{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 }{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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 31 "This is all the Maple syntax in" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "Chapter 4." }}{PARA 4 "" 0 "" {TEXT -1 75 "An Introduction to the Mathematics of Biology, with Computer Algebra Models" }}{PARA 4 "" 0 "" {TEXT -1 34 "by Yeargers, Shonkwiler, & Herod. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "The Syntax is written for Maple 6." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Section 4.3" }}{PARA 0 "" 0 "" {TEXT -1 8 "Page 113" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "dsolve(\{diff(y(t),t)=r*y(t)*(1-y(t)/k),y(0 )=yo\},y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Page 114" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=1; k:=3;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 78 "dsolve(\{diff(y(t),t) = r*y(t)*(1-y(t)/k),y(0) =1\},y(t));\ny1:=unapply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "dsolve(\{diff(y(t),t) = r*y(t)*(1-y(t)/k),y(0)=2\},y( t));\ny2:=unapply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "dsolve(\{diff(y(t),t) = r*y(t)*(1-y(t)/k),y(0)=4\},y(t));\ny4:=u napply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot( \{y1(t),y2(t),y4(t)\},t=0..5,y=0..5,color=black);" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Page 115" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=1 ; theta:=1/5; K:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot ([y,r*(y/theta-1)*(1-y/K),y=0..1],-.2..1,-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=1; theta:=1/5; K:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "inits:=\{[0,.05],[0,.1],[0,0.3],[0,.5],[0,1], [0,0.7],[0,1.5]\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "DEplot(diff(y( t),t)=r*y*(y/theta-1)*(1-y/K),y(t),t=0..3,inits, arrows=NONE,stepsize= 0.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 4.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercis e 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Anderfit:=t->alpha/(1+beta*exp(-delta*t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "dsolve(\{diff(y(t),t)-delta* y(t)*(1-y(t)/alpha)=0,\n\011\011y(0)=alpha/(1+beta)\},y(t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "alpha:=387.980205; beta:=54. 0812024; delta:=0.02270347337;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "J:=plot(Anderfit(t),t=0..200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "tt:=[seq(i*10,i=0..20)];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 264 "pop:=[ 3.929214, 5.308483, 7.239881, 9.63845 3, 12.866020, \n\011\011 17.069453, 23.191876, 31.433321, 39.818449 , 50.155783, \n\011\011 62.947714, 75.994575, 91.972266, 105.710620, 122.775046,\n\011\011131.669275, 151.325798,179.323175, 203.302031, 2 26.545805,\n\011\011248.709873];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "data:= [seq([tt[i],pop[i]],i=1..21)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "K:=plot(data,style=POINT):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plots[display](\{J,K\});" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expfit:=t->exp(0.0207538439 3*t+1.766257672);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "L:=plo t(expfit(t),t=0..200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p lots[display](\{J,K,L\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(Anderfit(t-1790),t=1790..2150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "k :=15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{.48*(1-mu/k ),2*mu/(4+mu^2)\},mu=0..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "h:=(t,P)->.48*P*(1-P/k)-2*P^2/(4+P^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "inits:=\{[0,1],[0,2],[0,4],[0,5],[0,6],[0,8],[0, 10],[0,12],[0,14],[0,16]\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "DEplo t(diff(y(t),t)=h(t,y(t)),y(t),t=0..30,inits, arrows=NONE,stepsize=0.1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Section 4.4" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Page 120" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "predprey:=diff(x(t),t)=r*x -a*x*y, diff(y(t),t)=-m*y+b*x*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=1; a:=1; m:=1; b:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "sol:=dsolve(\{predprey,x(0)=3/2,y(0)=1/2\},\{x(t),y(t )\},\n type=numeric, output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "xsol:=subs(sol,x(t)); ysol:=subs(sol,y(t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([xsol,ysol],0..10,-1..3 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Pag e 121." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve(diff(y(x),x)=(-y(x)+x*y(x))/(x-x*y(x)),y(x),im plicit);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Page 122." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(plots): with(DEtools):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "predprey:=[diff(x(t),t)=r*x-a*x*y,diff(y(t),t) =-m*y+b*x*y];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=1; a:=1 ; m:=1; b:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "inits:=\{[ 0,3/2,1/2],[0,4/5,3/2]\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "phaseportrait(predprey,[x,y],t=0..10,inits,stepsize=.1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Page 122, footnote 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "predprey:=diff(x(t),t)=r*x-a*x*y, diff(y(t),t)=-m*y +b*x*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=1; a:=1; m:=1 ; b:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "sol2:=dsolve(\{p redprey,x(0)=4/5,y(0)=3/2\},\{x(t),y(t)\},\n type=numeric, output=li stprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xsol2:=sub s(sol2,x(t)); ysol2:=subs(sol2,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot(\{['xsol(t)','ysol(t)','t'=0..10],\n ['xso l2(t)','ysol2(t)','t'=0..10]\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Page 125." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "V:=vector([r*x-a*x*y,-m*y+b*x*y]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{V[1]=0,V[2]=0\},\{x, y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "jacobian(V,[x,y]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(\{x=0,y=0\},%);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{x=m/b,y=r/a\},%%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eigenvals(%%); eigenval s(%%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 4.4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Exerc ise 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rsx:=-x+x*y;\nrsy:= -y+2*y*z-x*y;\nrsz:=2*z-z^2-y*z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 14 "For just grass" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "sol:=dsolve(\{diff(x(t),t)= rsx,diff(y(t),t)=rsy,diff(z(t),t)=rsz,\n x(0)=0,y(0)=0,z(0)=1.5\} ,\{x(t),y(t),z(t)\},\n type=numeric, output=listprocedure);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "zsol:=subs(sol,z(t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(zsol,0..20);" }}}{PARA 0 "" 0 "" {TEXT -1 19 "For grass and sheep" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 153 "sol:=dsolve(\{diff(x(t),t)=rsx,diff(y(t),t)=rsy,di ff(z(t),t)=rsz,\n x(0)=0,y(0)=.5,z(0)=1.5\},\{x(t),y(t),z(t)\},\n type=numeric, output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ysol:=subs(sol,y(t));zsol:=subs(sol,z(t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot([ysol,zsol],0..20);" }} }{PARA 0 "" 0 "" {TEXT -1 29 "For grass, sheep, and wolves." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "sol:=dsolve(\{diff(x(t),t)=rsx,dif f(y(t),t)=rsy,diff(z(t),t)=rsz,\n x(0)=.2,y(0)=.5,z(0)=1.5\},\{x( t),y(t),z(t)\},\n type=numeric, output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xsol:=subs(sol,x(t));ysol:=subs(sol ,y(t));zsol:=subs(sol,z(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot([xsol,ysol,zsol],0..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "part (b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=1; a:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "sol:=ds olve(\{diff(SU(t),t)=-r*SU(t)*IN(t),\n diff(IN(t),t)=r*SU( t)*IN(t)-a*IN(t),\n diff(R(t),t)=a*IN(t), SU(0)=2.8,IN(0)= 0.2,R(0)=0\},\n \{SU(t),IN(t),R(t)\},type=numeric,output=listproce dure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f:=subs(sol,SU(t) ): g:=subs(sol,IN(t)): h:=subs(sol,R(t)):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "plot(\{f,g,h\},0..20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Part (c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "readlib(spline): #If you are using Maple 6, \n #this li ne can be ommitted. JHerod,09,00" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r:=1; a:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "f[0]:=t->2.8; g[0]:=t->0.2; h[0]:=t->0;\n J[0]:=plot(\{f[0],g[0] ,h[0]\},-1..0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x0seq:=[ seq(j/5,j=0..5)]: N:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 658 "for n from 1 to N do\n sol:=dsolve(\{diff(SU(t),t)=-r*SU(t)*IN(t)+'h[ n-1](t-1)',\n\011\011diff(IN(t),t)=r*SU(t)*IN(t)-a*IN(t),\n\011\011dif f(R(t),t)=a*IN(t)-'h[n-1](t-1)',\n\011\011\011SU(n-1)=f[n-1](n-1),\n \011\011\011IN(n-1)=g[n-1](n-1),\n\011\011\011R(n-1)=h[n-1](n-1)\}, \n \011\011\{SU(t),IN(t),R(t)\}, numeric, output=listprocedure):\n\011f:= subs(sol,SU(t)): g:=subs(sol,IN(t)): h:=subs(sol,R(t)):\n\011xseq:=map (t->t+n-1,x0seq):\n fxseq:=map(f,xseq): gxseq:=map(g,xseq): hxseq:=map (h,xseq):\n\011\011F:=spline(xseq,fxseq,t,cubic): \n\011\011G:=spline (xseq,gxseq,t,cubic):\n\011\011H:=spline(xseq,hxseq,t,cubic): \n\011f[ n]:=unapply(F,t): \n\011g[n]:=unapply(G,t): \n\011h[n]:=unapply(H,t): \n\011J[n]:= plot(\{f[n],g[n],h[n]\}, (n-1)..n,color=BLACK):\n\011od: \011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plots[display](\{J[ 1],J[2],J[3],J[4],J[5]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }