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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Lotka-Volterra type mode
l of interacting populations" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"N
G6#,&%\"tG\"\"\"F)F)*&-F%6#F(\"\"\"-%$expG6#,(%#b1GF)*&%$a11GF)F+F)!\"
\"*&%$a12GF)-%\"MGF,F)F5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"MG6
#,&%\"tG\"\"\"F)F)*&-F%6#F(\"\"\"-%$expG6#,(%#b2GF)*&%$a21GF)-%\"NGF,F
)!\"\"*&%$a22GF)F+F)F7F)" }}{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 "" {TEXT -1 24 "First, load two
 packages" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg):with(plot
s):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 35 "Specify the parameters of the model" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a11:=1;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a12:=0.2;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "a21:=0.7;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a22:=1;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b1:=2.1;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "b2:=5;" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 21 "Analysis of the model" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "###Characteristics of the first population" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "lprint(`\\nFirst population:`);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "if (b1>0) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "   if (a11>0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "      if (b1<2) then Stab:=`, asymptoticaly stable.`" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 67 "                elif (b1>2) then Stab:=`, uns
table.` else Stab:=`.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "      fi:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "      lprint(`   Producer with
 intraspecific competition; carryin cappacity =`,   evalf(b1/a11),Stab
)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   elif (a11<0) then" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 63 "         lprint(`   Producer with intrasp
ecific mutualism.`)   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "       el
se lprint(`   Producer.`) fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "if (b1<=0) then" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "   if (a11>0) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "      lprint(`   Consumer with intraspecific competit
ion.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   elif (a11<0) then" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "         lprint(`   Consumer with i
ntraspecific mutualism.`)   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "   \+
    else lprint(`   Consumer.`) fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "###Characteristics of the second population" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "lprint(`\\nSecond population:`);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "if (b2>0) then" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "   if (a22>0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "      if (b2<2) then Stab:=`, asymptoticaly stable.`" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 67 "                elif (b2>2) then Stab:=`, u
nstable.` else Stab:=`.`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "      fi
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "      lprint(`   Producer wit
h intraspecific competition; carryin cappacity =`,   evalf(b2/a22),Sta
b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   elif (a22<0) then" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "         lprint(`   Producer with i
ntraspecific mutualism.`)   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "   \+
    else lprint(`   Producer.`) fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "if (b2<=0) then" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "   if (a22>0) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "      lprint(`   Consumer with intraspecific competit
ion.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   elif (a22<0) then" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "         lprint(`   Consumer with i
ntraspecific mutualism.`)   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "   \+
    else lprint(`   Consumer.`) fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "### Characteris
tic of interaction" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "if ((a12>0) a
nd (a21>0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "   lprint(`\\nI
nteraction: Competition.`) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "i
f ((a12<0) and (a21<0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "   \+
lprint(`\\nInteraction: Mutualism.`) fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "if (a12*a21<0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "   lprint(`\\nInteraction: Predator-prey.`) fi;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "if ((a12*a21=0) and (a12+a21>0)) then" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "   lprint(`\\nInteraction: Ammensalism.`)
 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if ((a12*a21=0) and (a12+a2
1<0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "   lprint(`\\nInterac
tion: Commensalism.`) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "if (ab
s(a12)+abs(a21)=0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "   lprin
t(`\\nInteraction: None.`) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "### Analysis of equilibria" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"lprint(`\\nEquilibria:`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Eq1:=
[0,0]:Stab:=``:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "if ((b1>0) and (
b2>0)) then Stab:=`repulsive`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "  \+
 elif ((b1<0) and (b2<0)) then Stab:=`asymptoticaly stable`" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "   elif (b1*b2<0) then Stab:=`unstable`" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "lprint(`   `,evalf(Eq1),Stab);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if ((b2>0) and (a22>0)) \+
then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  Eq2:=[0,b2/a22]:Stab:=``:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "  lambda1:=evalf(exp((b1*a22-b2
*a12)/a22)):lambda2:=evalf(abs(1-b2)):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 63 "  if (max(lambda1,lambda2)<1) then Stab:=`asymptoticaly stable
`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "     elif(min(lambda1,lambda2)
>1) then Stab:=`repulsive`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "     \+
elif(max(lambda1,lambda2)>1) then Stab:=`unstable`" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "  fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  lprint
(`   `,evalf(Eq2),Stab)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " else Eq
2:=[0,0]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if ((b1>0) an
d (a11>0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  Eq3:=[b1/a11,0
]:Stab:=``:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "  lambda1:=evalf(exp
((b2*a11-b1*a21)/a11)):lambda2:=evalf(abs(1-b1)):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 63 "  if (max(lambda1,lambda2)<1) then Stab:=`asymptoti
caly stable`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "     elif(min(lambd
a1,lambda2)>1) then Stab:=`repulsive`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "     elif(max(lambda1,lambda2)>1) then Stab:=`unstable`" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "  lprint(`   `,evalf(Eq3),Stab)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 " else Eq3:=[0,0]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "if (a11*a22-a12*a21<>0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 
"   Eq4:=[(b1*a22-b2*a12)/(a11*a22-a12*a21),(b2*a11-b1*a21)/(a11*a22-a
12*a21)]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "   else Eq4:=[0,0]" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "IntEquil:=((Eq4[1]>0) and (Eq4[2]>0)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "if (IntEquil) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "   Stab:=``:A:=array([[1-a11*Eq4[1],-a12*Eq4[1]],[-a21*Eq4[2],1-a2
2*Eq4[2]]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "   lambda:=[evalf(e
igenvals(A))]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "   lambda1:=abs(l
ambda[1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "   if (nops(lambda)>1
) then lambda2:=abs(lambda[2]) else lambda2:=lambda1 fi:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 64 "   if (max(lambda1,lambda2)<1) then Stab:=`as
ymptoticaly stable`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "     elif(mi
n(lambda1,lambda2)>1) then Stab:=`repulsive`" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "     elif(max(lambda1,lambda2)>1) then Stab:=`unstabl
e`" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  fi:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "  lprint(`   `,evalf(Eq4),Stab)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 41 "### Check of dissipativity and permanence" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "Dissip:=((a11>0) and (a22>0) and \+
(max(a12,a21)>0)) or ((a11>0) and (a12>0) and (a21<0) and (a22<=0)) or
 ((a22>0) and (a21>0) and (a12<0) and (a11<=0)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "Perm:=Dissip and IntEquil and (a11*a22-a12*a21>0):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "if Perm then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "   lprint(`\\nThe system is permanent.`)" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "   elif Dissip then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "     lprint(`\\nThe system is dissipative but it is n
ot permanent.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "### Plo
t phase portrait" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "if (a21<>0) the
n h:=b2/a21 else h:=0 fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Nmax:=
1.1*max(Eq3[1],Eq4[1],h):if (Nmax=0) then Nmax:=1 fi:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "if (a12<>0) then h:=b1/a12 else h:=0 fi:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Mmax:=1.1*max(Eq2[2],Eq4[2],h):if (
Mmax=0) then Mmax:=1 fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "if (a12
<>0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " p1:=plot((b1-a11*x)/a
12,x=0..Nmax,view=0..Mmax,labels=[`N`,`M`],color=red):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "  elif (a11<>0) then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "  p1:=implicitplot(x=b1/a11,x=0..Nmax,view=0..Mmax,la
bels=[`N`,`M`],color=red)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "    el
se p1:=plot(-1,x=0..Nmax,view=0..Mmax,labels=[`N`,`M`],color=red)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "if (a22<>0) then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "   p2:=p
lot((b2-a21*x)/a22,x=0..Nmax,color=blue)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "   elif (a21<>0) then" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 67 "       p2:=implicitplot(x=b2/a21,x=0..Nmax,view=0..Mmax,color=
blue)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "p3:=pointplot([[0,0],Eq2,Eq3,Eq4],color=black,symbol=
BOX,title=`Phase portrait`):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P:=
[p1,p2,p3]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if ((b1>2) and (a11>
0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "   H:=evalf(exp(b1-1)/a
11):h:=evalf(H*exp(b1-a11*H)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " \+
  p4:=pointplot([[h,0],[H,0]],style=line,color=red,thickness=2):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "   P:=[op(P),p4]" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "if ((b2>2
) and (a22>0)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "   H:=evalf(
exp(b2-1)/a22):h:=evalf(H*exp(b2-a22*H)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "   p5:=pointplot([[0,h],[0,H]],style=line,color=blue,
thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "   P:=[op(P),p5]" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display(P);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 61 "unassign('h','H','Nmax','Mmax','P','p1','p2'
,'p3','p4','p5'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 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 67 "Specify the length of simulation tmax and/or initial values N0,
 M0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "tmax:=50:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N0:=1.
27906:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "M0:=4.10465:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tmaxP:=
0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N:=array(0..tmax):N[0]:=N0:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M:=array(0..tmax):M[0]:=M0:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "   for i from 0 while ((i<tmax) and
 (N[i]<1e20) and (M[i]<1e20)) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"   N[i+1]:=evalf(N[i]*exp(b1-a11*N[i]-a12*M[i])):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "   M[i+1]:=evalf(M[i]*exp(b2-a21*N[i]-a22*M[i])):" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "   if ((N[i+1]>=1e20) or (M[i+1]>
=1e20)) then tmaxP:=i fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "if (tmaxP>0) then tmax:=tmaxP f
i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if (tmaxP>0) then" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 91 "   lprint(`\\nWarning: At least one of the \+
populations is unbounded. Simulation aborted.\\n`)" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 6 "   fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Nmax:=m
ax(N[t]$t=0..tmax):Mmax:=max(M[t]$t=0..tmax):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "for i from 0 to tmax do N[i]:=N[i]/Nmax:M[i]:=M[i]/Mm
ax od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "p1:=pointplot([[t,N[t]]$
t=0..tmax],style=point,symbol=circle,color=red,labels=[`t`,``],ytickma
rks=0,view=0..1,title=`Simulated evolution of interacting populations`
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p2:=pointplot([[t,N[t]]$t=0..
tmax],style=line,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "p3:
=pointplot([[t,M[t]]$t=0..tmax],style=point,symbol=circle,color=blue):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p4:=pointplot([[t,M[t]]$t=0..tm
ax],style=line,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p5:=
textplot([-tmax/100,0.5,`N, M `],color=black,align=LEFT):" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "display(p1,p2,p3,p4,p5);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 83 "lprint(`\\nRed:  the first population (N).  Scale o
n the vertical axis: 0 ..`,Nmax);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
81 "lprint(`Blue: the second population (M). Scale on the vertical axi
s: 0 ..`,Mmax);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "unassign('N','M'
,'Nmax','Mmax','p1','p2','p3','p4','p5'):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6+-%'POINTSG6X7$\"\"!$
\"+#)p/EV!#57$$\"\"\"F'$\"+W-4EVF*7$$\"\"#F'$\"+f(\\gK%F*7$$\"\"$F'$\"
+j5=EVF*7$$\"\"%F'$\"+*3UdK%F*7$$\"\"&F'$\"+D-@FVF*7$$\"\"'F'$\"++0IAV
F*7$$\"\"(F'$\"+(fC(QVF*7$$\"\")F'$\"+\\&RQG%F*7$$\"\"*F'$\"+dD&yY%F*7
$$\"#5F'$\"+V*>!fQF*7$$\"#6F'$\"+L-05fF*7$$\"#7F'$\"+P.M]q!#67$$\"#8F'
$\"+S_+sYF*7$$\"#9F'$\"+>o`]!*F*7$$\"#:F'$\"+I*)=sWFao7$$\"#;F'$\"+%*z
z)>$F*7$$\"#<F'$\"+F,Ha(*F*7$$\"#=F'$\"+PkBhPFao7$$\"#>F'$\"+!>_wu#F*7
$$\"#?F'$\"+/a0w'*F*7$$\"#@F'$\"+1Y^\"f&Fao7$$\"#AF'$\"+W%fc'QF*7$$\"#
BF'$\"+fP.&e)F*7$$\"#CF'$\"+Qcn3Y!#77$$\"#DF'$\"+%)ze7PFao7$$\"#EF'$\"
+hSe;FF*7$$\"#FF'$\"+\\$f^$**F*7$$\"#GF'$\"+fx&[E%F*7$$\"#HF'$\"+z4O+&
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