{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 46 "Second Order Systems of Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 236 "This section is designed to show how Maple can be u sed to numerically solve second order differential equations. It also \+ shows how Maple provides tools to view the direction field of these eq uations and observe the qualitative behavior." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Consider a predator-prey system with rabbits, R(t), and foxes, F(t). The classic Lotka-Volterr a system describing this interaction is given by the following system \+ of differential equations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Rab := diff(R(t),t) = 2*R(t) - 1.2*R(t)*F(t);\nFox := diff(F(t),t) = -F(t) + 0.9*R(t)*F(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RabG/- %%diffG6$-%\"RG6#%\"tGF,,&F)\"\"#*($\"#7!\"\"\"\"\"F)F3-%\"FGF+F3F2" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FoxG/-%%diffG6$-%\"FG6#%\"tGF,,&F) !\"\"*($\"\"*F.\"\"\"-%\"RGF+F2F)F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "The way you solve this system takes advantage of Maple's dsolve with the numeric option. The dsolve begins with the system of \+ equations and initial conditions inside the set brackets. Then the var iables solved are listed, which are followed by the numeric solving op tion and output being a procedure." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "sys := dsolve(\{Rab, Fox, R(0)=1,F(0) = 0.5\}, \{R(t) ,F(t)\}, type = numeric, output = listprocedure); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$sysG7%/%\"tGR6#F'6\"6#%aoCopyright~(c)~1993~by~the ~University~of~Waterloo.~All~rights~reserved.GF*9$F*F*F*/-%\"FGF)RF)6& %(rkf45_sG%)outpointG%#r1G%#r2GF+F*C&>8%-%&evalfG6#F-@$52-%$absG6#,$F9 !\"\"-FA6#,&&%,loc_controlG6#\"\"#\"\"\"F9FD4-%'memberG6$&FI6#\"\"'<*! \"#FDFLFK$FL\"\"!$FKFW$FDFW$FUFWC%>FI-%%copyG6#=F*6#;FL\"#LE\\[lBFLFKF K$FWFW\"\"$F_o\"\"%$FL!\")\"\"&FboFSFL\"\"($FL!\"*\"\")\"&++$\"\"*\"%+ 5\"#5FW\"#6FW\"#7FW\"#8FW\"#9FW\"#:FW\"#;FW\"#FW\"#?$FdoF D\"#@FV\"#AFW\"#BFW\"#CFW\"#DFW\"#EFW\"#FFW\"#GFW\"#HFW\"#IFW\"#JFW\"# KFWF]oFW>%'loc_y0G-Fhn6#=F*6#;FLFKE\\[l#FLFgpFKFV>%'loc_y1G-Fhn6#=F*Fi qE\\[l!@$0F@FWC$>&FI6#F`oF9@%1%'DigitsG-%'evalhfG6#FjrC$>8$-%*traperro rG6#-F\\s6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FI-F[t6# Feq-F[t6#F]r-F[t6#%'loc_F1G-F[t6#%'loc_F2G-F[t6#%'loc_F3G-F[t6#%'loc_F 4G-F[t6#%'loc_F5G-F[t6#%)loc_workG@$/F`s%*lasterrorGC%>8&-%+searchtext G6$.F\\s-%(convertG6$-%#opG6$FL7#F`s%%nameG>8'-Fju6$.%)hardwareGF]v@%5 0FhuFW0FfvFW-Fgs6,FisFIFeqF]rFctFftFitF\\uF_uFbu-%&ERRORG6#F`sF_w&Feq6 #FLF*6%FIFeqF]rF*/-%\"RGF)RF)F2F+F*C&>F9F:@$F>C%>FI-Fhn6#=F*F[oE\\[lBF LFKFKF_oF`oF_oFaoFboFdoFboFSFLFeoFfoFhoFioFjoF[pF\\pFWF]pFWF^pFWF_pFWF `pFWFapFWFbpFWFcpFWFdpFWFepFWFfpFgpFhpFVFipFWFjpFWF[qFWF\\qFWF]qFWF^qF WF_qFWF`qFWFaqFWFbqFWFcqFWF]oFW>Feq-Fhn6#=F*FiqE\\[l#FLFgpFKFV>F]r-Fhn 6#=F*FiqE\\[l!@$FcrC$>FfrF9@%FirC$>F`sFas@$FduC%>FhuFiu>FfvFgv@%F\\wF_ wFawF_w&FeqFJF*FfwF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Next we w ant to isolate the solutions for R(t) and F(t) for further analysis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "pR := subs(sys, R(t)); \n pF := subs(sys, F(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pRGR6#%\" tG6&%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCopyright~(c)~1993~by~the~Univ ersity~of~Waterloo.~All~rights~reserved.G6\"C&>8%-%&evalfG6#9$@$52-%$a bsG6#,$F2!\"\"-F;6#,&&%,loc_controlG6#\"\"#\"\"\"F2F>4-%'memberG6$&FC6 #\"\"'<*!\"#F>FFFE$FF\"\"!$FEFQ$F>FQ$FOFQC%>FC-%%copyG6#=F/6#;FF\"#LE \\[lBFFFEFE$FQFQ\"\"$Fin\"\"%$FF!\")\"\"&F\\oFMFF\"\"($FF!\"*\"\")\"&+ +$\"\"*\"%+5\"#5FQ\"#6FQ\"#7FQ\"#8FQ\"#9FQ\"#:FQ\"#;FQ\"# FQ\"#?$F^oF>\"#@FP\"#AFQ\"#BFQ\"#CFQ\"#DFQ\"#EFQ\"#FFQ\"#GFQ\"#HFQ\"#I FQ\"#JFQ\"#KFQFgnFQ>%'loc_y0G-FX6#=F/6#;FFFEE\\[l#FFFapFEFP>%'loc_y1G- FX6#=F/FcqE\\[l!@$0F:FQC$>&FC6#FjnF2@%1%'DigitsG-%'evalhfG6#FdrC$>8$-% *traperrorG6#-Ffr6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6# FC-Fes6#F_q-Fes6#Fgq-Fes6#%'loc_F1G-Fes6#%'loc_F2G-Fes6#%'loc_F3G-Fes6 #%'loc_F4G-Fes6#%'loc_F5G-Fes6#%)loc_workG@$/Fjr%*lasterrorGC%>8&-%+se archtextG6$.Ffr-%(convertG6$-%#opG6$FF7#Fjr%%nameG>8'-Fdu6$.%)hardware GFgu@%50FbuFQ0F`vFQ-Fas6,FcsFCF_qFgqF]tF`tFctFftFitF\\u-%&ERRORG6#FjrF iv&F_qFDF/6%FCF_qFgqF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pFGR6#%\" tG6&%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCopyright~(c)~1993~by~the~Univ ersity~of~Waterloo.~All~rights~reserved.G6\"C&>8%-%&evalfG6#9$@$52-%$a bsG6#,$F2!\"\"-F;6#,&&%,loc_controlG6#\"\"#\"\"\"F2F>4-%'memberG6$&FC6 #\"\"'<*!\"#F>FFFE$FF\"\"!$FEFQ$F>FQ$FOFQC%>FC-%%copyG6#=F/6#;FF\"#LE \\[lBFFFEFE$FQFQ\"\"$Fin\"\"%$FF!\")\"\"&F\\oFMFF\"\"($FF!\"*\"\")\"&+ +$\"\"*\"%+5\"#5FQ\"#6FQ\"#7FQ\"#8FQ\"#9FQ\"#:FQ\"#;FQ\"# FQ\"#?$F^oF>\"#@FP\"#AFQ\"#BFQ\"#CFQ\"#DFQ\"#EFQ\"#FFQ\"#GFQ\"#HFQ\"#I FQ\"#JFQ\"#KFQFgnFQ>%'loc_y0G-FX6#=F/6#;FFFEE\\[l#FFFapFEFP>%'loc_y1G- FX6#=F/FcqE\\[l!@$0F:FQC$>&FC6#FjnF2@%1%'DigitsG-%'evalhfG6#FdrC$>8$-% *traperrorG6#-Ffr6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6# FC-Fes6#F_q-Fes6#Fgq-Fes6#%'loc_F1G-Fes6#%'loc_F2G-Fes6#%'loc_F3G-Fes6 #%'loc_F4G-Fes6#%'loc_F5G-Fes6#%)loc_workG@$/Fjr%*lasterrorGC%>8&-%+se archtextG6$.Ffr-%(convertG6$-%#opG6$FF7#Fjr%%nameG>8'-Fdu6$.%)hardware GFgu@%50FbuFQ0F`vFQ-Fas6,FcsFCF_qFgqF]tF`tFctFftFitF\\u-%&ERRORG6#FjrF iv&F_q6#FFF/6%FCF_qFgqF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We wa nt to view the time varying solutions for the rabbits and foxes." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Rabt := [seq([0.1*n, pR(0.1* n)], n = 0..200)]:\nplot(Rabt, color = RED);" }}{PARA 13 "" 1 "" {GLPLOT2D 497 327 327 {PLOTDATA 2 "6&-%'CURVESG6#7ew7$$\"\"!F)$\"\"\"F )7$$\"3/+++++++5!#=$\"3AG*\\?Io/:\"!#<7$$\"35+++++++?F/$\"3$o8u@k#GB8F 27$$\"3))**************HF/$\"3u*>\\%>_F?:F27$$\"3A+++++++SF/$\"3/8$4>F fDu\"F27$$\"3++++++++]F/$\"3C=#RBF)*)*)>F27$$\"3w**************fF/$\"3 9\"yDM&[ufAF27$$\"3a**************pF/$\"3q?L^t,%ea#F27$$\"3U+++++++!)F /$\"3,'3))[F7j$GF27$$\"3A+++++++!*F/$\"3&Q[cyi>96$F27$F*$\"3`k9`!pT;M$ F27$$\"33+++++++6F2$\"3mB)y!)[=z[$F27$$\"3%**************>\"F2$\"3Cy%>K)e?b>F/7$$\"3!***************GF2$\"3rK]t(p%)G%=F/7$$ \"\"$F)$\"3?@.JiMSt3THCX'RF\"e;X@F/7$$ \"3#)*************z$F2$\"3'p)R`RF-=BF/7$$\"3!***************QF2$\"3Dks gL)f#GDF/7$$\"\"%F)$\"3YSsw'ze3y#F/7$$\"3k*************4%F2$\"33W=S:R# >3$F/7$$\"3;+++++++UF2$\"34[ZTl.!)QMF/7$$\"3#)*************H%F2$\"3e*4 X\")z8-'QF/7$$\"3M+++++++WF2$\"3IZXI\"zCkN%F/7$$\"3++++++++XF2$\"3%4Q# )4b)QR\\F/7$$\"3k*************f%F2$\"3R$GZ%*o/Hi&F/7$$\"3;+++++++ZF2$ \"3Xkex&[o#HW%)F/7$$\"\"&F)$\"3&HJ:Y()ypq*F/7$$\"3k*************4& F2$\"3eHJV'o)o;6F27$$\"3;+++++++_F2$\"3QPq@:[e%G\"F27$$\"3#)********** ***H&F2$\"3Qrj10NJw9F27$$\"3M+++++++aF2$\"3#Rw0]U%=$p\"F27$$\"3+++++++ +bF2$\"3#)3R0(\\:`$>F27$$\"3k*************f&F2$\"3:MG&*z7v+AF27$$\"3;+ ++++++dF2$\"3=5in-S;%[#F27$$\"3#)*************z&F2$\"3WD&4w\"p0vFF27$$ \"3M+++++++fF2$\"3?J]\"Re]b0$F27$$\"\"'F)$\"3+i1:x\\H)H$F27$$\"3k***** ********4'F2$\"39G4Vb(GgY$F27$$\"3;+++++++iF2$\"3&p\"f**y`q:NF27$$\"3# )*************H'F2$\"3*yRz$[C_5MF27$$\"3M+++++++kF2$\"3P>`i/a=RJF27$$ \"3++++++++lF2$\"33C:/(>\"pJFF27$$\"3k*************f'F2$\"3mSl^7duaAF2 7$$\"3;+++++++nF2$\"3Wb%p8!z!Ry\"F27$$\"3#)*************z'F2$\"3]fa?v? 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AF/7$$\"3')*************)=F][m$\"3KT!G!zbl*Q#F/7$$\"#>F)$\"3plbC%R7Yh# F/7$$\"39++++++5>F][m$\"37\"yW6)y+%)GF/7$$\"3$*************>>F][m$\"3% )))[4C6Q/KF/7$$\"32++++++I>F][m$\"3akke80c$e$F/7$$\"3')************R>F ][m$\"3E*)3P'e43.%F/7$$\"3+++++++]>F][m$\"3!*QxWAZ)pb%F/7$$\"39++++++g >F][m$\"3OO!)=)Q$pu^F/7$$\"3%*************p>F][m$\"3]Y]_i\\V)*eF/7$$\" 33++++++!)>F][m$\"3!HQbod&pWnF/7$$\"3')*************)>F][m$\"3Pd^#f'=& >t(F/7$$\"#?F)$\"3!z)*=e580)))F/-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-% +AXESLABELSG6$Q!6\"Fijn-%%VIEWG6$%(DEFAULTGF^[o" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Foxt := [seq([0.1*n, pF(0.1*n)], n = 0..200)]:\n plot(Foxt, color = BLUE);" }}{PARA 13 "" 1 "" {GLPLOT2D 489 322 322 {PLOTDATA 2 "6&-%'CURVESG6#7ew7$$\"\"!F)$\"3++++++++]!#=7$$\"3/+++++++ 5F,$\"3a\"fY?A&3$)\\F,7$$\"35+++++++?F,$\"3wanK:c#*Q]F,7$$\"3))******* *******HF,$\"3!zVE:YM3=&F,7$$\"3A+++++++SF,$\"3x]))o!Hs\"GaF,7$F*$\"3i ?(4$R*Q)3eF,7$$\"3w**************fF,$\"3Wl&>x6lFO'F,7$$\"3a*********** ***pF,$\"3Mm[2dj`YrF,7$$\"3U+++++++!)F,$\"3+*HX\"R**yQ#)F,7$$\"3A+++++ ++!*F,$\"3=c9gI\")fW(*F,7$$\"\"\"F)$\"3#*RWP<%y$z6!#<7$$\"33+++++++6Fg n$\"3AV@d!QWAX\"Fgn7$$\"3%**************>\"Fgn$\"3kkt3Mb<-=Fgn7$$\"3/+ ++++++8Fgn$\"3)yZm?s(fCAFgn7$$\"3!**************R\"Fgn$\"3-z<3^`B\"p#F gn7$$\"3++++++++:Fgn$\"3')3V>ber[JFgn7$$\"33+++++++;Fgn$\"3%472z$>-MNF gn7$$\"3%**************p\"Fgn$\"3C9\\kGV&)*z$Fgn7$$\"3/+++++++=Fgn$\"3 AgHxo9`IRFgn7$$\"3!***************=Fgn$\"3B@]6\"H())QRFgn7$$\"\"#F)$\" 3yo,9z4Z_QFgn7$$\"33+++++++@Fgn$\"3#[rWEQz4q$Fgn7$$\"3;+++++++AFgn$\"3 BwV9^6i4NFgn7$$\"3#)*************H#Fgn$\"3!f'))eVUH(H$Fgn7$$\"3!****** ********R#Fgn$\"3CVTL=8:xIFgn7$$\"3++++++++DFgn$\"3]k'z=)G*y&GFgn7$$\" 33+++++++EFgn$\"3sd?!3PI]k#Fgn7$$\"3;+++++++FFgn$\"3d@2#Q:&)=W#Fgn7$$ \"3#)*************z#Fgn$\"3Kr?Fgn7$$\"\"$F)$\"3LCJ%oX$y/>Fgn7$$\"33+++++++JFgn$\"3ag$='*3 R4v\"Fgn7$$\"3;+++++++KFgn$\"3@twT*Gh#4;Fgn7$$\"3#)*************H$Fgn$ \"3KW?ILY?z9Fgn7$$\"3!**************R$Fgn$\"3k'y'*RoV,O\"Fgn7$$\"3++++ ++++NFgn$\"33gT6V'=9D\"Fgn7$$\"33+++++++OFgn$\"37@29qeO_6Fgn7$$\"3;+++ ++++PFgn$\"3'yxCVAQB1\"Fgn7$$\"3#)*************z$Fgn$\"3*z47@$f=2)*F,7 $$\"3!***************QFgn$\"3kv\"eF=*Gp!*F,7$$\"\"%F)$\"30CtaAMO/%)F,7 $$\"3k*************4%Fgn$\"3w365?md2yF,7$$\"3;+++++++UFgn$\"3?Hy.(**HY F(F,7$$\"3#)*************H%Fgn$\"3$z5\"Q&e%z,oF,7$$\"3M+++++++WFgn$\"3 ?s,$p$[&fQ'F,7$$\"3++++++++XFgn$\"3M\\a&\\(pmCgF,7$$\"3k*************f %Fgn$\"3s-41.8C;dF,7$$\"3;+++++++ZFgn$\"3sqBG0/')faF,7$$\"3#)********* ****z%Fgn$\"3eG#p>\"*\\dD&F,7$$\"3M+++++++\\Fgn$\"3&[*G5n`U0^F,7$$\"\" &F)$\"3UP?$*Qi#=d pF,7$$\"3#)*************z&Fgn$\"31'fQ<`8g(zF,7$$\"3M+++++++fFgn$\"3') \\WjF&yPQ*F,7$$\"\"'F)$\"3/AbEB*z08\"Fgn7$$\"3k*************4'Fgn$\"3# *QbL\"ewzQ\"Fgn7$$\"3;+++++++iFgn$\"3?(Q-dQ\"G@Qa*o#Fgn7$$\"3;+++++++xFgn$\"3'p%3&eWPT[#Fgn7$$\"3#)****** *******z(Fgn$\"3?F+v?H+!H#Fgn7$$\"3M+++++++zFgn$\"3`)GVV4)>3@Fgn7$$\" \")F)$\"3!oq@A*)*3R>Fgn7$$\"3k*************4)Fgn$\"3%G$\\\"R*=g#y\"Fgn 7$$\"3G*************>)Fgn$\"3Z53[H,QQ;Fgn7$$\"3q+++++++$)Fgn$\"3j,0)4. /f]\"Fgn7$$\"3M+++++++%)Fgn$\"3YP4R;2c%Q\"Fgn7$$\"3++++++++&)Fgn$\"3U; 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For both solutions together, we n eed the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "with( plots):\nA := plot(Rabt, color = RED):\nB := plot(Foxt, color = BLUE): \ndisplay(\{A, B\});" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 13 "" 1 "" {GLPLOT2D 526 330 330 {PLOTDATA 2 "6&-%'CURVESG6$7ew7$$\"\"!F)$\"3++++++++]!#=7$$\"3 /+++++++5F,$\"3a\"fY?A&3$)\\F,7$$\"35+++++++?F,$\"3wanK:c#*Q]F,7$$\"3) )**************HF,$\"3!zVE:YM3=&F,7$$\"3A+++++++SF,$\"3x]))o!Hs\"GaF,7 $F*$\"3i?(4$R*Q)3eF,7$$\"3w**************fF,$\"3Wl&>x6lFO'F,7$$\"3a*** ***********pF,$\"3Mm[2dj`YrF,7$$\"3U+++++++!)F,$\"3+*HX\"R**yQ#)F,7$$ \"3A+++++++!*F,$\"3=c9gI\")fW(*F,7$$\"\"\"F)$\"3#*RWP<%y$z6!#<7$$\"33+ ++++++6Fgn$\"3AV@d!QWAX\"Fgn7$$\"3%**************>\"Fgn$\"3kkt3Mb<-=Fg n7$$\"3/+++++++8Fgn$\"3)yZm?s(fCAFgn7$$\"3!**************R\"Fgn$\"3-z< 3^`B\"p#Fgn7$$\"3++++++++:Fgn$\"3')3V>ber[JFgn7$$\"33+++++++;Fgn$\"3%4 72z$>-MNFgn7$$\"3%**************p\"Fgn$\"3C9\\kGV&)*z$Fgn7$$\"3/++++++ +=Fgn$\"3AgHxo9`IRFgn7$$\"3!***************=Fgn$\"3B@]6\"H())QRFgn7$$ \"\"#F)$\"3yo,9z4Z_QFgn7$$\"33+++++++@Fgn$\"3#[rWEQz4q$Fgn7$$\"3;+++++ ++AFgn$\"3BwV9^6i4NFgn7$$\"3#)*************H#Fgn$\"3!f'))eVUH(H$Fgn7$$ \"3!**************R#Fgn$\"3CVTL=8:xIFgn7$$\"3++++++++DFgn$\"3]k'z=)G*y &GFgn7$$\"33+++++++EFgn$\"3sd?!3PI]k#Fgn7$$\"3;+++++++FFgn$\"3d@2#Q:&) =W#Fgn7$$\"3#)*************z#Fgn$\"3Kr?Fgn7$$\"\"$F)$\"3LCJ%oX$y/>Fgn7$$\"33+++++++JFgn$ \"3ag$='*3R4v\"Fgn7$$\"3;+++++++KFgn$\"3@twT*Gh#4;Fgn7$$\"3#)********* ****H$Fgn$\"3KW?ILY?z9Fgn7$$\"3!**************R$Fgn$\"3k'y'*RoV,O\"Fgn 7$$\"3++++++++NFgn$\"33gT6V'=9D\"Fgn7$$\"33+++++++OFgn$\"37@29qeO_6Fgn 7$$\"3;+++++++PFgn$\"3'yxCVAQB1\"Fgn7$$\"3#)*************z$Fgn$\"3*z47 @$f=2)*F,7$$\"3!***************QFgn$\"3kv\"eF=*Gp!*F,7$$\"\"%F)$\"30Ct aAMO/%)F,7$$\"3k*************4%Fgn$\"3w365?md2yF,7$$\"3;+++++++UFgn$\" 3?Hy.(**HYF(F,7$$\"3#)*************H%Fgn$\"3$z5\"Q&e%z,oF,7$$\"3M+++++ ++WFgn$\"3?s,$p$[&fQ'F,7$$\"3++++++++XFgn$\"3M\\a&\\(pmCgF,7$$\"3k**** *********f%Fgn$\"3s-41.8C;dF,7$$\"3;+++++++ZFgn$\"3sqBG0/')faF,7$$\"3# )*************z%Fgn$\"3eG#p>\"*\\dD&F,7$$\"3M+++++++\\Fgn$\"3&[*G5n`U0 ^F,7$$\"\"&F)$\"3UP?$*Qi#=dpF,7$$\"3#)*************z&Fgn$\"31'fQ<`8g(zF,7$$\"3M+++++++ fFgn$\"3')\\WjF&yPQ*F,7$$\"\"'F)$\"3/AbEB*z08\"Fgn7$$\"3k************* 4'Fgn$\"3#*QbL\"ewzQ\"Fgn7$$\"3;+++++++iFgn$\"3?(Q-dQ\"G@Qa*o#Fgn7$$\"3;+++++++xFgn$\"3'p%3&eWPT[#Fgn7$$\" 3#)*************z(Fgn$\"3?F+v?H+!H#Fgn7$$\"3M+++++++zFgn$\"3`)GVV4)>3@ Fgn7$$\"\")F)$\"3!oq@A*)*3R>Fgn7$$\"3k*************4)Fgn$\"3%G$\\\"R*= g#y\"Fgn7$$\"3G*************>)Fgn$\"3Z53[H,QQ;Fgn7$$\"3q+++++++$)Fgn$ \"3j,0)4./f]\"Fgn7$$\"3M+++++++%)Fgn$\"3YP4R;2c%Q\"Fgn7$$\"3++++++++&) Fgn$\"3U;OI7Zpt7Fgn7$$\"3k*************f)Fgn$\"3kjR\"z?TE<\"Fgn7$$\"3G *************p)Fgn$\"3#y=*zL#[23\"Fgn7$$\"3q+++++++))Fgn$\"3!y+Q:;9R(* *F,7$$\"3M+++++++*)Fgn$\"3fVeTY!Q)>#*F,7$$\"\"*F)$\"3idj[/_#)R&)F,7$$ \"3k*************4*Fgn$\"3m6kQe[$*GzF,7$$\"3G*************>*Fgn$\"3Q*Q D,%>v#Q(F,7$$\"3q+++++++$*Fgn$\"3OH;-xIU(*oF,7$$\"3M+++++++%*Fgn$\"3=! Q!e*3)ppkF,7$$\"3++++++++&*Fgn$\"3?eR<3X)p4'F,7$$\"3k*************f*Fg n$\"3)*Q-V0RUxdF,7$$\"3G*************p*Fgn$\"3x`m'4E,+^&F,7$$\"3q+++++ ++)*Fgn$\"3AP_gvGq%H&F,7$$\"3M+++++++**Fgn$\"3=vfU1YuK^F,7$$\"#5F)$\"3 \"QXYhD-p-&F,7$$\"3(*************45!#;$\"3wE1YT3*>)\\F,7$$\"3$******** *****>5F][m$\"3VuY>z;c0]F,7$$\"32++++++I5F][m$\"3a<)HVG=*3^F,7$$\"3/++ ++++S5F][m$\"3CP*=ax&f3`F,7$$\"3+++++++]5F][m$\"3)f%ph<3]GcF,7$$\"3)** ***********f5F][m$\"3$p/qv\"4&H5'F,7$$\"3%*************p5F][m$\"3oa&Hv *)=3y'F,7$$\"33++++++!3\"F][m$\"3WYr2>hqIxF,7$$\"3/++++++!4\"F][m$\"3m =aQ7.8Y!*F,7$$\"#6F)$\"3))fW%y;NZ3\"Fgn7$$\"3(*************46F][m$\"37 LOWCt6F][m$\"3)z60!zO'Qk\"Fgn7$$\"32++++++I 6F][m$\"3\">N10Qnt.#Fgn7$$\"3/++++++S6F][m$\"3aGl3*>F/\\#Fgn7$$\"3++++ +++]6F][m$\"3!RVAJe#\\fHFgn7$$\"3)*************f6F][m$\"37?1lA0t#Q$Fgn 7$$\"3%*************p6F][m$\"3B:Lm_=B.PFgn7$$\"33++++++!=\"F][m$\"3uDZ 9Q?=\"*QFgn7$$\"3/++++++!>\"F][m$\"3W64x=tn[RFgn7$$\"#7F)$\"3)=>kGO@)) *QFgn7$$\"3(*************47F][m$\"3C(\\Ws\\n:x$Fgn7$$\"3$************* >7F][m$\"3_18O_mZ%f$Fgn7$$\"32++++++I7F][m$\"3A)R@T0\\\"*Q$Fgn7$$\"3/+ +++++S7F][m$\"3A,qLZ!Q5<$Fgn7$$\"3+++++++]7F][m$\"3m5Ve'>r0&HFgn7$$\"3 )*************f7F][m$\"3>Hm>'RwWt#Fgn7$$\"3%*************p7F][m$\"3L$> Tzd-p_#Fgn7$$\"33++++++!G\"F][m$\"3\\,Gi-YDIBFgn7$$\"3/++++++!H\"F][m$ \"3)zd/fjzd9#Fgn7$$\"#8F)$\"3#p9G#Gt'R(>Fgn7$$\"3(*************48F][m$ \"3F0`+A$>[\"=Fgn7$$\"3$*************>8F][m$\"3yZE8?)H!o;Fgn7$$\"32+++ +++I8F][m$\"3GE,DAk5L:Fgn7$$\"3/++++++S8F][m$\"3Y0)[%>5X49Fgn7$$\"3+++ ++++]8F][m$\"3c@CIYRT'H\"Fgn7$$\"3)*************f8F][m$\"31`2r5-L$>\"F gn7$$\"3%*************p8F][m$\"3%e?ZAnV&*4\"Fgn7$$\"33++++++!Q\"F][m$ \"3D]!QN8BW,\"Fgn7$$\"3/++++++!R\"F][m$\"3+S)4H[IPP*F,7$$\"#9F)$\"3]=A .ksSy')F,7$$\"3(*************49F][m$\"3%*))Q^IQ@`!)F,7$$\"3$********** ***>9F][m$\"3i!HP[U?O\\(F,7$$\"32++++++I9F][m$\"3;&3\"p<6l&*pF,7$$\"3/ ++++++S9F][m$\"3?&zX*HN#fb'F,7$$\"3+++++++]9F][m$\"3)[4[o!**prhF,7$$\" 3)*************f9F][m$\"33&*>axz&4%eF,7$$\"3%*************p9F][m$\"3s2 7+s@\\ibF,7$$\"33++++++![\"F][m$\"3&\\My(f'fgL&F,7$$\"3/++++++!\\\"F][ m$\"3)eN:1P\"fi^F,7$$\"#:F)$\"3oHz`it\\W]F,7$$\"3(*************4:F][m$ \"3E-jvh&4h)\\F,7$$\"3$*************>:F][m$\"3E!o]$e`K%*\\F,7$$\"32+++ +++I:F][m$\"3KM$4IHX&z]F,7$$\"3/++++++S:F][m$\"3+*32VGRqD&F,7$$\"3++++ +++]:F][m$\"3w%4%))Hr#*[bF,7$$\"3)*************f:F][m$\"39zXsBi+()fF,7 $$\"3%*************p:F][m$\"3>%f))\\y^mh'F,7$$\"33++++++!e\"F][m$\"3\" zc9iW4=](F,7$$\"3/++++++!f\"F][m$\"3`$G;Xh_.t)F,7$$\"#;F)$\"3ny2hd*4;F][m$\" 3a30>S'))*p:Fgn7$$\"32++++++I;F][m$\"3]Q0iIFgn7$$\"3')************ R;F][m$\"3gP`.9e$4R#Fgn7$$\"3+++++++];F][m$\"3!yB6)>WVhGFgn7$$\"39++++ ++g;F][m$\"3+f&\\/M_(*H$Fgn7$$\"3%*************p;F][m$\"33L1(Ro7dk$Fgn 7$$\"33++++++!o\"F][m$\"3%=*)>>gUE'QFgn7$$\"3')*************o\"F][m$\" 3/et)o(3TYRFgn7$$\"#=F ][m$\"3%*\\RH]e@)p\"Fgn7$$\"32++++++I=F][m$\"3%Hnm#\\z\"3c\"Fgn7$$\"3' )************R=F][m$\"3QkT*p)4#[V\"Fgn7$$\"3+++++++]=F][m$\"3dT#3Ns#e> 8Fgn7$$\"39++++++g=F][m$\"3aH$GO1RW@\"Fgn7$$\"3%*************p=F][m$\" 31$ooKTI(=6Fgn7$$\"33++++++!)=F][m$\"3=YlbG#>=.\"Fgn7$$\"3')********** ***)=F][m$\"3n*[-'zo,J&*F,7$$\"#>F)$\"3f`L*f9b,#))F,7$$\"39++++++5>F][ m$\"3_*4#H+RX!=)F,7$$\"3$*************>>F][m$\"3U(4!4%Gqsg(F,7$$\"32++ ++++I>F][m$\"3Y*pz(zv]'4(F,7$$\"3')************R>F][m$\"36E\"=g[`Yk'F, 7$$\"3+++++++]>F][m$\"3mQkyD\"G)[iF,7$$\"39++++++g>F][m$\"3%>.Wo*)\\o! 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Then the cycle repeats itself." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 14 "Phase Portrait" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Next we want to view the phase portrait. We will first view it us ing the numeric solution above, then use the special package in Maple \+ that includes the direction field." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "RabFox := [seq([pR(0.1*n), pF(0.1*n)], n = 0..200)]: \nplot(RabFox, color = MAGENTA);" }}{PARA 13 "" 1 "" {GLPLOT2D 460 387 387 {PLOTDATA 2 "6&-%'CURVESG6#7ew7$$\"\"\"\"\"!$\"3++++++++]!#=7$ $\"3AG*\\?Io/:\"!#<$\"3a\"fY?A&3$)\\F-7$$\"3$o8u@k#GB8F1$\"3wanK:c#*Q] F-7$$\"3u*>\\%>_F?:F1$\"3!zVE:YM3=&F-7$$\"3/8$4>FfDu\"F1$\"3x]))o!Hs\" GaF-7$$\"3C=#RBF)*)*)>F1$\"3i?(4$R*Q)3eF-7$$\"39\"yDM&[ufAF1$\"3Wl&>x6 lFO'F-7$$\"3q?L^t,%ea#F1$\"3Mm[2dj`YrF-7$$\"3,'3))[F7j$GF1$\"3+*HX\"R* *yQ#)F-7$$\"3&Q[cyi>96$F1$\"3=c9gI\")fW(*F-7$$\"3`k9`!pT;M$F1$\"3#*RWP <%y$z6F17$$\"3mB)y!)[=z[$F1$\"3AV@d!QWAX\"F17$$\"3Cyber[JF17$$\"3mQ-DP%f=:#F1$\"3%472z$>- MNF17$$\"3=[=W2$*R!p\"F1$\"3C9\\kGV&)*z$F17$$\"3#)4Iw=zo'H\"F1$\"3AgHx o9`IRF17$$\"3KL=&*yJRm)*F-$\"3B@]6\"H())QRF17$$\"3*H_vY1'zWvF-$\"3yo,9 z4Z_QF17$$\"3gN[c]C2aeF-$\"3#[rWEQz4q$F17$$\"3%fB&3&Gbwj%F-$\"3BwV9^6i 4NF17$$\"3?=?<$HnYw$F-$\"3!f'))eVUH(H$F17$$\"3/%>K)e?b>F-$\"3Kr?F17$$\"3?@.JiMStF 17$$\"3\">3THCX'RF\"e;X@F-$\"3'yxCVAQB1\"F17$$\"3'p)R`RF-=BF-$\"3*z47@$f =2)*F-7$$\"3DksgL)f#GDF-$\"3kv\"eF=*Gp!*F-7$$\"3YSsw'ze3y#F-$\"30CtaAM O/%)F-7$$\"33W=S:R#>3$F-$\"3w365?md2yF-7$$\"34[ZTl.!)QMF-$\"3?Hy.(**HY F(F-7$$\"3e*4X\")z8-'QF-$\"3$z5\"Q&e%z,oF-7$$\"3IZXI\"zCkN%F-$\"3?s,$p $[&fQ'F-7$$\"3%4Q#)4b)QR\\F-$\"3M\\a&\\(pmCgF-7$$\"3R$GZ%*o/Hi&F-$\"3s -41.8C;dF-7$$\"3Xke\"*\\dD&F-7$$\"3L>x&[o#HW%)F-$\"3&[*G5n`U0^F-7$$\"3&HJ:Y()ypq*F-$\"3U P?F1$\"3eEw*=p.]r&F-7$$\"3:MG&*z7v+AF1$\"3 bR#)ztM,GiF-7$$\"3=5in-S;%[#F1$\"3K$>$*Qi#=dpF-7$$\"3WD&4w\"p0vFF1$\"3 1'fQ<`8g(zF-7$$\"3?J]\"Re]b0$F1$\"3')\\WjF&yPQ*F-7$$\"3+i1:x\\H)H$F1$ \"3/AbEB*z08\"F17$$\"39G4Vb(GgY$F1$\"3#*QbL\"ewzQ\"F17$$\"3&p\"f**y`q: NF1$\"3?(Q-dQ\"G@`i/a=RJF 1$\"37\")*HtO;2f#F17$$\"33C:/(>\"pJFF1$\"3'QQX=kia0$F17$$\"3mSl^7duaAF 1$\"3i`05/1'4Y$F17$$\"3Wb%p8!z!Ry\"F1$\"3%e+P(=1kaPF17$$\"3]fa?v?[t8F1 $\"3G:S98Bv8RF17$$\"3K?m#>\"3hX5F1$\"3qC8U3I2YRF17$$\"3YXJUbB\"*zzF-$ \"3%ob1LLXs(QF17$$\"3)yle&prpohF-$\"3Z%H#p#Rrst$F17$$\"3/>%)znr^j[F-$ \"3W5(Ree!f_NF17$$\"3^**HBVd%o#RF-$\"34GxmjLWVLF17$$\"3+8D`rz^`KF-$\"3 gf!pbY.T7$F17$$\"3iv/\\hs&yw#F-$\"3&*e6Pe]5/HF17$$\"3E:rdI&4qT#F-$\"3' e1Q&>Qa*o#F17$$\"3CaDY8Zbk@F-$\"3'p%3&eWPT[#F17$$\"31Eo^U)Hb)>F-$\"3?F +v?H+!H#F17$$\"3%>r$y3@F17$$\"3'R=@:l%)\\y\"F-$\"3 !oq@A*)*3R>F17$$\"321**>`.4WF-$ \"3kjR\"z?TE<\"F17$$\"3?VS&Ra]G6#F-$\"3#y=*zL#[23\"F17$$\"3'3,$=GbIyAF -$\"3!y+Q:;9R(**F-7$$\"3iu/5]s@![#F-$\"3fVeTY!Q)>#*F-7$$\"3vw#HMmTLs#F -$\"3idj[/_#)R&)F-7$$\"3'\\@MTQNN,$F-$\"3m6kQe[$*GzF-7$$\"3c(3nO\")oyN $F-$\"3Q*QD,%>v#Q(F-7$$\"3500!fmhZw$F-$\"3OH;-xIU(*oF-7$$\"3tY=![#\\8W UF-$\"3=!Q!e*3)ppkF-7$$\"3:\"f?!p/d2[F-$\"3?eR<3X)p4'F-7$$\"3A$ehBhk%o aF-$\"3)*Q-V0RUxdF-7$$\"3q0p)on`@C'F-$\"3x`m'4E,+^&F-7$$\"3CQ!yo%o(f9( F-$\"3AP_gvGq%H&F-7$$\"33\"eu_t@#*>)F-$\"3=vfU1YuK^F-7$$\"3.sRSbv'GU*F -$\"3\"QXYhD-p-&F-7$$\"31e6rO%**Q3\"F1$\"3wE1YT3*>)\\F-7$$\"3'y]r)zc'p C\"F1$\"3VuY>z;c0]F-7$$\"3PN&QWm\"\\L9F1$\"3a<)HVG=*3^F-7$$\"3&)p6p(*y &\\k\"F1$\"3CP*=ax&f3`F-7$$\"3ETK()Rtz\")=F1$\"3)f%ph<3]GcF-7$$\"3[bg= W\")eU@F1$\"3$p/qv\"4&H5'F-7$$\"3x(4%p*H_GU#F1$\"3oa&Hv*)=3y'F-7$$\"3E 4p2zlO8FF1$\"3WYr2>hqIxF-7$$\"3khyKRT,)*HF1$\"3m=aQ7.8Y!*F-7$$\"3!*GxS 7[e^KF1$\"3))fW%y;NZ3\"F17$$\"3-#fyP>)yQMF1$\"37LOWCt^ 0r^$F1$\"3)z60!zO'Qk\"F17$$\"3yQ5eY%epW$F1$\"3\">N10Qnt.#F17$$\"3QQ3< \"*Q$*4KF1$\"3aGl3*>F/\\#F17$$\"3w@)\\*)QMn#GF1$\"3!RVAJe#\\fHF17$$\"3 !392ef&*zN#F1$\"37?1lA0t#Q$F17$$\"3)\\x)ex1F!)=F1$\"3B:Lm_=B.PF17$$\"3 kNchN\\4a9F1$\"3uDZ9Q?=\"*QF17$$\"3bt%4-,Y#36F1$\"3W64x=tn[RF17$$\"3sa )*4^p6X%)F-$\"3)=>kGO@))*QF17$$\"3v!z$*H!y31lF-$\"3C(\\Ws\\n:x$F17$$\" 3lv5*)QC!f5&F-$\"3_18O_mZ%f$F17$$\"3Agf)44735%F-$\"3A)R@T0\\\"*Q$F17$$ \"3ah5\\L3vyLF-$\"3A,qLZ!Q5<$F17$$\"3-NxG`[HeGF-$\"3m5Ve'>r0&HF17$$\"3 !3D#y7GJ#[#F-$\"3>Hm>'RwWt#F17$$\"3!yu(3n,O6AF-$\"3L$>Tzd-p_#F17$$\"3g _'4Fo!Q=?F-$\"3\\,Gi-YDIBF17$$\"3=Xo_])p[)=F-$\"3)zd/fjzd9#F17$$\"3qwn //sA)z\"F-$\"3#p9G#Gt'R(>F17$$\"3KKO>Y<(*\\[\"=F17$$\"3! 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{Req,Feq\},\{R,F \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"RG$\"\"!F'/%\"FGF&<$/F%$ \"+66666!\"*/F)$\"+nmmm;F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT 258 29 "Modified Lotka-Volterra Model" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "Consider a modification to the Lotka-Volterra model that includes logistic gro wth of the rabbits. We repeat most of the calculations above, and obse rve the differences for this system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RabM := diff(R(t),t) = 2*R(t)*(1 - R(t)/2) - 1.2*R(t )*F(t);\nFoxM := diff(F(t),t) = -F(t) + 0.9*R(t)*F(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RabMG/-%%diffG6$-%\"RG6#%\"tGF,,&*&F)\"\"\",&F/ F/*&#F/\"\"#F/F)F/!\"\"F/F3*($\"#7F4F/F)F/-%\"FGF+F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FoxMG/-%%diffG6$-%\"FG6#%\"tGF,,&F)!\"\"*($\"\" *F.\"\"\"-%\"RGF+F2F)F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Once again we find the numerical solution to this system of equations." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "sysM := dsolve(\{RabM, Fox M, R(0)=1,F(0) = 0.5\}, \{R(t),F(t)\}, type = numeric, output = listpr ocedure); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sysMG7%/%\"tGR6#F'6\" 6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~rese rved.GF*9$F*F*F*/-%\"FGF)RF)6&%(rkf45_sG%)outpointG%#r1G%#r2GF+F*C&>8% -%&evalfG6#F-@$52-%$absG6#,$F9!\"\"-FA6#,&&%,loc_controlG6#\"\"#\"\"\" F9FD4-%'memberG6$&FI6#\"\"'<*!\"#FDFLFK$FL\"\"!$FKFW$FDFW$FUFWC%>FI-%% copyG6#=F*6#;FL\"#LE\\[lBFLFKFK$FWFW\"\"$F_o\"\"%$FL!\")\"\"&FboFSFL\" \"($FL!\"*\"\")\"&++$\"\"*\"%+5\"#5FW\"#6FW\"#7FW\"#8FW\"#9FW\"#:FW\"# ;FW\"#FW\"#?$FdoFD\"#@FV\"#AFW\"#BFW\"#CFW\"#DFW\"#EFW\"# FFW\"#GFW\"#HFW\"#IFW\"#JFW\"#KFWF]oFW>%'loc_y0G-Fhn6#=F*6#;FLFKE\\[l# FLFgpFKFV>%'loc_y1G-Fhn6#=F*FiqE\\[l!@$0F@FWC$>&FI6#F`oF9@%1%'DigitsG- %'evalhfG6#FjrC$>8$-%*traperrorG6#-F\\s6#-%=dsolve/numeric_solnall_rkf 45G6,%&loc_FG-%$varG6#FI-F[t6#Feq-F[t6#F]r-F[t6#%'loc_F1G-F[t6#%'loc_F 2G-F[t6#%'loc_F3G-F[t6#%'loc_F4G-F[t6#%'loc_F5G-F[t6#%)loc_workG@$/F`s %*lasterrorGC%>8&-%+searchtextG6$.F\\s-%(convertG6$-%#opG6$FL7#F`s%%na meG>8'-Fju6$.%)hardwareGF]v@%50FhuFW0FfvFW-Fgs6,FisFIFeqF]rFctFftFitF \\uF_uFbu-%&ERRORG6#F`sF_w&Feq6#FLF*6%FIFeqF]rF*/-%\"RGF)RF)F2F+F*C&>F 9F:@$F>C%>FI-Fhn6#=F*F[oE\\[lBFLFKFKF_oF`oF_oFaoFboFdoFboFSFLFeoFfoFho FioFjoF[pF\\pFWF]pFWF^pFWF_pFWF`pFWFapFWFbpFWFcpFWFdpFWFepFWFfpFgpFhpF VFipFWFjpFWF[qFWF\\qFWF]qFWF^qFWF_qFWF`qFWFaqFWFbqFWFcqFWF]oFW>Feq-Fhn 6#=F*FiqE\\[l#FLFgpFKFV>F]r-Fhn6#=F*FiqE\\[l!@$FcrC$>FfrF9@%FirC$>F`sF as@$FduC%>FhuFiu>FfvFgv@%F\\wF_wFawF_w&FeqFJF*FfwF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We isolate the solutions for R(t) and F(t) for \+ further analysis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "pRM := subs(sysM, R(t)); \npFM := subs(sysM, F(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pRMGR6#%\"tG6&%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCo pyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserved.G6 \"C&>8%-%&evalfG6#9$@$52-%$absG6#,$F2!\"\"-F;6#,&&%,loc_controlG6#\"\" #\"\"\"F2F>4-%'memberG6$&FC6#\"\"'<*!\"#F>FFFE$FF\"\"!$FEFQ$F>FQ$FOFQC %>FC-%%copyG6#=F/6#;FF\"#LE\\[lBFFFEFE$FQFQ\"\"$Fin\"\"%$FF!\")\"\"&F \\oFMFF\"\"($FF!\"*\"\")\"&++$\"\"*\"%+5\"#5FQ\"#6FQ\"#7FQ\"#8FQ\"#9FQ \"#:FQ\"#;FQ\"#FQ\"#?$F^oF>\"#@FP\"#AFQ\"#BFQ\"#CFQ\"#DFQ \"#EFQ\"#FFQ\"#GFQ\"#HFQ\"#IFQ\"#JFQ\"#KFQFgnFQ>%'loc_y0G-FX6#=F/6#;FF FEE\\[l#FFFapFEFP>%'loc_y1G-FX6#=F/FcqE\\[l!@$0F:FQC$>&FC6#FjnF2@%1%'D igitsG-%'evalhfG6#FdrC$>8$-%*traperrorG6#-Ffr6#-%=dsolve/numeric_solna ll_rkf45G6,%&loc_FG-%$varG6#FC-Fes6#F_q-Fes6#Fgq-Fes6#%'loc_F1G-Fes6#% 'loc_F2G-Fes6#%'loc_F3G-Fes6#%'loc_F4G-Fes6#%'loc_F5G-Fes6#%)loc_workG @$/Fjr%*lasterrorGC%>8&-%+searchtextG6$.Ffr-%(convertG6$-%#opG6$FF7#Fj r%%nameG>8'-Fdu6$.%)hardwareGFgu@%50FbuFQ0F`vFQ-Fas6,FcsFCF_qFgqF]tF`t FctFftFitF\\u-%&ERRORG6#FjrFiv&F_qFDF/6%FCF_qFgqF/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$pFMGR6#%\"tG6&%(rkf45_sG%)outpointG%#r1G%#r2G6#%ao Copyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserved. G6\"C&>8%-%&evalfG6#9$@$52-%$absG6#,$F2!\"\"-F;6#,&&%,loc_controlG6#\" \"#\"\"\"F2F>4-%'memberG6$&FC6#\"\"'<*!\"#F>FFFE$FF\"\"!$FEFQ$F>FQ$FOF QC%>FC-%%copyG6#=F/6#;FF\"#LE\\[lBFFFEFE$FQFQ\"\"$Fin\"\"%$FF!\")\"\"& F\\oFMFF\"\"($FF!\"*\"\")\"&++$\"\"*\"%+5\"#5FQ\"#6FQ\"#7FQ\"#8FQ\"#9F Q\"#:FQ\"#;FQ\"#FQ\"#?$F^oF>\"#@FP\"#AFQ\"#BFQ\"#CFQ\"#DF Q\"#EFQ\"#FFQ\"#GFQ\"#HFQ\"#IFQ\"#JFQ\"#KFQFgnFQ>%'loc_y0G-FX6#=F/6#;F FFEE\\[l#FFFapFEFP>%'loc_y1G-FX6#=F/FcqE\\[l!@$0F:FQC$>&FC6#FjnF2@%1%' DigitsG-%'evalhfG6#FdrC$>8$-%*traperrorG6#-Ffr6#-%=dsolve/numeric_soln all_rkf45G6,%&loc_FG-%$varG6#FC-Fes6#F_q-Fes6#Fgq-Fes6#%'loc_F1G-Fes6# %'loc_F2G-Fes6#%'loc_F3G-Fes6#%'loc_F4G-Fes6#%'loc_F5G-Fes6#%)loc_work G@$/Fjr%*lasterrorGC%>8&-%+searchtextG6$.Ffr-%(convertG6$-%#opG6$FF7#F jr%%nameG>8'-Fdu6$.%)hardwareGFgu@%50FbuFQ0F`vFQ-Fas6,FcsFCF_qFgqF]tF` tFctFftFitF\\u-%&ERRORG6#FjrFiv&F_q6#FFF/6%FCF_qFgqF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We view the time varying solutions for th e rabbits and foxes." 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*************48F][m$\"3IIkggok4uF,7$$\"3$*************>8F][m$\"3SV9@vh i4uF,7$$\"32++++++I8F][m$\"3_M!**y+*e4uF,7$$\"3/++++++S8F][m$\"3.o)3\\ SP&4uF,7$$\"3+++++++]8F][m$\"3,CVk=LZ4uF,7$$\"3)*************f8F][m$\" 36%)e6,')R4uF,7$$\"3%*************p8F][m$\"3LM/#*)*\\J4uF,7$$\"33+++++ +!Q\"F][m$\"3#f,P'\\TA4uF,7$$\"3/++++++!R\"F][m$\"3lh,_zv74uF,7$$\"#9F )$\"3o:NP-n-4uF,7$$\"3(*************49F][m$\"3\"z$))[>G#*3uF,7$$\"3$** ***********>9F][m$\"3U/'Q07<)3uF,7$$\"32++++++I9F][m$\"3(f=9`o5(3uF,7$ $\"3/++++++S9F][m$\"3-7e?'[/'3uF,7$$\"3+++++++]9F][m$\"3\"GED8R*\\3uF, 7$$\"3)*************f9F][m$\"3!fBx\"ohR3uF,7$$\"3%*************p9F][m$ \"3#[m!z([&H3uF,7$$\"33++++++![\"F][m$\"3:CnAHz>3uF,7$$\"3/++++++!\\\" F][m$\"37.fQ%)R53uF,7$$\"#:F)$\"3qo@/jS,3uF,7$$\"3(*************4:F][m $\"31]z.)\\GzS(F,7$$\"3$*************>:F][m$\"3X)Hz0bZyS(F,7$$\"32++++ ++I:F][m$\"3]Ohd:9x2uF,7$$\"3/++++++S:F][m$\"3SKz(pA+xS(F,7$$\"3++++++ +]:F][m$\"3;B/1jSj2uF,7$$\"3)*************f:F][m$\"3Y)3C;&Hd2uF,7$$\"3 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{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "RMeq := 2*R*(1 \+ - R/2) - 1.2*R*F = 0;\nFMeq := -F + 0.9*R*F= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RMeqG/,&*&%\"RG\"\"\",&F)F)*&#F)\"\"#F)F(F)!\"\"F)F- *($\"#7F.F)F(F)%\"FGF)F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FM eqG/,&%\"FG!\"\"*($\"\"*F(\"\"\"%\"RGF,F'F,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{RMeq,FMeq\},\{R,F\});" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%<$/%\"FG$\"\"!F'/%\"RGF&<$/F)$\"\"#F'F$<$/F)$\"+ 66666!\"*/F%$\"+2uS2u!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "There are now 3 equilibria." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "29 1 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }