Population dynamics is the branch of mathematics that studies the size and age composition of populations as dynamical systems, the biological and environmental processes driving them such as birth and death rates and by immigration and emigration. In this paper, we are discussed how to read mathematical models and how to analyze them with the ultimate aim that we can critically judge the assumptions and the contributions of such models whenever we encounter them in your future biological research. Mathematical models are used in all areas of biology. All models in this paper are formulated in ordinary differential equations (ODEs). These will be analyzed by computing steady states. We developed the differential equations by ourselves following a simple graphical procedure, depicting each biological process separately. Experience with an approach for writing models will help us to evaluate models proposed by others.

Keywords: general equation of population growth, logistic equation, logistic, model for given data; solution of logistic model, comparing logistic model with actual data

In 1798, English economist Thomas Malthus was stated that population would grow at a geometric rate while the food supply grows at an arithmetic rate. The theory has been seen as flawed because of the limited factors observed when he developed the Law. It does not include factors, such as technology, disease, poverty, international conflict and natural disasters.

Malthusian models have the form $P(t)=P_0{e}^{kt}$  where P0 is the initial number of population, k is population growth rate (Malthusian parameter) and t is the time. Sometimes this model is called simple exponential growth model.

The rate of change of quantity = the rate of births - The rate of deaths.

Suppose$P(t)$  is the population, $\alpha$ is the per capital births rate and $\beta$  is the per capital number of deaths population.

 $\begin{array}{l}\frac{dP(t)}{dt}=\alpha P(t)-\beta P(t)\\ =P(t)(\alpha -\beta )\\ =P(t)KwhereK=\alpha -\beta \end{array}$

This is the first order first degree ordinary differential equation.1 The solution of (1) is $P(t)=c{e}^{kt}$ . If $t=0,P=P_0$  $\therefore P_0=CandP=P_0{e}^{Kt}$

If $\alpha ={\alpha }_{0}and\beta ={\beta }_0+{\beta }_{1}P(t)$ then we have$\frac{dP}{dt}={\alpha }_{0}P(t)-\left\{{\beta }_0+{\beta }_{1}P(t)\right\}P(t)$ $=({\alpha }_0+{\beta }_0)P(t)-{\beta }_1{P}^2(t)$

If$\beta ={\beta }_1{e}^{kt}$  and $\alpha ={\alpha }_0$ then $\frac{dp}{dt}={\alpha }_{0}P(t)-{\beta }_1{e}^{kt}P(t)$

If$\beta ={\beta }_1\sin t$ and $\alpha ={\alpha }_0$ then $\frac{dP}{dt}={\alpha }_{0}P(t)-{\beta }_{1}Sin(t)P(t)$

If $\alpha ={\alpha }_0+{\alpha }_{1}P(t)$  and $\beta =\beta$ then $\frac{dP(t)}{dt}=({\alpha }_0+{\alpha }_{1}P(t))P(t)-\beta P(t)$

If $\alpha ={\alpha }_1{e}^{kt}$  and $\beta ={\beta }_0$ then $\frac{dP(t)}{dt}={\alpha }_1{e}^{kt}P(t)-{\beta }_{0}P(t)$

If$\alpha ={\alpha }_0+{\alpha }_{1}P(t)$  and $\beta ={\beta }_0+{\beta }_{1}P(t)$ then $\begin{array}{l}\frac{dP(t)}{dt}=({\alpha }_0+{\alpha }_{1}P(t))P-({\beta }_0+{\beta }_{1}P)P\\ =({\alpha }_0-{\beta }_0)P+({\alpha }_1-{\beta }_1)P^2\end{array}$

In real population growth is not always unlimited but may have an upper limit L where population can no longer be sustained as time increase. The logistic ODE is

$\frac{dP}{dt}=KP(1-\frac{P}{L})$ .(1)²

Since we have discrete data, then we describe the model using a difference equation. The equation (A) can be written as

$\begin{array}{l}P(t+1)-P(t)=KP\left(1-\frac{P}{L}\right)\\ \Rightarrow \frac{\Delta P}{P}=K\left(1-\frac{P}{L}\right)\mathrm{.............................}(2)\end{array}$

The equation says that the ratio of to P is linear function of P. First of all, let’s consider the left hand side (LHS) of equation (2). We calculate the difference of the populations for two consecutive years, and then use those differences against the corresponding function values.³

Determining the value of K and L: In the Least Square Approximation graph, we know the equation for the line, which is, $y=a+bx\mathrm{..............................}(3)$

Substituting the point P(1950) and P(1951) in (10) we have

Equation (2) can be written as $\begin{array}{l}K(1-P_1/L)=y_1\mathrm{.............................}(4)\\ andK(1-P_2/L)=y_2\mathrm{.............................}(5)\end{array}$ Solving (3) and (4) we have $L=\frac{P_1{y}_2-P_2{y}_1}{y_2-y_1}andK=\frac{y_1}{1-P_1/L}$

Equation (1) is Bernoulli equation,⁴ we have

$\begin{array}{l}\frac{dP}{dt}=KP(1-\frac{P}{L})\\ \Rightarrow \frac{dP}{dt}=KP-\frac{K}{L}{P}^2\\ \Rightarrow \frac{dP}{dt}-KP=-\frac{K}{L}{P}^2\\ \Rightarrow -\frac{1}{P^2}\frac{dP}{dt}+\frac{K}{P}=\frac{K}{L}\mathrm{......................}(6)\end{array}$ Put $\frac{1}{P}=V$ $\therefore -\frac{1}{P^2}\frac{dP}{dt}=\frac{dV}{dt}$ From (13) we have $\frac{dV}{dt}+KV=\frac{K}{L}$ Now this equation is exact. $\begin{array}{l}IF=e^{{\displaystyle \int Kdt}}\\ =\text{}{e}^{Kt}\end{array}$ Hence integrating factor

$\begin{array}{l}V.e^{Kt}={\displaystyle \int \frac{K}{L}{e}^{Kt}dt}\\ \Rightarrow V.e^{Kt}=\frac{K}{L}\frac{e^{Kt}}{K}+c\\ \Rightarrow \frac{1}{P}{e}^{Kt}=\frac{1}{L}{e}^{Kt}+c\\ \Rightarrow \frac{1}{P}=\frac{1}{L}+c{e}^{-Kt}\\ \Rightarrow P=\frac{L}{1+Lc{e}^{-Kt}}\mathrm{...............................}(7)\end{array}$

Hence the solution is

If $t\to \infty ,$ then P=L.

$\begin{array}{l}\frac{dP}{dt}=P(K-\frac{PK}{L})\\ \Rightarrow \frac{dp}{dt}=KP\left(\frac{L-P}{L}\right)\\ \Rightarrow \frac{1}{K}\frac{L}{P(L-P)}dP=dt\end{array}$

Integrating we have, $\frac{1}{K}Ln\left(\frac{P}{L-P}\right)=t+c$

If t = 0 then find the value of c

Putting the values of c in (7), we have

Figure 1 Comparing graph of theoretical data with original data of Bangladesh.

Figure 2 Comparing graph of theoretical data with original data of India.

Figure 3 Comparing graph of theoretical data with original data of Pakistan.

Figure 4 Comparing graph of theoretical data with original data of Canada.

The carrying capacity of Bangladesh is 85415102.72 but at this moment total number of population is 164827718. It is the biggest problem. The government of Bangladesh needs to take necessary step otherwise socio economic system is breakdown. Every country of Subcontinent, the total population of these countries is grater twice of carrying capacity. In Canada, total number of population is greater the carrying capacity.

None.

The author declares no conflict of interest.

1. Dreyer TP. Modeling with Ordinary Differential Equations. USA: CRC Press; 1993. 304 p.
2. Kelley W, Peterson A. Theory of Differential Equations Classical and Qualitative. Springer; 2004.
3. Mooney DD, Swift RJ. A Course in Mathematical Modeling. UK: Cambridge University Press; 1999. 431 p.
4. Zill DD. A First Course in Differential Equations. 1993.