{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 "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 "" 11 257 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 "" 11 258 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 "" 11 259 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 "" 11 260 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 "" 4 261 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 "" 4 262 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 "" 3 263 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 "" 11 264 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 "" 11 265 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 "" 11 266 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 "" 11 267 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 "" 11 268 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 "" 11 269 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 "" 4 270 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 "" 4 271 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 263 "" 0 "" {TEXT -1 31 "Model of a prey-predator system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"KG6#,&%\"tG\"\"\"F)F),(*&%\"RGF)-F%6#F(F)F)*&*&,&F,F)!\"\"F )F))F-\"\"#\"\"\"F5%\"kG!\"\"F2*&*(%\"CGF)F-F5-%\"PGF.F)F5F6F7F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#,&%\"tG\"\"\"F)F)*&*(%\"rGF)- %\"KG6#F(F)-F%F/F)\"\"\"%\"kG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 1 "" {TEXT -1 0 "" } {XPPMATH 20 "6#%\"KG" }{TEXT -1 30 " ... prey population density" }} {PARA 265 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#%\"PG" }{TEXT -1 34 " \+ ... predator population density" }}{PARA 266 "" 1 "" {TEXT -1 0 "" } {XPPMATH 20 "6#%\"RG" }{TEXT -1 51 " ... intrisic growth rate for th e prey population" }}{PARA 267 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#% \"kG" }{TEXT -1 49 " ... carrying cappacity for the prey population " }}{PARA 268 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#%\"CG" }{TEXT -1 26 " ... foraging efficiency" }}{PARA 269 "" 1 "" {TEXT -1 0 "" } {XPPMATH 20 "6#%\"rG" }{TEXT -1 98 " ... growth rate for the predato r population provided the prey density equals carrying cappacity" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "First, load a package." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Specify the parameters of the model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R:=3.1;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "C:=0.1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=2.1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# c heck inputs" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "if (evalf(R)<=1) the n lprint(`Error! The parameter R should be greater than 1.`) fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "if (evalf(r)<=1) then lprint(`Error ! The parameter r should be greater than 1.`) fi;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "if (evalf(k)<=0) then lprint(`Error! The parameter \+ k should be positive.`) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "if ( evalf(C)<=0) then lprint(`Error! The parameter C should be positive.`) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 25 "Analysis \+ of steady states" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "xmax:=m ax(10,1.1*R):ymax:=max(2.2,1.1*r):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "p1:=plot(\{2,3*(x-1)/(x+3)\},x=1..9,color=red,labels=[` R `,`r \+ `],title=`Plot of the actual values of the growth rates and the cr itical lines`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p2:=plot(\{(1+sq rt(x))/2,2*(1-1/x)\},x=1..4,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "p3:=pointplot([R,r],style=point,symbol=circle,view=[1 ..xmax,1..ymax]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(p1,p2, p3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lprint(`\\nSteady states:`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "lprint(` Extinction of both species ( 0 , 0 )`);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lprint(` - unstable.`);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "lprint(` Extinction of the preda tor with survival of the prey at its carrying capacity (`,k,`, 0 )`): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "if (R<=3) then Stab:=`unstable. ` else Stab:=`repulsive.` fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lp rint(` -`,Stab);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "SSprey:=evalf(k/r):SSpredator:=evalf(k*(R -(R-1)/r-1)/C):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "lprint(` Coexi stence of both species (`,SSprey,`,`,SSpredator,`)`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "beta:=(R-1)/(2*r):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Diskr:=(1+beta)^2-R:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "lambda:=[1-beta+sqrt(Diskr),1-beta-sqrt(Diskr)]:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "Compl:=evalb(evalf(R,40)>evalf((1+beta)^2,40)) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "if Compl" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " abslambda :=evalf(abs(lambda[1]),40):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " Os cil:=`oscilate.`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " if (evalf(r) <2) then Stab:=`asymptoticaly stable.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " elif (evalf(r)>2) then Stab:=`repulsive.`" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 " else Stab:=`critical case between asympto tical stability and repulsivity.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " if evalb(evalf(r) " 0 "" {MPLTEXT 1 0 36 " minlambda:=evalf(lambda[2],40) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " maxlambda:=evalf(lambda[1 ],40)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " else minlambda:=evalf(la mbda[1]):maxlambda:=evalf(lambda[1])" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " maxabslambda:=max(a bs(maxlambda),abs(minlambda)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " \+ if evalb(maxabslambda>1.0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " Stab:=`repulsive.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " el if evalb(maxabslambda<1.0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " Stab:=`asymptoticaly stable.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 " else Stab:=`critical case between asymptotical stability \+ and repulsivity.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " if evalb(minlambda<0.0) then Oscil:=`os cilate.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " else Oscil:=`do no t oscilate.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " fi" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lprin t(` -`,Stab);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "lprint(` \+ Solutions in neighbourhood of this equilibrium`,Oscil);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda:=evalf(lambda,40):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "lprint(`\\n Eigenvalues:`,evalf(lambda,10)) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "if Compl then lprint(` \+ absolute 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e&e0Ik(3SFfu7$Fj_l$\"1)*pK>!R%)y%Ffu7$F_`l$\"1L&o45:jW&Ffu7$Fd`l$\"1xY EQF6qgFfu7$Fi`l$\"13PsdQQhmFfu7$F^al$\"1#R\\mCLG?(Ffu7$Fcal$\"1@q[Ddv: xFfu7$Fhal$\"1im[!=UY8)Ffu7$F]bl$\"1P8QH,Ht&)Ffu7$Fbbl$\"1jv%QfuA)*)Ff u7$Fgbl$\"1/`ym#H'\\$*Ffu7$F\\cl$\"1e\"HO+BEm*Ffu7$Facl$\"18)*Htn;,5F0 7$Ffcl$\"1&)3YcD%*G5F07$F[dl$\"15GQs'p$f5F07$F`dl$\"1Da;A[v%3\"F07$Fed l$\"1u(p!y`266F07$Fjdl$\"1h>Vh\"pZ8\"F07$F_el$\"1)4=r2\"=e6F07$Fdel$\" 1G;99;fy6F07$Fiel$\"1B6\"F07$F^fl$\"1ld8(**p,A\"F07$Fcfl$\"1_5># *GHP7F07$Fhfl$\"1B/ZR5'\\D\"F07$F]gl$\"1dV4-WPs7F07$Fbgl$\"1sj:,*R')G \"F07$Fggl$\"1NWhp+q.8F07$F\\hl$\"1urL))=p>8F07$Fahl$\"10TghtWL8F07$Ff hl$\"1u'R+@LvM\"F07$F[il$\"11\"3_f#zf8F07$F`il$\"1>T;4)yEP\"F07$Feil$ \"1H()R\\)QVQ\"F07$Fjil$\"1e/!3WsgR\"F07$F_jl$\"1XqP\"p>rS\"F07$Fdjl$ \"1'*z0M-E=9F07$Fijl$\"1(fz'R6gG9F07$F^[m$\"1JnkzF!)Q9F07$Fc[m$\"1T%y+ Wm&[9F07$Fh[m$\"1*oc " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 " " 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT -1 29 "Plot of a simula ted evolution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Specify \+ the length of simulation tmax and/or initial values K0, P0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K0:= k:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P0:=.001*SSpredator:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tmax:=60:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "K:=array(0..tmax):K[0]:=K0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P:=array(0..tmax):P[0]:=P0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tmaxP:=tmax:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "fo r i from 0 while ((i0)) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " K[i+1]:=evalf(max(0,R*K[i]-(R-1)/k*K[i]^2-C*K[i]*P [i]/k)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " P[i+1]:=evalf(r*K[i] *P[i]/k):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " if (P[i+1]=0) then \+ tmaxP:=i+1 fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "tmax:=tmaxP:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Kmax:=max(K[t]$t=0..tmax):Pmax:=max(P[t]$t=0..tmax):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# Phase portrait" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "p1:=pointplot([[K[t],P[t]]$t=0..tmax],style=line, view=[0..Kmax,0..Pmax],color=magenta,labels=[`prey`,`predator`],title= `Simulated evolution of the prey-predator system at the phase plane`): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p2:=pointplot([K0,P0],style=poi nt,symbol=circle):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p1,p2 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "unassign('p1','p2'):lprint(` \\n`):lprint(`\\n`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "# Time dependence" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 0 to tmax do K[i]:=K[i]/Kmax:P[i]:=P[i]/Pm ax od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "p1:=pointplot([[t,K[t]]$ t=0..tmax],style=point,symbol=circle,color=blue,labels=[`t`,``],ytickm arks=0,view=0..1,title=`Simulated evolution of the prey-predator syste m`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p2:=pointplot([[t,K[t]]$t=0 ..tmax],style=line,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 " p3:=pointplot([[t,P[t]]$t=0..tmax],style=point,symbol=circle,color=red ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p4:=pointplot([[t,P[t]]$t=0.. tmax],style=line,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p5: =textplot([-tmax/100,0.5,`K, P `],color=black,align=LEFT):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display(p1,p2,p3,p4,p5);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "lprint(`\\nBlue: the prey population. Scale on vertical axis: 0 ..`,Kmax);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "lpr int(`Red: the predator population. 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Scale on vertical axis: 0 .. 1" }}{PARA 6 "" 1 "" {TEXT -1 73 "Red: the predator population. Scale on vertical axis: 0 \+ .. 14.29105629" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 33 "Model of a host-parasitoid system" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6#,&%\"tG\"\"\"F)F)*(%\"rGF)-F% 6#F(F)-%$expG6#,$*&%\"aGF)-%\"PGF-F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"PG6#,&%\"tG\"\"\"F)F)*&-%\"HG6#F(F),&F)F)-%$expG6# ,$*&%\"aGF)-F%F-F)!\"\"F6F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#%\"HG" }{TEXT -1 30 " ... host \+ population density" }}{PARA 258 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6# %\"PG" }{TEXT -1 36 " ... parasitoid population density" }}{PARA 259 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#%\"rG" }{TEXT -1 52 " ... \+ intrinsic growth rate for the host population" }}{PARA 260 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#-%$expG6#,$*&%\"aG\"\"\"%\"PGF)!\"\"" } {TEXT -1 54 " ... probability that a host pupa is not parasitized" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Specify the parameters of the model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=1.1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a:=1.25; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# check inputs" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "if ((r<=0 ) or (a<=0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " lprint(`Erro r! The both parameters should be positive.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Compute the steady state of popula tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "if (r>1) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " Hs:=r*ln( r)/((r-1)*a):Ps:=ln(r)/a:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " lpr int(`\\nUnstable steady state - host population: `,Hs);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " lprint(` - parasi toid population:`,Ps);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "else lpri nt(`\\nThere is no positive steady state.`); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 29 "Plot of a simulated evolution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "You may specify the initial values H0 and P0 and/or the length \+ of simulation tmax." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "H0:=1: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P0:=0.05:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tmax:=60:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H:=array(0..tmax):H[0]:=H0:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P:=array(0..tmax):P[0]:=P0:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "if (H0>0) then HP:=H0 else HP:=1 fi :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "if (P0>0) then PP:=H0 else PP: =1 fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i from 0 while ((i " 0 "" {MPLTEXT 1 0 38 " H[i+1]:=evalf(r*H[i]*exp(-a*P[i])):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 " P[i+1]:=evalf(H[i]*(1-exp(-a*P[i])))" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tmax:=i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# Phase port rait" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "p1:=pointplot([[H[t],P[t]] $t=0..tmax],style=line,color=magenta,labels=[`host`,`parasitoid`],titl e=`Simulated evolution of the host-parasitoid system at the phase plan e`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p2:=pointplot([H0,P0],style =point,symbol=circle):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p 1,p2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "unassign('p1','p2'):lprin t(`\\n`):lprint(`\\n`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "# Time dependence" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Hmax:=max(H[t]$t=0..tmax):Pmax:=max(P[t]$t=0..tm ax):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 0 to tmax do H[i ]:=H[i]/Hmax:P[i]:=P[i]/Pmax od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "p1:=pointplot([[t,H[t]]$t=0..tmax],style=point,symbol=circle,colo r=blue,labels=[`t`,``],ytickmarks=0,view=0..1,title=`Simulated evoluti on of the host-parasitoid system`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p2:=pointplot([[t,H[t]]$t=0..tmax],style=line,color=blue):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p3:=pointplot([[t,P[t]]$t=0..tmax], style=point,symbol=circle,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p4:=pointplot([[t,P[t]]$t=0..tmax],style=line,color=red):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p5:=textplot([-tmax/100,0.5,`H, P ` ],color=black,align=LEFT):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "displ ay(p1,p2,p3,p4,p5);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "lprint(`\\nB lue: the host population. Scale on vertical axis: 0 ..`,Hmax);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "lprint(`Red: the parasitoid popu lation. 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