{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }1 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 0 0 0 8 4 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 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE " " -1 -1 "" 0 1 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 "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 30 "Linear Discrete Dynamic \+ Models" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 57 "Fibonacci model of hypothetical growing rabbit population" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Fibonacci equation and its solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Fibonacci:=N(t+2)=N(t+1)+N(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Fibonacci;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "rsolve(\{Fibonacc i,N(0)=0,N(1)=1\},N(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"NG6#, &%\"tG\"\"\"\"\"#F),&-F%6#,&F(F)F)F)F)-F%6#F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*&,&\"\"\"F'*$-%%sqrtG6#\"\"&\"\"\"#!\"\"F,F'),$*&F -F-,&F/F'F(F'!\"\"\"\"#%\"tGF'F-F3F4F'*&*&,&F(F.F/F'F'),$*&F-F-,&F'F'F (F'F4!\"#F6F'F-F=F4F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Simulation of the rab bit population:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tmax:=12: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "N:=array(0..tmax):N[0]:=0:N[1]: =1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " for i from 2 to tmax do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " N[i]:=N[i-1]+N[i-2]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p1:=pointplot([ [t,N[t]]$t=0..tmax],style=point,symbol=circle,color=red):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 103 "p2:=pointplot([[t,N[t]]$t=0..tmax],style=li ne,labels=[`month`,`No. of pairs`],title=`Fibonacci model`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p1,p2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "lprint(`Fibonacci sequence:`,seq(N[t],t=0..tmax));" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('N'):" }}{PARA 13 "" 1 " " {INLPLOT "6&-%'POINTSG627$\"\"!F'7$$\"\"\"F'F)7$$\"\"#F'F)7$$\"\"$F' F,7$$\"\"%F'F/7$$\"\"&F'F57$$\"\"'F'$\"\")F'7$$\"\"(F'$\"#8F'7$F:$\"#@ F'7$$\"\"*F'$\"#MF'7$$\"#5F'$\"#bF'7$$\"#6F'$\"#*)F'7$$\"#7F'$\"$W\"F' -%'COLOURG6&%$RGBG$\"*++++\"!\")F'F'-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%& POINTG-%'CURVESG6$7/F&F(F+F.F1F4F7F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 28 "Propagation of a nnual plants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "unassign('g','alpha','beta','sigma'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "PlantPropagation:=N(t+2)=alpha*g*sigma*N( t+1)+beta*g*sigma^2*(1-alpha)*N(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lprint(`\\n Model:`);PlantPropagation;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lprint(`\\n Parameters of the model:\\n`," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " g,`... number of seeds produced per plant \+ in autumn\\n`," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " alpha,`... fract ion of one-year-old seeds that germinate in spring\\n`," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 " beta,`... fraction of two-year-old seeds that germinate in spring\\n`," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " sigma ,`... fraction of seeds that survive a given winter\\n`);" }}{PARA 6 " " 1 "" {TEXT -1 8 "\n Model:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"N G6#,&%\"tG\"\"\"\"\"#F),&**%&alphaGF)%\"gGF)%&sigmaGF)-F%6#,&F(F)F)F)F )F)*,%%betaGF)F.\"\"\")F/F*F5,&F)F)F-!\"\"F)-F%6#F(F)F)" }}{PARA 6 "" 1 "" {TEXT -1 285 "\n Parameters of the model:\n g ... number of s eeds produced per plant in autumn\n alpha ... fraction of one-year -old seeds that germinate in spring\n beta ... fraction of two-yea r-old seeds that germinate in spring\n sigma ... fraction of seeds that survive a given winter\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Specification of param eters and solution of the model:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g:=11;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "alpha:=0. 2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "beta:=0.9;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sigma:=0.25;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "lprint(`\\n Concrete model:`);P lantPropagation;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "lprint(`\\n \+ solution N(t)=`);rsolve(PlantPropagation,N(t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lprint(`\\n\\n\\n Characteristic roots:`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solve(x^2-alpha*g*sigma*x-beta*g*sigma^2* (1-alpha),x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "if (g>1/(alpha*sig ma+beta*sigma^2*(1-alpha)))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " t hen lprint(`\\n\\nThe plant population propagates.\\n`)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 " else lprint(`\\n\\nThe plant population dec ays.\\n`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&a lphaG$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG$\"\"*!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"#D!\"#" }}{PARA 6 " " 1 "" {TEXT -1 17 "\n Concrete model:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"NG6#,&%\"tG\"\"\"\"\"#F),&-F%6#,&F(F)F)F)$\"$]&!\"$-F%6#F($ \"'+]\\!\"'" }}{PARA 6 "" 1 "" {TEXT -1 19 "\n solution N(t)=" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&*(-%%sqrtG6#\"$8*\"\"\",*-%\"NG6# \"\"\"\"$?#*&F,F/F&F*!#?-F-6#\"\"!!$<&*&F3F/F&F*\"#6F/),$*&F*F*,&\"#bF /*$F&F*!\"&!\"\"!#**%\"tGF/F*FFDF@FAFBF/F*FPF@F/" }}{PARA 6 "" 1 "" {TEXT -1 25 "\n\n\n Characteristic roots:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+lC(R![!#5$\"+YsRI5!\"*" }}{PARA 6 "" 1 "" {TEXT -1 35 "\n\nThe plant population propagates.\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Simulation \+ of the plant population evolution" }}{PARA 0 "" 0 "" {TEXT -1 68 "Spec ify the time of simulation tmax and/or initial values N[0], N[1]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tmax:=10:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "N:=array(0..tmax):N[0]:=0:N[1]:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " for i from 2 to tmax do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " N[i]:=alpha*g*sigma*N[i-1]+beta*g*sigma^2*(1-alpha )*N[i-2]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p1:=pointplot([[t,N[t]]$t=0..tmax],style=point,symbol=circle,color =red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "p2:=pointplot([[t,N[t]]$ t=0..tmax],style=line,labels=[`year`,`No. of plants`],title=`Propagati on of annual plants`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p 1,p2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('N'):" }}{PARA 13 "" 1 "" {INLPLOT 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