Comparing Theoretical and Practical Solution of the First Order First Degree Ordinary Differential Equation of Population Model

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1 Open Access Journal of Mahemaical and Theoreical Physics Comparing Theoreical and Pracical Soluion of he Firs Order Firs Degree Ordinary Differenial Equaion of Populaion Model Absrac Populaion dynamics is he branch of mahemaics ha sudies he size and age composiion of populaions as dynamical sysems, he biological and environmenal processes driving hem such as birh and deah raes and by immigraion and emigraion. In his paper, we are discussed how o read mahemaical models and how o analyze hem wih he ulimae aim ha we can criically judge he assumpions and he conribuions of such models whenever we encouner hem in your fuure biological research. Mahemaical models are used in all areas of biology. All models in his paper are formulaed in ordinary differenial equaions (ODEs). These will be analyzed by compuing seady saes. We developed he differenial equaions by ourselves following a simple graphical procedure, depicing each biological process separaely. Experience wih an approach for wriing models will help us o evaluae models proposed by ohers. Volume 1 Issue Research Aricle Deparmen of Compuer Science & Engineering, Shano- Mariam Universiy of Creaive Technology, India *Corresponding auhor: Abdullah Bin Masud, Deparmen of Compuer Science & Engineering, Shano-Mariam Universiy of Creaive Technology, Dhaka-130, Bangladesh, India, Received: January 5, 018 Published: February 1, 018 Keywords: General equaion of populaion growh; Logisic equaion; Logisic; Model for given daa; Soluion of logisic model; Comparing logisic model wih acual daa Inroducion In 1798, English economis Thomas Malhus was saed ha populaion would grow a a geomeric rae while he food supply grows a an arihmeic rae. The heory has been seen as flawed because of he limied facors observed when he developed he Law. I does no include facors, such as echnology, disease, povery, inernaional conflic and naural disasers. Malhusian models have he form P() = Pe 0 where P 0 is he iniial number of populaion, k is populaion growh rae (Malhusian parameer) and is he ime. Someimes his model is called simple exponenial growh model. General Equaion of Populaion Growh The rae of change of quaniy = he rae of birhs - The rae of deahs. Suppose P () is he populaion, α is he per capial birhs rae and β is he per capial number of deahs populaion. = αp () βp () = P ( )( α β) = P() K where K = α β This is he firs order firs degree ordinary differenial equaion [1]. The soluion of (1) is P() = ce. If = 0, P= P0 K P = C and P= P e 0 0 Birh Rae is Consan and Deah Rae is Linearly Increasing If α= α0 and β= β0+ β1 P() hen we have dp = α0p () { β0+ β1p ()} P () = ( α + β ) P () β P() Birh Rae Consan and Deah Rae is Exponenially Increasing dp If β= β1e and α= α0 hen = α0p () β1e P () Birh Rae Consan and Deah Rae is Sine Funcion dp If β= β1sin and α= α0 hen = α0p () β1sinp () () Deah Rae Consan and Birh Rae Linearly Increasing If α= α0+ α1 P () and β= β hen = ( α0+ α1p ()) P () βp () Deah Rae Consan and Birh Rae Exponenially Increasing If α= α1e and β= β0 hen = α1e P () β0p () Deah Rae and Birh Rae are Linearly Increasing If α= α0+ α1 P () and β= β0+ β1 P () hen = ( α + α P ( )) P ( β + β PP ) = ( α β ) P + ( α β ) P Submi Manuscrip hp://medcraveonline.com Open Acc J Mah Theor Phy 018, 1(1): 00003

2 1 Logisic Equaion: In real populaion growh is no always unlimied bu may have an upper limi L where populaion can no longer be susained as ime increase. The logisic ODE is dp P = KP(1 ) [].(1) Logisic Model for given daa: Since we have discree daa, hen we describe he model using a difference equaion. The equaion (A) can be wrien as P P ( + 1) P ( ) = KP 1 L P P = K P L 1...() The equaion says ha he raio of P o P is linear funcion of P. Firs of all, le s consider he lef hand side (LHS) of equaion (). We calculae he difference of he populaions for wo consecuive years, and hen use hose differences agains he corresponding funcion values [3]. Year Bangladesh India Pakisan Canada P() A P() a P() a P() a E

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4 14 Deermining he value of K and L: In he Leas Square Approximaion graph, we know he equaion for he line, which is, Subsiuing he poin P(1950) and P(1951) in (10) we have y = a + bx...(3) Variable/Counry Bangladesh India Pakisan Canada y= a+ bx. Equaion () can be wrien as P P y y K(1 P / L) = y...(4) 1 1 and K P L y (1 / ) =...(5) Solving (3) and (4) we have Py L= P y and K = y y y 1 P/ L 1 1 Variable/ Counry Bangladesh India Pakisan Canada L(Caring Capaciy ) Exp(Exp(L)) K (Consan) Soluion of Logisic Model Equaion (1) is Bernoulli equaion [4], we have dp P = KP(1 ) dp K = KP P dp K KP= P 1 dp K K + = P P L Pu 1 V P =...(6) 1 dp dv = P From (13) we have dv + KV = K Now his equaion is exac. Hence inegraing facor Hence he soluion is IF = e = e K K K K K V. e = e L K K Ke Ve. = + c L K 1 K 1 K e = e + c P L 1 1 K = + ce P L L P =.. K 1 + Lce If, hen P=L....(7) Comparing Logisic Model Wih Acual dp PK = PK ( ) dp L P = KP L 1 L dp = KPL ( P) Inegraing we have, 1 Ln P = + c K L P If = 0 hen find he value of c

5 15 Variable/ Counry Bangladesh India Pakisan Canada L(Caring Capaciy ) Exp(Exp(L)) K (Consan) Puing he values of c in (7), we have Year Time Theoreical Bangladesh India Pakisan Canada Theoreical Theoreical Theoreical

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7 17 Figure 4: Comparing graph of heoreical daa wih original daa of Canada. Figure 1: Comparing graph of heoreical daa wih original daa of Bangladesh. Figure : Comparing graph of heoreical daa wih original daa of India. Conclusion The carrying capaciy of Bangladesh is bu a his momen oal number of populaion is I is he bigges problem. The governmen of Bangladesh needs o ake necessary sep oherwise socio economic sysem is breakdown. Every counry of Subconinen, he oal populaion of hese counries is graer wice of carrying capaciy. In Canada, oal number of populaion is greaer he carrying capaciy. Acknowledgmen None. Conflic of Ineres None. References 1. Dreyer TP (1993) Modelling wih Ordinary Differenial Equaions. CRC Press, USA, pp Kelley W, Peerson A (004) Theory of Differenial Equaions Classical and Qualiaive. Springer. 3. Mooney DD, Swif RJ (1999) A Course in Mahemaical Modeling. Cambridge Universiy Press, UK, pp Zill DD (1993) A Firs Course in Differenial Equaions. Figure 3: Comparing graph of heoreical daa wih original daa of Pakisan.