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# Class Note for MATH 3331 with Professor He at UH 2

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This 15 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Principle Principle MOM1 Principle Develop model function ftx for the rate of change of a variable a gt First order ODE x ftx I Iii F Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 2 15 l Section 31 Malthusian n Cass Exercises Loistic Loistic ln Class Exer Modeling Population Growth Malthusian Model v a zc Gv Y09 1Alb DJ 5 M quotJain Io 1325 Population of species 4 P 7 Vs lt bacteria US pop 4 0 i rt 2 lnPtP Model dpj dt fP O r n 39 Use this to determine 0 Malthusian model rm39d wk 7 if PO Pt1 P1 are 0 fP 7 g dzconst 5quot given 19 birth rate d death rate 15 if 7313073 are given AJ quot go a cm I dP ti and t is sought st 4 v v J o 5 v Mhquot Pgtllt o Solution 0 V49 V b at 2 quot 7o Pt Poe PO 130 P ha lt f G9 F1 39 o gl tu 3 balm Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 3 15 a Section 31 Malthusian In Class Exercises Lo istic Loistic ln Class Exe Example 039quot 5 H Ex rO1 1302103 gt WUP t w P10 27 x 103 03 evol39t39 6 P100 22 x 10 rapid growth gt see text p 125 I 3039 3w 3 39l39w g 2 quotE20 8 393 C1 0 g 1 E 1039 e vurmh a 8 c3 0 5 10 0 50 100 Time in hours Time in hours Figure 1 Exponential growth of a Figure 2 Exponential growth over H population a longer period of time Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 4 15 Section 31 Malthusl39an ln Class Exercises Loistic Loistic lnClass Exe Example 314 Evaluating the Parameters EX At t 0 P0 10 cells After 1 day P1 25 cells Q number of cells after 10 days 7quot 11 n2510 09163day gt P10 10610X0399613 m 954 cells 01 l wscfi Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 5 15 Section 31 Malthusian ln Class Exercises Loistic Loistic ln Class Exe Section 32 Models ande al v orld 20 1 h b y f Halquotthou Vii Ir 4 My Popuh onintn ons 0 I I I I I 1790 1800 1810 1820 1830 1840 Year Figure 1 Fitting a Malthusian model to early US population Iii Jiwen He University of Houston Math 3331 VSection 19470 Lecture 10 February 11 2009 6 15 7 Section 31 Exercise 312 Ex 2 A cell culture is grown at t 0 After 751 1 day P1 1000 After 752 2 days P2 3000 Q P0 2 m2 t1 nP2P1 in 11 n30001000 1099day P0 P1a7 xl 100051099 3 333 90 14 J 5030i Jiwen He University of Houston Math 3331 Section 19470 Lecture IQ February 11 2009 7 15 Section 31 Malthusian ln Class Exercises Loistic Loistic In Class Exer Exercise 314 Ex 4 Doubling Time Given td st Ptd 2P0 gt Poem 2P0 gt rtd In 2 gt 7quot In 2td Q Given td 10 days and P0 1000 find ti st Pt 10000 E P td 10days gt 7 In 210 00693day gt ti 1r InPP0 In 1000693 m 33 days Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 8 15 ii Section 31 Modeling Population Growth Logistic Mod Model dPdtP r39 aP l 4 49 axienc 539 oifa 0 Set K 7401 gt dP Janna DE E TPlt1 PK E fPC duau O 5 3 an L 1 f s v P a p CQ alitative Analysis 0 Equilibria P o gt P P in P o f O 7 gt 0 macaw a gt unstable K P PZKI fK 7 ltO 6 gt asympt stabe K carrying capacity or H eventual population A Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 9 15 Section 31 Malthusian n Cass Exercises Loistic Loistic n Cass Exer Solution of the Logistic Equation Model dPdtP 7 CLP Set K Ta Solutlon of 1 dP Pt KPO rP1 PK 2 fP Po K Poe rt dt 1 2 Derivation of 2 SoV dPP1 PK 1p 1P KdPrdt nP nK PnPK Prt C gt PK P49rt Fort0 PQK Pg 2A PoK PPK Po e 7 t 2 Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 10 15 1 Section 31 Malthusian In Class Exercises Loistic Loistic nCass Exer Evaluating the Parameters in the Logistic Equation Computing Parameters If K P0 15 h P1 Ph 39 If Po t h P1 PM are known P2 P2h are known KP P1 P K OP m T 1nP2P1 Po e 10 P K OP h 130032 131 1 O gt In rh PO Plerh P1P21 e rh P1 PQe rh ill Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 11 15 Section 31 Malthusian lnClass Exercises Lo istic Loistic ln Class Exer Section 32 Models and the Real World 100 80 60 40 Population in millions 20 0 I I I I 1790 1820 1850 1880 1910 Year Figure 2 Fitting the logistic model to US population F1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 12 15 739 Section 31 Malthusian ln Class Exercises Loistic Lo isti39c ln Class Exer Section 32 Models and the Real World Popula onlnln hons G c k 31 O I I I I 1800 1850 1900 1950 2000 Year Figure 3 Logistic model projection of US population I Jiwen He University of Houston Math 3331 Section 10470 Lecture 10 February 11 2009 13 15 v Section 31 Malthusian ln Class Exercises Loistic Loistic InaClass Exe39 Exercise 3112 EX 12 Given K 20000 P0 1000 and P1 P8 hrs 2 1200 find 7 and 75 st 1305 3K4 15 000 1220 1106 7quot 18nm 00241hr p KPoP0K PO W s 75 lt1rgtlnltgt 1 I 1520 1106 00241 n120 15106 z 7222hrs 3911 Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 14 15 Section 31 Malthusian In Class Exercises Loistic Loistic In Class Exer Exercise 3114 Ex 14 modified Given P0 100 P1 P2O hrs 2 476 P2 P4O hrs 1986 find r and K 1 1986476 100 In T 20 1001986 476 00799 476 1001 600820 gt1 100 476eO0820 10136 22 Jiwen He University of Houston Math 3331 Section 19470 Lecture 10 February 11 2009 15 39 15 Legtuve 10 31mm mum ow Wm H Wm M WM WW M mm quotA divlavamZ mm

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