{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 "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "" 4 256 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 "" 3 258 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 258 "" 0 "" {TEXT -1 41 "Delay Discrete Model for a Single Species" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"NG6#,&%\"tG \"\"\"F)F)*(%\"rGF)-F%6#F(F)-%\"fG6$%+parametersG-F%6#,&F(F)%\"TG!\"\" F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Pre defined functions f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Gom pertz:=(b,N)->1/N^b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Pielou:=(be ta,N)->1/(1+beta*N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Ricker:=(de lta,N)->exp(-delta*N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Verhulst: =(delta,N)->max(0,(1-delta*N));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " Hassell:=(beta,b,N)->1/(1+beta*N)^b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fundamental:=(beta,b,N)->1/(1+(beta*N)^b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Speci fy the function f, the intrinsic growth rate r, and the delay T" }} {PARA 7 "" 0 "" {TEXT -1 62 "Write parameters as floating point number (2.0 rather than 2)." }}{PARA 7 "" 0 "" {TEXT -1 85 "You need not use one of the predefined functions; you can also write a different one. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=N->fundamental(1.0,1 .3,N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=3.3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 27 "Analysis of a steady state:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S:=[solve(r*f(x)=1,x)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "equ il:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 by 1 to nops (S) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " if (type(S[i],float) and (S[i]>equil)) then equil:=S[i] fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PositiveEquilibri um:=evalb(equil>0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "if PositiveE quilibrium then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " fprime:=N->di ff(f(N),N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " CharEq:=lambda^(T +1)-lambda^T-r*equil*subs(x=equil,fprime(x))=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " S:=[solve(CharEq,lambda)]:DominantLambda:=0:abs(S[ 1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " for i from 1 by 1 to nop s(S) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " DL:=abs(S[i]):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " if (DL>DominantLambda) then DominantLambda:=DL fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " od :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " Stab:=evalb(DominantLambda< 1)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ts:=convert(T,string):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "print(lprint(`\\nModel:`),`N(t+1)`= r*N(t)*f(`N(t-`.Ts.`)`)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "if PositiveEquilibrium" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " if Stab" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " the n lprint(`\\nSteady state`,evalf(equil),`is assymptoticaly stable.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " else lprint(`\\nSteady state`, evalf(equil),`is not assymptoticaly stable.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " lprint(` \\nThe eigenvalues are`,S,`.`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " lprint(`\\nAbsolute value of the dominant eigenvalue is`,evalf(Domin antLambda),`.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " else lprint(`\\ nThere is no positive steady state.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 39 "Simulation of the population evolution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Specify the length of simulation t max. You can also set the initial value N0." }{TEXT 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "tmax:=50:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "if PositiveEquilibrium then N0:=0.01*equil" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " else N0:=1" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 6 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N:=array(-T..tmax):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "for i from -T to -1 do N[i]:=0 od:" }}{PARA 0 "" 0 "" {TEXT -1 47 "The assigment of \"tail\" values can be changed." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "N[0]:=N0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for i from 1 to tmax do N[i]:=r*N [i-1]*f(N[i-T-1]) od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p1:=pointp lot([[t,N[t]]$t=0..tmax],style=point,symbol=circle,color=red):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "p2:=pointplot([[t,N[t]]$t=0..tmax] ,style=line,labels=[`t`,`N `],title=`Simulated evolution of the popula tion`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "if PositiveEquilibrium" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " then p3:=plot(equil,t=0..tmax, color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " P:=[p1,p2,p 3]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " else P:=[p1,p2]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display(P);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "unassign('N','P' ):" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'POINTSG6X7$\"\"!$\"++w!y*=!#67$$ \"\"\"F'$\"+!3lFE'F*7$$\"\"#F'$\"+wCrm?!#57$$\"\"$F'$\"+z^(4y'F57$$\" \"%F'$\"+#H/$y@!\"*7$$\"\"&F'$\"+C*o#ojF@7$$\"\"'F'$\"+%o&e58!\")7$$\" \"(F'$\"+bU)G:\"FK7$$\"\")F'$\"+))='[9$F@7$$\"\"*F'$\"+%\\lZ`$F57$$\"# 5F'$\"+F'$\"+Y3RyCF@7$$\"#?F'$ \"+8a1\\EF57$$\"#@F'$\"+0(z&\\RF*7$$\"#AF'$\"+f;$Q1$F*7$$\"#BF'$\"+0F3 %e)F*7$$\"#CF'$\"+<'Q4z#F57$$\"#DF'$\"+dv(>6*F57$$\"#EF'$\"+NhD))GF@7$ $\"#FF'$\"+qKK2!)F@7$$\"#GF'$\"+YU(4S\"FK7$$\"#HF'$\"+7\"H;I*F@7$$\"#I F'$\"+R?#\\#>F@7$$\"#JF'$\"+#o>&*)>F57$$\"#KF'$\"+$RWlU$F*7$$\"#LF'$\" +O>m#Q$F*7$$\"#MF'$\"+PE)R%**F*7$$\"#NF'$\"+(f[6C$F57$$\"#OF'$\"+(>Qm0 \"F@7$$\"#PF'$\"+]4k@LF@7$$\"#QF'$\"+C'QL!*)F@7$$\"#RF'$\"+!fmkT\"FK7$ $\"#SF'$\"+.M$G6)F@7$$\"#TF'$\"+EWcu9F@7$$\"#UF'$\"+.H3.:F57$$\"#VF'$ \"+9:IhIF*7$$\"#WF'$\"+p*oC!QF*7$$\"#XF'$\"+aPPc6F57$$\"#YF'$\"+2CUvPF 57$$\"#ZF'$\"+cTPG7F@7$$\"#[F'$\"+7RCAQF@7$$\"#\\F'$\"+/b\")R)*F@7$$\" #]F'$\"+FMy29FK-%'COLOURG6&%$RGBG$\"*++++\"FKF'F'-%'SYMBOLG6#%'CIRCLEG -%&STYLEG6#%&POINTG-%'CURVESG6$7UF&F+F0F6F;FAFFFLFQFVFenFjnF_oFdoFioF^ pFcpFhpF]qFbqFgqF\\rFarFfrF[sF`sFesFjsF_tFdtFitF^uFcuFhuF]vFbvFgvF\\wF awFfwF[xF`xFexFjxF_yFdyFiyF^zFczFhzF][l-F]\\l6#%%LINEG-Fa\\l6$7S7$F'$ \"1++++w!y*=!#:7$$\"1LLL3x&)*3\"F]]lF[]l7$$\"1nm\"H2P\"Q?F]]lF[]l7$$\" 1LL$eRwX5$F]]lF[]l7$$\"1ML$3x%3yTF]]lF[]l7$$\"1nm\"z%4\\Y_F]]lF[]l7$$ \"1LLeR-/PiF]]lF[]l7$$\"1++DcmpisF]]lF[]l7$$\"1MLe*)>VB$)F]]lF[]l7$$\" 1,+DJbw!Q*F]]lF[]l7$$\"1nm;/j$o/\"!#9F[]l7$$\"1ML3_>jU6F\\_lF[]l7$$\"1 ++]i^Z]7F\\_lF[]l7$$\"1++](=h(e8F\\_lF[]l7$$\"1++]P[6j9F\\_lF[]l7$$\"1 L$e*[z(yb\"F\\_lF[]l7$$\"1nm;a/cq;F\\_lF[]l7$$\"1nmm;t,mF\\_lF[]l7$$\"1+]i!f#=$3#F\\_lF[]l 7$$\"1+](=xpe=#F\\_lF[]l7$$\"1nm\"H28IH#F\\_lF[]l7$$\"1n;zpSS\"R#F\\_l F[]l7$$\"1ML3_?`(\\#F\\_lF[]l7$$\"1M$e*)>pxg#F\\_lF[]l7$$\"1+]Pf4t.FF \\_lF[]l7$$\"1LLe*Gst!GF\\_lF[]l7$$\"1+++DRW9HF\\_lF[]l7$$\"1++DJE>>IF \\_lF[]l7$$\"1+]i!RU07$F\\_lF[]l7$$\"1++v=S2LKF\\_lF[]l7$$\"1nmm\"p)=M LF\\_lF[]l7$$\"1++](=]@W$F\\_lF[]l7$$\"1L$e*[$z*RNF\\_lF[]l7$$\"1,+]iC $pk$F\\_lF[]l7$$\"1m;H2qcZPF\\_lF[]l7$$\"1+]7.\"fF&QF\\_lF[]l7$$\"1nm; /OgbRF\\_lF[]l7$$\"1+]ilAFjSF\\_lF[]l7$$\"1MLL$)*pp;%F\\_lF[]l7$$\"1LL 3xe,tUF\\_lF[]l7$$\"1n;HdO=yVF\\_lF[]l7$$\"1,++D>#[Z%F\\_lF[]l7$$\"1nm T&G!e&e%F\\_lF[]l7$$\"1MLL$)Qk%o%F\\_lF[]l7$$\"1,]iSjE!z%F\\_lF[]l7$$ \"1,]P40O\"*[F\\_lF[]l7$F^[lF[]l-Fc[l6&Fe[lF'F'Ff[l-%&TITLEG6#%FSimula ted~evolution~of~the~populationG-%+AXESLABELSG6$%\"tG%#N~G-%%VIEWG6$;F 'F^[l%(DEFAULTG" 2 766 294 294 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }