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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 47 "Discrete Population Mode
ls for a Single Species" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"NG6#,&
%\"tG\"\"\"F)F)-%\"FG6$%+parametersG-F%6#F(" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 23 "Predefined functions F:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Malthus:=(r,N)->r*N;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "Gompertz:=(r,b,N)->r*N^(1-b);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "Pielou:=(alpha,beta,N)->N*(alpha+1)/(1+beta*N);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Ricker:=(alpha,beta,N)->N*exp(a
lpha-beta*N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Verhulst:=(alpha,b
eta,N)->max(0,N*(alpha+1-beta*N));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "Hassell:=(alpha,beta,b,N)->(alpha+1)/(1+beta*N)^b;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "fundamental:=(alpha,beta,b,N)->N*(alpha+1)/(1+
(beta*N)^b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(MalthusGR6$%\"rG%\"
NG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"9%F/F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)GompertzGR6%%\"rG%\"bG%\"NG6\"6$%)operatorG%&arrowGF
**&9$\"\"\")9&,&F0F09%!\"\"F0F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%'PielouGR6%%&alphaG%%betaG%\"NG6\"6$%)operatorG%&arrowGF**&*&9&\"\"
\",&9$\"\"\"F4F4F4F1,&F4F4*&9%F4F0F4F4!\"\"F*F*F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'RickerGR6%%&alphaG%%betaG%\"NG6\"6$%)operatorG%&arro
wGF**&9&\"\"\"-%$expG6#,&9$\"\"\"*&9%F6F/F6!\"\"F6F*F*F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)VerhulstGR6%%&alphaG%%betaG%\"NG6\"6$%)operat
orG%&arrowGF*-%$maxG6$\"\"!*&9&\"\"\",(9$\"\"\"F7F7*&9%F7F3F7!\"\"F7F*
F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(HassellGR6&%&alphaG%%betaG%
\"bG%\"NG6\"6$%)operatorG%&arrowGF+*&,&9$\"\"\"F2F2\"\"\"),&F2F2*&9%F2
9'F2F29&!\"\"F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,fundamentalG
R6&%&alphaG%%betaG%\"bG%\"NG6\"6$%)operatorG%&arrowGF+*&*&9'\"\"\",&9$
\"\"\"F5F5F5F2,&F5F5)*&9%F5F1F59&F5!\"\"F+F+F+" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Specif
y the function F" }}{PARA 7 "" 0 "" {TEXT -1 62 "Write parameters as f
loating 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 wri
te a different one." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F:=N
->fundamental(5.3,1.0,3.5,N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 22 "Analysis of the model:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "S:=[solve(F(x)=x,x)]:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "equil:=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 equ
il:=S[i] fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "    od:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "PositiveEquilibrium:=evalb(equil>0):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "if PositiveEquilibrium " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "   then xmax:=1.5*equil" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "   else S:=sort([solve(F(x)=x/3,x)]):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 59 "        if type(S[-1],float) and (S[-1]>0) th
en xmax:=S[-1]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "                 \+
                          else xmax:=1" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "        fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   fi:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "p1:=plot([F(x),x],x=0..xmax,labels=[`N`,`y`]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "t1:=textplot([xmax,xmax,` y=N`],ali
gn=RIGHT):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "t2:=textplot([xmax,F(
xmax),` y=F(N)`],align=RIGHT):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d
isplay(p1,t1,t2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "print(lprint(`
Model:`),`N(t+1)`= F(`N(t)`));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "i
f PositiveEquilibrium then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "   Fp
rime:=(N)->diff(F(N),N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "   lamb
da:=subs(x=equil,Fprime(x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   f
i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "if PositiveEquilibrium" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " then" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 57 "   if (abs(lambda)<1) then s:=`is assymptoticaly stable;`" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "      elif (abs(lambda)>1) then s:=
`is unstable;`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "                 \+
          else s:=`, critical case;`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "      fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "   lprint(`\\n St
eady state`,evalf(equil),s,`eigenvalue=`,evalf(lambda));" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "   if (lambda<-1) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "      upper:=eval(maximize(F(x),\{x\},\{x=0..equil\})
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "      lower:=F(upper):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "      fi:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "   if (lambda<-1) then" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "      lprint(`\\n lower bound=`,evalf(lower)," }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "             `upper bound=`,evalf(upper),`\\n`)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "      fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "   if (lambda>1) then" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "      lprint(`\\n the population either goes to extinction`," 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "             `or it is unbounded
\\n`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "      fi" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 " else lprint(`\\n No positive steady state`)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "if (not PositiveEquilibrium) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "   Fprime:=(N)->diff(F(N),N):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "   lambda:=subs(x=0,Fprime(x))" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "if ((not Pos
itiveEquilibrium) and (lambda<1)) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "   then lprint(` the population decays\\n`) fi;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 46 "if ((not PositiveEquilibrium) and (lambda>1)) " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "   then lprint(` the population gro
ws unlimitedly\\n`) fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 
-1 39 "Simulation of the population evolution:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 0 "" {TEXT 
-1 77 "Specify the length of simulation tmax. You can also set the ini
tial value N0." }{TEXT 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "tmax:=50:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "if PositiveEquili
brium 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 27 "N:=array(0..tmax):N[0]:=N0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "   for i from 1 to tmax do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " \+
  N[i]:=F(N[i-1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p1:=pointplot([[t,N[t]]$t=0..tmax],
style=point,symbol=circle,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 107 "p2:=pointplot([[t,N[t]]$t=0..tmax],style=line,labels=[`t`,`N `]
,title=`Simulated evolution of population`):" }}{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,p3]" }}{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 
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