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

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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 1-018 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, Email: 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() 0 0 1 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 ) 0 1 0 1 = ( α β ) P + ( α β ) P 0 0 1 1 Submi Manuscrip hp://medcraveonline.com Open Acc J Mah Theor Phy 018, 1(1): 00003

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 1950.859358 0.00045.98949 0.00064.85883 0.0003.79945 0.000566 1951.860573 0.00040.983737 0.00076.859481 0.00067.801036 0.000579 195.86173 0.000405.98456 0.00086.86044 0.0003.80656 0.00059 1953.8688 0.00045.985416 0.00094.861101 0.00039.804309 0.000598 1954.864098 0.000455.98694 0.0003.8604 0.000355.805987 0.000601 1955.865401 0.000488.987191 0.000306.863058 0.000379.807674 0.000597 1956.8668 0.000519.988106 0.000311.864143 0.0004.80935 0.000584 1957.86887 0.00054.989036 0.000316.86589 0.0004.81099 0.000563 1958.869843 0.000556.98998 0.0003.866491 0.000437.81573 0.00053 1959.871439 0.00056.990945 0.00038.867745 0.000454.814068 0.000494 1960.873049 0.00056.99195 0.000333.869047 0.000468.815458 0.000454 1961.874659 0.00056.9993 0.000338.87039 0.000481.816737 0.000418 196.87669 0.000567.993935 0.00034.87177 0.000491.817913 0.000391 1963.87790 0.000584.99496 0.000344.873181 0.000499.819015 0.000377 1964.879585 0.000604.99599 0.000346.874615 0.000505.80079 0.00037 1965.88136 0.00063.99709 0.000347.876068 0.00051.8119 0.000369 1966.883143 0.000644.998069 0.000348.877539 0.000517.817 0.000363 1967.885001 0.00065.999113 0.00035.87907 0.00051.83194 0.000358 1968.886806 0.000567 3.000168 0.000358.88058 9.55E-09.8404 0.00035 1969.888444 0.000487 3.0014 0.000366.88058 0.001048.85197 0.000345 1970.88985 0.0004 3.00339 0.000374.883547 0.00055.86173 0.00034 1971.891009 0.000333 3.003461 0.00038.88506 0.00058.87135 0.000335 197.89197 0.000304 3.004603 0.000383.886585 0.000534.8808 0.00035 1973.8985 0.00036 3.005755 0.000383.88817 0.000546.89 0.000308 1974.893793 0.00038 3.006908 0.000381.88970 0.00056.8987 0.00088 1975.894898 0.000446 3.008055 0.000378.89131 0.000573.830687 0.00068 1976.896189 0.000495 3.009191 0.000376.89979 0.000585.831444 0.0005 1977.8976 0.00056 3.01034 0.00037.89467 0.000596.83153 0.00036 1978.899147 0.000533 3.011443 0.000374.896396 0.000607.838 0.0006 1979.900694 0.00054 3.0157 0.000375.898155 0.000617.833463 0.000

13 1980.9015 0.00051 3.013701 0.000377.899945 0.00066.834085 0.00013 1981.90370 0.000505 3.014838 0.000377.901759 0.000631.834688 0.00007 198.905169 0.000534 3.015975 0.000375.903589 0.000631.83575 0.0001 1983.9067 0.000466 3.017106 0.000371.90541 0.00066.835871 0.0003 1984.908076 0.00050 3.0185 0.000365.9074 0.000617.836503 0.0004 1985.909535 0.000503 3.01935 0.000358.909035 0.000608.837191 0.00064 1986.911 0.000501 3.00407 0.00035.910804 0.000598.837941 0.0008 1987.91459 0.000493 3.0147 0.000346.91544 0.00058.838741 0.00091 1988.913895 0.000478 3.0514 0.000339.9144 0.00056.839566 0.00087 1989.91588 0.000458 3.0354 0.000333.915877 0.000538.84038 0.00074 1990.91664 0.000437 3.04548 0.00037.917446 0.000514.841157 0.00059 1991.917898 0.000418 3.05537 0.00031.918945 0.00049.841894 0.00047 199.919117 0.000404 3.06507 0.000315.9038 0.000475.84597 0.00035 1993.9098 0.000398 3.0746 0.00031.91767 0.000466.84365 0.0004 1994.91459 0.000395 3.08401 0.000307.9318 0.000461.843901 0.00013 1995.9614 0.000393 3.0939 0.000303.94474 0.000458.844507 0.0000 1996.93763 0.000388 3.03047 0.00099.95813 0.000453.845083 0.00019 1997.94899 0.000381 3.03115 0.00094.97139 0.000444.84563 0.000186 1998.96014 0.00037 3.0304 0.00088.98439 0.00043.846161 0.000185 1999.97098 0.000495 3.03915 0.00081.99697 0.00041.846688 0.000188 000.98547 0.00005 3.033768 0.00075.930905 0.000395.8473 0.000191 001.99148 0.000331 3.034601 0.00068.93061 0.00038.847767 0.000194 00.930118 0.000314 3.035415 0.0006.933176 0.000371.8483 0.000199 003.931039 0.00035 3.0361 0.00055.93464 0.000367.848887 0.00006 004.93071 0.000144 3.036985 0.00049.935341 0.000356.849474 0.00014 005.9349 0.000371 3.037741 0.00043.936387 0.000381.850083 0.000 006.933579 0.00015 3.03848 0.00037.937506 0.00037.850717 0.0009 007.93405 0.00004 3.039199 0.0003.93859 0.000371.85137 0.00033 008.93464 0.00001 3.0399 0.0003.939683 0.000374.85033 0.00031 009.93513 0.00005 3.040578 0.00016.940781 0.000376.8569 0.0006 010.935816 0.0001 3.04135 0.00009.941887 0.000378.853337 0.000 011.936438 0.00017 3.04187 0.0000.94999 0.00038.853965 0.00014 01.937075 0.00019 3.04483 0.000196.944117 0.000379.854577 0.00009 013.9377 0.00019 3.04308 0.00019.9453 0.000375.85517 0.00003 014.938363 0.00016 3.043665 0.000189.946337 0.00037.85575 0.000198 015.938997 0.00013 3.04441 0.000187.94747 0.000364.856318 0.000193 016.93963 0.0001 3.04481 0.000184.948499 0.000357.85687 0.000187

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 1.859358064.9894918.8588666.79944991 P 1.86057649.98373741.859481009.80103555 y 1 0.045499 0.033148173 0.045835 0.034733 y 0.04485 0.0335856 0.045177 0.033148 K(1 P / L) = y...(4) 1 1 and K P L y (1 / ) =...(5) 1 1 1 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 ).904878989 3.016041509.90468156.834196893 Exp(Exp(L)) 8541510.7 7313066 85107708.4 45648.95 K (Consan).903479866 3.0116376.9031681.83305651 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

15 Variable/ Counry Bangladesh India Pakisan Canada L(Caring Capaciy ).904878989 3.016041509.90468156.834196893 Exp(Exp(L)) 8541510.7 7313066 85107708.4 45648.95 K (Consan).903479866 3.0116376.9031681.83305651 Puing he values of c in (7), we have Year Time Theoreical Bangladesh India Pakisan Canada Theoreical Theoreical Theoreical 1950 0 0.56491.859358 0.548991.98949 0.565691.85883 0.5574.79945 1951 1 0.338815.860573.474081.983737.367713.859481.5355.801036 195 0.507008.86173.984176.98456.868994.86044.791768.80656 1953 3 0.73477.8688 3.01447.985416.90701.861101.831665.804309 1954 4 1.0145.864098 3.015965.98694.904573.8604.834048.805987 1955 5 1.349886.865401 3.016038.987191.904676.863058.834188.807674 1956 6 1.689356.8668 3.016041.988106.904681.864143.834196.80935 1957 7.004061.86887 3.01604.989036.90468.86589.834197.81099 1958 8.6787.869843 3.01604.98998.90468.866491.834197.81573 1959 9.470983.871439 3.01604.990945.90468.867745.834197.814068 1960 10.617359.873049 3.01604.99195.90468.869047.834197.815458 1961 11.717896.874659 3.01604.9993.90468.87039.834197.816737 196 1.784688.87669 3.01604.993935.90468.87177.834197.817913 1963 13.88086.87790 3.01604.99496.90468.873181.834197.819015 1964 14.855879.879585 3.01604.99599.90468.874615.834197.80079 1965 15.87351.88136 3.01604.99709.90468.876068.834197.8119 1966 16.884634.883143 3.01604.998069.90468.877539.834197.817 1967 17.8916.885001 3.01604.999113.90468.87907.834197.83194 1968 18.896004.886806 3.01604 3.000168.90468.88058.834197.8404 1969 19.898746.888444 3.01604 3.0014.90468.88058.834197.85197 1970 0.900461.88985 3.01604 3.00339.90468.883547.834197.86173 1971 1.901533.891009 3.01604 3.003461.90468.88506.834197.87135 197.9003.89197 3.01604 3.004603.90468.886585.834197.8808 1973 3.9061.8985 3.01604 3.005755.90468.88817.834197.89 1974 4.90883.893793 3.01604 3.006908.90468.88970.834197.8987 1975 5.903046.894898 3.01604 3.008055.90468.89131.834197.830687 1976 6.903148.896189 3.01604 3.009191.90468.89979.834197.831444 1977 7.9031.8976 3.01604 3.01034.90468.89467.834197.83153 1978 8.90351.899147 3.01604 3.011443.90468.896396.834197.838 1979 9.90376.900694 3.01604 3.0157.90468.898155.834197.833463

16 1980 30.9039.9015 3.01604 3.013701.90468.899945.834197.834085 1981 31.903301.90370 3.01604 3.014838.90468.901759.834197.834688 198 3.903307.905169 3.01604 3.015975.90468.903589.834197.83575 1983 33.903311.9067 3.01604 3.017106.90468.90541.834197.835871 1984 34.903313.908076 3.01604 3.0185.90468.9074.834197.836503 1985 35.903315.909535 3.01604 3.01935.90468.909035.834197.837191 1986 36.903316.911 3.01604 3.00407.90468.910804.834197.837941 1987 37.903316.91459 3.01604 3.0147.90468.91544.834197.838741 1988 38.903317.913895 3.01604 3.0514.90468.9144.834197.839566 1989 39.903317.91588 3.01604 3.0354.90468.915877.834197.84038 1990 40.903317.91664 3.01604 3.04548.90468.917446.834197.841157 1991 41.903317.917898 3.01604 3.05537.90468.918945.834197.841894 199 4.903317.919117 3.01604 3.06507.90468.9038.834197.84597 1993 43.903317.9098 3.01604 3.0746.90468.91767.834197.84365 1994 44.903317.91459 3.01604 3.08401.90468.9318.834197.843901 1995 45.903317.9614 3.01604 3.0939.90468.94474.834197.844507 1996 46.903317.93763 3.01604 3.03047.90468.95813.834197.845083 1997 47.903317.94899 3.01604 3.03115.90468.97139.834197.84563 1998 48.903317.96014 3.01604 3.0304.90468.98439.834197.846161 1999 49.903317.97098 3.01604 3.03915.90468.99697.834197.846688 000 50.903317.98547 3.01604 3.033768.90468.930905.834197.8473 001 51.903317.99148 3.01604 3.034601.90468.93061.834197.847767 00 5.903317.930118 3.01604 3.035415.90468.933176.834197.8483 003 53.903317.931039 3.01604 3.0361.90468.93464.834197.848887 004 54.903317.93071 3.01604 3.036985.90468.935341.834197.849474 005 55.903317.9349 3.01604 3.037741.90468.936387.834197.850083 006 56.903317.933579 3.01604 3.03848.90468.937506.834197.850717 007 57.903317.93405 3.01604 3.039199.90468.93859.834197.85137 008 58.903317.93464 3.01604 3.0399.90468.939683.834197.85033 009 59.903317.93513 3.01604 3.040578.90468.940781.834197.8569 010 60.903317.935816 3.01604 3.04135.90468.941887.834197.853337 011 61.903317.936438 3.01604 3.04187.90468.94999.834197.853965 01 6.903317.937075 3.01604 3.04483.90468.944117.834197.854577 013 63.903317.9377 3.01604 3.04308.90468.9453.834197.85517 014 64.903317.938363 3.01604 3.043665.90468.946337.834197.85575 015 65.903317.938997 3.01604 3.04441.90468.94747.834197.856318 016 66.903317.93963 3.01604 3.04481.90468.948499.834197.85687

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 8541510.7 bu a his momen oal number of populaion is 16487718. 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. 304.. 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. 431. 4. Zill DD (1993) A Firs Course in Differenial Equaions. Figure 3: Comparing graph of heoreical daa wih original daa of Pakisan.

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