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" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 17 "Solved Example 1:" }} {PARA 0 "" 0 "" {TEXT -1 23 "Suppose that we have a " }{TEXT 278 13 "p redator-prey" }{TEXT -1 85 " situation described by the system of nonl inear first-order differential equations " }}{PARA 0 "" 0 "" {TEXT -1 28 " " }{XPPEDIT 18 0 "dx/dt = .2*x-.2e- 2*xy,dy/dt = -.1*y+.1e-2*xy;" "6$/*&%#dxG\"\"\"%#dtG!\"\",&*&-%&FloatG 6$\"\"#F(F&%\"xGF&F&*&-F,6$F.!\"$F&%#xyGF&F(/*&%#dyGF&F'F(,&*&-F,6$F&F (F&%\"yGF&F(*&-F,6$F&F3F&F4F&F&" }{TEXT -1 43 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 142 "(a) Plot the trajectory corresponding to the conditio ns x(0) = 100 and y(0) = 25 . What does this say about the predator -prey interaction?" }}{PARA 0 "" 0 "" {TEXT -1 57 "(b) Plot x(t) ag ainst t for t going from 0 to 400." }}{PARA 0 "" 0 "" {TEXT -1 57 "(c) Plot y(t) against t for t going from 0 to 400." }}{PARA 0 "" 0 "" {TEXT -1 53 "(d) Plot x(t) and y(t) on the same set of ax es. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 25 "(a) We load the package " }{TEXT 275 7 "DEtools" } {TEXT -1 64 ", enter the system, define the initial conditions, and th en use " }}{PARA 0 "" 0 "" {TEXT 276 7 " " }{TEXT 281 0 "" } {TEXT -1 20 "a special function, " }{TEXT 282 0 "" }{TEXT 283 6 "pphas e" }{TEXT 284 0 "" }{TEXT -1 13 ", which uses " }{TEXT 280 7 "DEplot \+ " }{TEXT -1 39 "and allows us to vary the range of t " }}{PARA 0 "" 0 "" {TEXT -1 21 " conveniently: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "with(DEtools):\nsys:=d iff(x(t),t)=0.2*x(t)-0.002*x(t)*y(t),diff(y(t),t)=-0.1*y(t)+0.001*x(t) *y(t):\npphase:=trange->DEplot([sys],[x(t), y(t)], t = trange, \{[x(0) =100,y(0)=25]\}, stepsize = 0.1, linecolor = black, arrows = MEDIUM,s caling=unconstrained,title=`Predator vs. Prey` ):\npphase(0..55);\n" } }{PARA 13 "" 1 "" {GLPLOT2D 313 206 206 {PLOTDATA 2 "6)-%'CURVESG6]dl7 '7$$\"3Y@8tO31:t!#<$\"3r/*yc\"G)=/$!#;7$$\"3M?pBJ&[@e\"F-$\"3I&4@V=<\" e>F-7$$\"3]J3LpQs*>\"F-$\"3?B$)3[A5?@F-7$$\"3u-#z&3UB'f\"F-$\"3YRg![S; aE#F-F.7'7$$\"3u=bR#y=&y@F-$\"3muk!*>\"G*3GF-7$$\"3im=!pxYYt$F-$\"3MDN 4!)=2\">#F-7$$\"3QO[8X'4ZG$F-$\"31d3r1g#>>#F-7$$\"3_#ex7sg2^$F-$\"3-m& eKNdxX#F-FC7'7$$\"3!Q&H?!))Ht)QF-$\"3I'RYB;u'fEF-7$$\"36l<[BmMDcF-$\"3 s.OlPeKSBF-7$$\"3crx#\\+O1>&F-$\"3Owf<%>:5E#F-7$$\"3^&o%4;RZ2`F-$\"3!3 @^?O=zb#F-FX7'7$$\"3)f0UY[7Nm&F-$\"3qWy5m_9yDF-7$$\"3)o**HM'\\n[uF-$\" 3Jb@*Qta=U#F-7$$\"3d)fs\\W&eLqF-$\"35;90?T@.BF-7$$\"3_<3j`kw!4(F-$\"31 $ynL8r\"3EF-Fgo7'7$$\"3!pZB].1qX(F-$\"3c5s-IqZHDF-7$$\"3W3fVdBpa#*F-$ \"3W*ys*pH_qCF-7$$\"3Up+$3G(pa))F-$\"3\\E.nv'Q(HBF-7$$\"3S`#>CUmi())F- $\"3'Q$p5@^$oj#F-F\\q7'7$$\"3M'pej@LdD*F-$\"3eyw(4--x\\#F-7$$\"3A-)[?h Zb5\"!#:$\"3V@B-zzH-DF-7$$\"3=]K%)4hmm5Fdr$\"3M+@fS!zvM#F-7$$\"3K::\\g z\\m5Fdr$\"35[=C4o-bEF-Far7'7$$\"3$3TJ\"f(>c5\"Fdr$\"3_\\a+j&*[vCF-7$$ \"3M%*>**oAX&G\"Fdr$\"3\\]X*pV5X_#F-7$$\"3?X^/OTTZ7Fdr$\"33`.2BVHgBF-7 $$\"33`g&4k?cC\"Fdr$\"3S4O#)*))*\\nEF-Fgs7'7$$\"3)pB='>Es&G\"Fdr$\"3kE rDoj9fCF-7$$\"3],R%H]+`Y\"Fdr$\"3QtGuJO&3a#F-7$$\"3s\\3=Oa\"zU\"Fdr$\" 3;#pAX>y(pBF-7$$\"3q@![_1E\\U\"Fdr$\"3Cso.'y[ln#F-F\\u7'7$$\"3ggL(QxQe Y\"Fdr$\"3OE6oA@kYCF-7$$\"3A6v7Va8X;Fdr$\"3mt)=t(yN`DF-7$$\"3B^AwX(o#3 ;Fdr$\"3Uu)H,93rP#F-7$$\"3(fv)H0WO/;Fdr$\"3y9xDP$)R$o#F-Fav7'7$$\"3%\\ JDDz^fk\"Fdr$\"37y\"pXwxnV#F-7$$\"3?!H9*=N(\\#=Fdr$\"3)=#3VNAAjDF-7$$ \"3'3RJ$\\s_)y\"Fdr$\"35')ecYf$HQ#F-7$$\"3[*3cq5,Ry\"Fdr$\"3eD'GBYc()o #F-Ffw7'7$$\"3p1izvc0E=Fdr$\"3wyr@1H!)GCF-7$$\"3xJ@3I2#[+#Fdr$\"3C@Gy$ 4(>rDF-7$$\"3!p\"pj-=so>Fdr$\"3a)4%yCon(Q#F-7$$\"3)*)3n$=@^j>Fdr$\"3a( ***)3()eIp#F-F[y7'7$$\"3`^$)e&p[h+#Fdr$\"3_\"pA,DDAU#F-7$$\"3E?(GZ!)yY =#Fdr$\"3[3t()\\ZxxDF-7$$\"3Kk$foRr)[@Fdr$\"3)e?G\"\\pg\"R#F-7$$\"3^#4 qwS!=V@Fdr$\"3O\"of9k(e'p#F-F`z7'7$$\"3K&e%4\"*3B'=#Fdr$\"3?ZBv%**3nT# F-7$$\"3y>7m.xakBFdr$\"3#GlZ_+\"H$e#F-7$$\"3KS4`&H))*GBFdr$\"3GJ\\v\"y ;\\R#F-7$$\"3U7#*)pm$*GK#Fdr$\"33k\"*G-J`*p#F-Fe[l7'7$$\"3%Rn^%**HImBF dr$\"3&4,XF]:X;G?q#F-Fj\\l7'7$$\"3! )er7()fOYDFdr$\"31/0K_$zzS#F-7$$\"3z8h]'z9Vs#Fdr$\"3%f\\zwk??f#F-7$$\" 3Y_&)>\\W:*o#Fdr$\"3ou,p02=+CF-7$$\"3#3*f1\\5U#o#Fdr$\"3Fwc5r$oTq#F-F_ ^l7'7$$\"3`p2'[%3UEFFdr$\"3p\\1!QRnWS#F-7$$\"3nO7@L5@/HFdr$\"3K]$*>1E` &f#F-7$$\"3OQU7=N@pGFdr$\"3T*>&pEuI-CF-7$$\"3cs1ZQJAiGFdr$\"3AH^%=7Cgq #F-Fd_l7'7$$\"3S%)o!>]ok!HFdr$\"3F7(\\eO&Q,CF-7$$\"3qbQgqW6%3$Fdr$\"3s (G]Tj9')f#F-7$$\"33Kf:S4E\\IFdr$\"3U?mo\"3yTS#F-7$$\"3YVg;S]/UIFdr$\"3 G(*[D3%[wq#F-Fi`l7'7$$\"3SO;2&z4l3$Fdr$\"3op\\Dx!f')R#F-7$$\"3ePy(=FCS E$Fdr$\"3LI]uA4M,EF-7$$\"3(z@@_&*)HHKFdr$\"3**[[klh$eS#F-7$$\"32?wIlN) =A$Fdr$\"3Xf]Im=34FF-F^bl7'7$$\"3Y'eSFYXlE$Fdr$\"3_WqRR/B'R#F-7$$\"3T@ wk)pRRW$Fdr$\"3[bHgg&pPg#F-7$$\"3F,d\"*p#H$4MFdr$\"3w2QXveJ2CF-7$$\"3G I$*)*pht,MFdr$\"3#\\RI,y*)F*$\"3 FiM[P(y'=VF-7$$\"3K4wPw[$eT\"F-$\"3nB[g(e?49$F-7$$\"3?o5O@>B)3\"F-$\"3 ?aGhwnm^LF-7$$\"3wWZ$ppI\">:F-$\"3jB8DP'e,W$F-F]fl7'7$$\"3IYEO2,\">L#F -$\"3I\")GF)Q3C<%F-7$$\"3SRZ$>Xb7e$F-$\"3I/a\"o$4>(G$F-7$$\"3)\\(RZ^)H )[JF-$\"3SGDx1T8sLF-7$$\"3-we9\\')psMF-$\"3iEgFn\"ebe$F-Fbgl7'7$$\"3sD 4!GjL]%RF-$\"3Ee&oq9Be*RF-7$$\"3?$z$)3(GknbF-$\"3MF(>!yhxjMF-7$$\"3\\& ['zp$4!>^F-$\"3H&3A)*ft.W$F-7$$\"3yiNnhfm8`F-$\"3tp(==&=c!QF-F\\jl 7'7$$\"35N2pzPDfuF-$\"35P!QI%4F#y$F-7$$\"3E]'oFhWCD*F-$\"3[[-0#QGtn$F- 7$$\"3y@;4!35]%))F-$\"3ojZgp[)oa$F-7$$\"3w]i!yt/M)))F-$\"3]WDPcL@`QF-F a[m7'7$$\"3](p.Q%pub#*F-$\"3Zg8`)*))pDPF-7$$\"36-VIRia06Fdr$\"36DpbE/! 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Our starting point, corresponding to \+ t = 0, is (100, 25) and the direction of the trajectory is counterc lockwise. Both prey and predator populations increase as the number of prey increases; but when the prey population exceeds about 350, the p redators seem to overwhelm their prey to the extent that there are mor e and more predators but a declining prey population. The predators c ontinue to increase until their number is about 260, at which time the effect of a dwindling food supply catches up to the predators and the ir population begins to decline. Eventually, the predator population \+ is low enough for the prey population to recover, and the cycle begins again." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) We plot as follows:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "DEplot([sys],[x(t),y(t)],t= 0..400,\{[x(0)=100,y(0)=25]\},scene=[t,x],stepsize=0.1, linecolor=blac k,arrows=NONE,title=`Prey Population`);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 255 225 225 {PLOTDATA 2 "6'-%'CURVESG6%7]ez7$$\"\"!F)$\"$+\" F)7$$\"\"\"!\"\"$\"3%R,Us$H6:5!#:7$$\"\"#F/$\"3OG6j&\\`/.\"F27$$\"\"$F /$\"3A9!H))GCg/\"F27$$\"\"%F/$\"37Xz4?z#=1\"F27$$\"\"&F/$\"35[Drzp'y2 \"F27$$\"\"'F/$\"3ch5AUS9%4\"F27$$\"\"(F/$\"3$))z[Nmh16\"F27$$\"\")F/$ \"3G'=OzPAu7\"F27$$\"\"*F/$\"3a0Mq%pGW9\"F27$$\"#5F/$\"398L$[4$oh6F27$ $\"#6F/$\"3oXROF!)=z6F27$$\"#7F/$\"3iO:e0f%p>\"F27$$\"#8F/$\"34wX+.\"f \\@\"F27$$\"#9F/$\"34h*G\"F27$$\"#=F/$\"3QPtQ9Y$*38F27$ $\"#>F/$\"3C2tc!yE&G8F27$$\"#?F/$\"3!))e-6Q*Q[8F27$$\"#@F/$\"3%\\=g#)H 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