N\left(t\right)}{N\left(t\right)}=A{\text{e}}^{-rt}$

$\Rightarrow \frac{K}{N\left(t\right)}-1=A{\text{e}}^{-rt}$

$\Rightarrow \frac{K}{N\left(t\right)}=1+A{\text{e}}^{-rt}$

$\Rightarrow N\left(t\right)=\frac{K}{1+A{\text{e}}^{-rt}}$ (6)

Consider at the initial stage $t=0$, then density $N\left(0\right)=N_0$ and after replacing $N=N_0$ in (6) we get

$A=\frac{K-N_0}{N_0}$

Putting the value of A in Equation (6) we get the following equation

$N\left(t\right)=\frac{K}{1+\frac{K-N_0}{N_0}{\text{e}}^{-rt}}$

$N\left(t\right)=\frac{N_{0}K}{N_0+K{\text{e}}^{-rt}-N_0{\text{e}}^{-rt}}$

$N\left(t\right)=\frac{N_{0}K}{K{\text{e}}^{-rt}+N_0\left(1-{\text{e}}^{-rt}\right)}$(7)

Equation (7) is the solution of the autonomous logistic growth equation [27] [28] . Solution curve of logistic population model is represented bellow:

For Figure 2 , we consider artificial values $K=10$ (Carrying capacity), initial population, $N_0=1$ at time $t=0$ and a growth rate of $r=0.8$. Clearly, after a certain time population doesn’t increase realistically, as after that time the population becomes approximately equal to the carrying capacity, it tends to rise a little compared to the earlier time period.

In this section, we use the current population of Bangladesh to compare estimated population for the year 1974 to 2022 of both the Malthusian population model and the logistic population model ( Table 1 ).

The population estimated from the Malthusian population model and the logistic population model are compared to the census population in this table. Here, we considered 1.8315% as the growth rate for both models and 250,000,000 as Bangladesh’s highest possible capacity for the logistic population model. The inclusion of carrying capacity in the logistic population model has resulted in noticeable population variations between both models.

Figure 3 shows that the actual, Malthusian, and logistic population trajectories were initially at the point. However, over time, fluctuations between these three curves become visible, and we observe significant changes take place to the Malthusian and logistic model curves derived from census or real population curves. The Malthusian curve was slightly above the logistic curve from 1974 to 1990; however, as time passed, an upsurge in the logistic population model’s curve surpassed the Malthusian curve. The logistic curve moved above the Malthusian curve and began to approach the census’s population curve between 2010 and 2020.

As a result, the logistic population model’s accuracy improves with time far more than the Malthusian population models. The Malthusian population curve continues to grow exponentially even when the logistic curve approaches the true population curve. This leads us to the conclusion that the Logistic population model produces more accurate results than the Malthusian population model for an actual population of a system over a long period of time.

Here, we forecast the future population of Bangladesh along with the population that already estimated up to 2022 tabulated in Table 1 to compare the predictions of the Malthusian and logistic population to see which one act batter and gives more realistic prediction for population projection.

Table 2 presents a comparison between the population predictions derived from the Malthusian population model and the logistic population model for the future. Here, we took into consideration 1.8315% as the growth rate for both models and 250,000,000 as Bangladesh’s maximum capacity for the logistic population model. The addition of carrying capacity to the logistic population model has culminated in measurable population variations between the two models. In contrast, Bangladesh’s population will surpass the carrying capacity in 2061, according to the Malthusian population model, which assumes that the country’s population will grow exponentially and without limits in the future, reaching 633,280,981 by the year 2101. However, Bangladesh’s population is expected to reach 249,997,589 by the year 2101, approximately half of what the Malthusian population model predicts. This is due to the introduction of carrying capacity.