{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 259 34 "lab3.mws --- Logistic Growth Model" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart;\nwith( plots ):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Lab Overvi ew" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This lab introduces and inve stigates the " }{TEXT 256 21 "logistic growth model" }{TEXT -1 312 ". \+ The logistic model is a simple modification of the exponential growth \+ model. While the differential equations that define the exponential an d logisic models are quite similar, their solutions are noticeably dif ferent. These differences make the logistic model more appropriate for many biological applications." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 218 "We begin by reviewing the exponential mo del - using Maple. Then, the logistic model is motivated and formulate d. The questions will lead you to some of the important properties of \+ the solution to the logistic equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Exponential Growth Model with Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We fol low the approach used on the " }{URLLINK 17 "handout" 4 "http://www.ma th.sc.edu/~miller/142/expgr.pdf" "" }{TEXT -1 31 " for exponential gro wth models." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Model Formulation " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Beginning with the ``The Idea' ' section, let " }{XPPEDIT 18 0 "P(t)" "6#-%\"PG6#%\"tG" }{TEXT -1 31 " denote the population at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 30 ". Then, for the time interval " }{XPPEDIT 18 0 "[ t, t+dt ]" "6 #7$%\"tG,&F$\"\"\"%#dtGF&" }{TEXT -1 28 " the change in population is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dP := P(t+dt) - P(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 76 "The exponential growth model assumes the number of birt hs and deaths during " }{XPPEDIT 18 0 "[ t, t+dt ]" "6#7$%\"tG,&F$\"\" \"%#dtGF&" }{TEXT -1 87 " are proportional to the current size of the \+ population and the length of the interval:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "birth := beta*P(t)*dt;\ndeath := delta*P(t)*dt;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The balance law for this situatio n is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "q1 := dP = birth-de ath;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Eqauation (1) from the ha ndout is obtained when we use " }{XPPEDIT 18 0 "k=beta-delta" "6#/%\"k G,&%%betaG\"\"\"%&deltaG!\"\"" }{TEXT -1 32 " and the equation is divi ded by " }{XPPEDIT 18 0 "dt" "6#%#dtG" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "q2 := simplify( eval( q1, beta=k+de lta ) );\neq1 := q2/dt;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The di fferential equation is the limit of this equation as dt approaches 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q3 := limit( eq1, dt=0 ); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Note that the left-hand side of this equation stands for the deriviative of P at t. We can convert this to our standard notation as follows:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "odeE := convert( q3, diff );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Step -by-step derivation of the general solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "This differential equation can be solved by separating v ariables. Here, equation (2) is obtained by dividing the differential \+ equation by " }{XPPEDIT 18 0 "P(t)" "6#-%\"PG6#%\"tG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq2 := odeE / P(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 221 "To reproduce the steps in the solution of this different ial equation it is necessary to rewrite P(t) = P. Then, the left-hand \+ side is integrated with respect to P while the right-hand side is inte grated with respect to t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q4 := Int( 1/P, P ) = Int( k, t ) + C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The integrals are evaluated using" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "q5 := value( q4 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Applying the exponential function to both sides of this e quation yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q6 := exp ( lhs(q5) ) = exp( rhs(q5) );" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Two alternate approaches: the " }{HYPERLNK 17 "map" 2 "map" "" } {TEXT -1 5 " and " }{HYPERLNK 17 "isolate" 2 "isolate" "" }{TEXT -1 9 " commands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Note that the above command could also have been written more compactly - but somewhat le ss clearly - as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map( exp , q5 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "This command says to a pply the " }{TEXT 19 3 "exp" }{TEXT -1 47 " function to both sides of \+ the equation called " }{TEXT 19 2 "q5" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "A second alternative would be ask Maple to attempt to isolate P on one side of the equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "isolate( q5, P );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Note t hat all three approaches yield the exact same result." }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Initial conditions and the constant " } {XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "If an initial condition, " }{XPPEDIT 18 0 "P(0) = P[0]" " 6#/-%\"PG6#\"\"!&F%6#F'" }{TEXT -1 61 ", is given, this information ca n be used to find a value for " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 14 ". In general, " }{XPPEDIT 18 0 "exp(C) = P[0]" "6#/-%$expG6#%\" CG&%\"PG6#\"\"!" }{TEXT -1 8 " so that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q7 := C = ln( p[0] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "When this information is inserted into the formula for t he general solution (and simplified), we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify( eval( q6, q7 ) );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 8 " command" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The steps used to find these solut ions are combined into a single, very powerful, Maple command: " } {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 176 ". This command is u sed, in general, to attempt to find solutions to a differential equati on. For example, the general solution to the exponential growth model \+ can be found with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve ( odeE, P(t) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Note that Mapl e has written the solution in terms of an indpendent parameter, " } {TEXT 19 3 "_C1" }{TEXT -1 59 ". This parameter is related to our cons tant of integration " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 5 " by \+ " }{TEXT 19 3 "_C1" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "exp(C)" "6#-%$ex pG6#%\"CG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 57 " command can also handle initial conditions. For example," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ic := P(0) = p[0];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve( \{ odeE, ic \}, P(t) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This is obviously the same result as we obtained by more explicit steps." }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Logi stic Growth Model" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Description \+ of the Logistic Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "The fund amental difference between the exponential and logistic models can be \+ explained in terms of the per capita growth equation. For the exponent ial model the per capita growth rate is constant." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "per_capE := eq2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "However, in reality, the per capita growth rate decreases as the population increases" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(P(t),t)/P = g(P);" "6#/*&-%%DiffG6$-%\"PG6#%\"tGF+ \"\"\"F)!\"\"-%\"gG6#F)" }{TEXT -1 11 " where " }{XPPEDIT 18 0 "g( 0) = k" "6#/-%\"gG6#\"\"!%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "di ff( g(P), P ) *`<`* 0" "6#*(-%%diffG6$-%\"gG6#%\"PGF*\"\"\"%\" " 0 "" {MPLTEXT 1 0 457 "k := 2: \nL := 5:\np1 := plot( [ k, k-k/L*P ], P=0..L, tickmarks=[0,0], color= [red,blue], legend=[\"Exponential\",\"Logistic\"] ):\np2 := textplot( \+ [0,k+0.2,\"(0,k)\"], align=\{ABOVE\} ):\np3 := textplot( [L,0.2,\"(L,0 )\"], align=\{ABOVE\} ):\np4 := textplot( [-0.1,k/2,\"g(P)\"], align= \{LEFT\} ):\np5 := textplot( [L/2,-0.1,\"P\"], align=\{BELOW\} ):\n#di splay( [p1,p2,p3,p4,p5], title=\"Per capita growth rates\", scaling=co nstrained, view=[-0.2..6,-0.2..2.2] );\nunassign( 'k', 'L' ):" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 427 427 427 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"\"!F)$\"\"#F)7$$\"3GLLL3x&)* 3\"!#=F*7$$\"3umm\"H2P\"Q?F/F*7$$\"3MLL$eRwX5$F/F*7$$\"32ML$3x%3yTF/F* 7$$\"3emm\"z%4\\Y_F/F*7$$\"3`LLeR-/PiF/F*7$$\"3\\***\\il'pisF/F*7$$\"3 >MLe*)>VB$)F/F*7$$\"3X++DJbw!Q*F/F*7$$\"3%ommTIOo/\"!#j U6FKF*7$$\"37++]i^Z]7FKF*7$$\"32++](=h(e8FKF*7$$\"3/++]P[6j9FKF*7$$\"3 UL$e*[z(yb\"FKF*7$$\"3vmm;a/cq;FKF*7$$\"3$ommmJFKF*7$$\"3K+]i!f#=$3#FKF*7$$\"3>+](=xpe=#FKF*7$$ \"37nm\"H28IH#FKF*7$$\"3um;zpSS\"R#FKF*7$$\"3GLL3_?`(\\#FKF*7$$\"3fL$e *)>pxg#FKF*7$$\"33+]Pf4t.FFKF*7$$\"3uLLe*Gst!GFKF*7$$\"3/+++DRW9HFKF*7 $$\"39++DJE>>IFKF*7$$\"3F+]i!RU07$FKF*7$$\"3+++v=S2LKFKF*7$$\"3Jmmm\"p )=MLFKF*7$$\"3B++](=]@W$FKF*7$$\"35L$e*[$z*RNFKF*7$$\"3e++]iC$pk$FKF*7 $$\"3Zm;H2qcZPFKF*7$$\"3O+]7.\"fF&QFKF*7$$\"3Xmm;/OgbRFKF*7$$\"3w**\\i lAFjSFKF*7$$\"3yLLL$)*pp;%FKF*7$$\"3(RL$3xe,tUFKF*7$$\"3Cn;HdO=yVFKF*7 $$\"3`+++D>#[Z%FKF*7$$\"3RnmT&G!e&e%FKF*7$$\"3\"RLLL)Qk%o%FKF*7$$\"37+ ]iSjE!z%FKF*7$$\"3a+]P40O\"*[FKF*7$$\"\"&F)F*-%'COLOURG6&%$RGBG$\"*+++ +\"!\")F(F(-%'LEGENDG6#Q,Exponential6\"-F$6%7SF'7$F-$\"3xmmm\"p0k&>FK7 $F1$\"3FLL3FK7$F4$\"3bmm;Wp\"e(=FK7$F7$\"3hmm;4m(G$=FK7$F:$\"3QLL 3i.9!z\"FK7$F=$\"3dmmT!R=0v\"FK7$F@$\"3)****\\P8#\\4s%Ha\" FK7$FP$\"3/+++N*4)*\\\"FK7$FS$\"3z*****\\_&\\c9FK7$FV$\"3*)*****\\1aZT \"FK7$FY$\"3WmmT?)[oP\"FK7$Ffn$\"3CLLL=exJ8FK7$Fin$\"3=LLLtIf$H\"FK7$F \\o$\"3%****\\PYx\"\\7FK7$F_o$\"3KLLLB@')47FK7$Fbo$\"3&****\\P'psm6FK7 $Feo$\"3))***\\74_c7\"FK7$Fho$\"3(HLL3x%z#3\"FK7$F[p$\"37LL3s$QM/\"FK7 $F^p$\"3omm;zr)4+\"FK7$Fap$\"33mm;/K#*o&*F/7$Fdp$\"3!))***\\ih2&=*F/7$ Fgp$\"3]lmmT3^q()F/7$Fjp$\"3[)******HCAM)F/7$F]q$\"3k(****\\ZHK#zF/7$F `q$\"38(***\\P/$y^(F/7$Fcq$\"3b*****\\#RqnqF/7$Ffq$\"3uMLLL_CjmF/7$Fiq $\"3<)*****\\#*RJiF/7$F\\r$\"3enm;/E3SeF/7$F_r$\"3*e*****\\,F7aF/7$Fbr $\"3@LL$3(>t4]F/7$Fer$\"3w'***\\(ej*)e%F/7$Fhr$\"3=MLL$e&exTF/7$F[s$\" 3#4++v$4\"pu$F/7$F^s$\"3%Rmmm1?@L$F/7$Fas$\"3zimm\"\\Oz!HF/7$Fds$\"3@H L$3Pls[#F/7$Fgs$\"34'******H725#F/7$Fjs$\"31HLLe)ywl\"F/7$F]t$\"3(Hmmm YC9E\"F/7$F`t$\"3D&****\\PY$*Q)!#>7$Fct$\"3Ut***\\izbM%Fa^l7$FftF(-Fit 6&F[uF(F(F\\u-F`u6#Q)LogisticFcu-%%TEXTG6%7$F($\"#A!\"\"Q&(0,k)Fcu%+AL IGNABOVEG-F\\_l6%7$Fft$F+Fa_lQ&(L,0)FcuFc_l-F\\_l6%7$$Fa_lFa_l$\"\"\"F )Q%g(P)Fcu%*ALIGNLEFTG-F\\_l6%7$$\"+++++D!\"*F\\`lQ\"PFcu%+ALIGNBELOWG -%+AXESLABELSG6$Fg`lQ!Fcu-%&TITLEG6#Q8Per~capita~growth~ratesFcu-%*AXE STICKSG6$F)F)-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!\"#Fa_l$\"\"'F);F \\blF__l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 1 "Exp onential" "Logistic" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "That is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g := P -> k - k/L * P;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The logistic growth model can be written in per capita fo rm as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "per_capL := diff(P (t),t)/P(t) = g(P(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "or as \+ a differential equation in which the (absolute) growth rate is a quadr atic function of the current population size" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "odeL := per_capL * P(t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Solu tion of the Logistic Model" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 47 "Ste p-by-step derivation of the general solution" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 134 "This differential equation can be solved by separating variables. In this case it is necessary to divide the differential eq uation by " }{XPPEDIT 18 0 "P(t)*(1-P(t)/L)" "6#*&-%\"PG6#%\"tG\"\"\", &F(F(*&-F%6#F'F(%\"LG!\"\"F.F(" }{TEXT -1 45 " -- note that it is esse ntial that you write " }{XPPEDIT 18 0 "P(t)" "6#-%\"PG6#%\"tG" }{TEXT -1 75 " if you expect Maple to cancel common factors and to simplify t he equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "r1 := simpl ify( odeL / ( P(t) * (1-P(t)/L) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "The left-han d side is integrated with respect to P while the right-hand side is in tegrated with respect to t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r2 := Int( L/(P*(L-P)), P ) = Int( k, t ) + C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "Unlike the exponential model, we do not yet know how to evaluate the integral on the left-hand side of this equation. \+ But, using Maple, the integrals are found to be" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "r3 := value( r2 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Applying the exponential function to both sides of this e quation yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r4 := exp ( lhs(r3) ) = exp( rhs(r3) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Since the exponential and logarithm functions are inverse functions, \+ the combinations on the left-hand side of the equation should simplify :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r5 := simplify( r4 ); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "With a little more algebra it is possible to solve for P:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r6 := isolate( r5, P );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "Initial conditions and the constant " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "If an initial condition, " }{XPPEDIT 18 0 "P(0) = P[0]" "6#/-%\"PG6#\"\"!&F%6#F'" }{TEXT -1 61 ", is given, th is information can be used to find a value for " }{XPPEDIT 18 0 "C" "6 #%\"CG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " r7 := eval( r6, [P=p[0], t=0] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r8 := isolate( r7, C );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "When this information is inserted into the formula for t he general solution (and simplified), we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r9 := simplify( eval( r6, r8 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 8 " command " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 62 " command can find the general solutions to t he logistic model:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "s1 := dsolve( odeL, P(t) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This so lution looks somewhat similar to -- but also somewhat different from - - the one found above" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "r9; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 57 " command can also handle initial condition s. For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ic := P( 0) = p[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "s2 := dsolve( \{ odeL, ic \}, P(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "s3 := simplify( s2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Analysis of the Solution" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 382 "While the logistic growth model i s not much different from the exponential growth model, the logistic s olution is significantly more complicated. It is no longer possible to simply look at the solution and know almost everything about the solu tion. To begin to understand the solution to the logistic model, defin e a Maple function that returns the solution for a specific value of \+ " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L" "6# %\"LG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solP := un apply( rhs(r9), (k,L,p[0]) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " For example, the solution to the logistic model with intrinsic growth \+ rate " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 23 ", threshold population " }{XPPEDIT 18 0 "L=10" "6#/%\"LG\"#5" }{TEXT -1 25 ", and initial population " }{XPPEDIT 18 0 "p[0]=3" "6#/&%\"pG6#\"\"!\"\"$" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solP(1, 10,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "One way to use this solution function is to pl ot a collection of curves with different initial populations." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "p0_vals := [0, 50, 100, 150, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot( [ seq( solP(0.0005,100 0,p0), p0=p0_vals) ], t=0..20000 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "Notice that \+ solutions with the smaller initial values have one inflection point ( where the concavity changes from concave up to concave down). One int eresting property of the logistic growth model is that the " }{TEXT 257 4 "time" }{TEXT -1 73 " at which the population passes through thi s inflection point depends on " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 10 ", but the " }{TEXT 258 4 " size" }{TEXT -1 59 " of the population at the inflection point depends only on " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 100 ". To better und estand this property we need to work with the second derivative of the solution: P''(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 2 ")." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 "Approach # 1: Differentiate the e xplicit solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "One approach i s to differentiate the explicit solution." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "D2Pa := diff( rhs(r9), t, t );" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 105 "This involves the application of the quotient rul e twice and produces a real mess! The mess does simplify" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u1 := simplify( D2Pa );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "but I do not want to think about setting this equal to 0 and solving for t! We can ask Maple to attempt to " } {HYPERLNK 17 "factor" 2 "factor" "" }{TEXT -1 23 " the second derivati ve:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u2 := factor( D2Pa ) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "This is much better. I leav e it to you to solve for the times when solutions change concavity, i. e., change concavity. (Once you find the time when the solutions have \+ an inflection point, you should compute the corresponding population s ize. Does this agree with the result found below using Approach #2?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 53 "Approach \+ # 2: Differentiate the differential equation" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 332 "Many differential equations cannot be solved explicitl y. Even if an explicit solution is not known it is still possible to d o a lot of analysis of the solution. For example, critical points in a solution occur whenever the first derivative is zero. In the case of \+ the logistic growth model this means that critical points occur when" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "crit_pt_eq := rhs(odeL) = 0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The solution(s) to this eq uation are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "crit_pt := so lve( crit_pt_eq, \{P(t)\} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 " Note that these are the two constant solutions to the logistic growth \+ model. Similarly, inflection points can be found by differentiating th e differential equation with respect to t:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "d1 := diff( odeL, t );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "and then using the differential equation to replace all \+ occurrences of diff( P(t), t ) on the right-hand side:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d2 := lhs(d1) = eval( rhs(d1), odeL );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d3 := simplify( d2 ) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The possible inflection poin ts are now found to occur when" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "poss_infl_pt := solve( rhs(d3)=0, \{P(t)\} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "It is not surprising to see the two cons tant solutions. The last solution is new - and interesting. (Is this c onsistent with your findings from Approach #1?)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 13 "Lab Questions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "1. Consider the logistic growth model with " }{XPPEDIT 18 0 "k" "6 #%\"kG" }{TEXT -1 9 "=0.0005, " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 11 "=1000, and " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 63 "=200. Estimate the time at which the population reaches 95% of " } {XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "2. Consider the logistic growth model with " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 12 "=0.0005 and " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 51 "=1000. Plot the solutions with initial populations " }{XPPEDIT 18 0 "p[0]" " 6#&%\"pG6#\"\"!" }{TEXT -1 111 " = 200, 400, 600, 800, and 1000 on the time interval [0, 20000]. Copy and paste this plot into a Word docume nt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 43 "3. Consider the logistic growth model with " } {XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 10 "=1000 and " }{XPPEDIT 18 0 " p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 30 "=200. Plot the solutions with " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 114 " = 0.0005, 0.001, 0.003, a nd 0.005 on the time interval [0, 10000]. Copy and paste this plot int o a Word document." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "4. Consider the logistic growth mo del with " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 12 "=0.0005 and " } {XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 30 "=200. Plot the so lutions with " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 110 "=100, 200, \+ 300, 400, and 500 on the time interval [ 0, 10000 ]. Copy and paste th is plot into a Word document." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "5. Consider the five cur ves in Question 4. For each solution, what is the population size when " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 101 "=10000? Express each an swer as the population size and as a percentage of the corresponding v alue of " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "Essay Question. Find a formula for the time when the solution \+ to the logisitic growth model has an inflection point. Use this formul a to explain why solutions with " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\" \"!" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "L/2" "6#*&%\"LG\"\"\"\"\"#!\"\" " }{TEXT -1 31 " do not have inflection points." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }