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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
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{PARA 0 "" 0 "" {TEXT -1 69 "   Lets consider a predator-prey discrete
 dynamical population model." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "P[t+1]-P[t] = Delta*P" 
"6#/,&&%\"PG6#,&%\"tG\"\"\"F*F*F*&F&6#F)!\"\"*&%&DeltaGF*F&F*" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "F(P[t],Q[t]);" "6#-%\"FG6$&%\"PG6#%\"tG&%\"
QG6#F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 7 "       " }{XPPEDIT 18 0 "Q[t+1]-Q[t] = Delta*Q;" "6#/,&
&%\"QG6#,&%\"tG\"\"\"F*F*F*&F&6#F)!\"\"*&%&DeltaGF*F&F*" }{TEXT -1 3 "
 = " }{XPPEDIT 18 0 "G(P[t],Q[t]);" "6#-%\"GG6$&%\"PG6#%\"tG&%\"QG6#F)
" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "     or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 6 "      " }{XPPEDIT 18 0 "P[t+1] = P[t]+Delta*P;" "6#/&%\"
PG6#,&%\"tG\"\"\"F)F),&&F%6#F(F)*&%&DeltaGF)F%F)F)" }{TEXT -1 3 " = " 
}{XPPEDIT 18 0 "f(P[t],Q[t]);" "6#-%\"fG6$&%\"PG6#%\"tG&%\"QG6#F)" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "     " }{XPPEDIT 18 0 "Q[t+1] = Q[t]+Delta*Q;" "6#/&%\"QG6#,&%\"
tG\"\"\"F)F),&&F%6#F(F)*&%&DeltaGF)F%F)F)" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "g(P[t],Q[t]);" "6#-%\"gG6$&%\"PG6#%\"tG&%\"QG6#F)" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 7 "  where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{XPPEDIT 18 0 "f(P[t],Q[t]) = P[t]+r*P[t]*(1-P[t]/
K)-s*P[t]*Q[t];" "6#/-%\"fG6$&%\"PG6#%\"tG&%\"QG6#F*,(&F(6#F*\"\"\"*(%
\"rGF1&F(6#F*F1,&F1F1*&&F(6#F*F1%\"KG!\"\"F;F1F1*(%\"sGF1&F(6#F*F1&F,6
#F*F1F;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "g(P[t],Q[t]) = Q[t]-u*Q[t]+v*P[
t]*Q[t];" "6#/-%\"gG6$&%\"PG6#%\"tG&%\"QG6#F*,(&F,6#F*\"\"\"*&%\"uGF1&
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"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := (P,Q) -> P+r*P*(1-P/K)
-s*P*Q;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := (P,Q) -> Q-
u*Q+v*P*Q;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "   Solve \+
for the equilibrium populations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "solve(\{f(P,Q)=P,g(P,Q)=Q\},\{P,Q\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "  Plug in values for " }
{XPPEDIT 18 0 "r,K,v;" "6%%\"rG%\"KG%\"vG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "r := 0.7; K := 100; s := 0.05;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "u := 0.25; v := 0.025;" }}}
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "   Get the numerical valu
es of the equilibrium population." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "r*(v*K-u)/v/s/K; 1/v*u;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 30 "   Plot the equilibrium curves" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 
18 0 "P = 0;" "6#/%\"PG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Q = (r-
r/K*P)/s;" "6#/%\"QG*&,&%\"rG\"\"\"*(%\"rGF(%\"KG!\"\"%\"PGF(F,F(%\"sG
F," }{TEXT -1 5 " ;  (" }{XPPEDIT 18 0 "Delta*P = 0;" "6#/*&%&DeltaG\"
\"\"%\"PGF&\"\"!" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "Q = 0,P = u/v
;" "6$/%\"QG\"\"!/%\"PG*&%\"uG\"\"\"%\"vG!\"\"" }{TEXT -1 5 " ;  (" }
{XPPEDIT 18 0 "Delta*Q = 0;" "6#/*&%&DeltaG\"\"\"%\"QGF&\"\"!" }{TEXT 
-1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "g1 := p
lot((r-r/K*P)/s,P=0..2*K/r,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "g2 := i
mplicitplot(P-u/v=0,P=u/v-0.0001..u/v+0.0001,Q=0..4/v*u,color=red):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(g1,g2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "   Check the signs of   \+
" }{XPPEDIT 18 0 "Delta*P;" "6#*&%&DeltaG\"\"\"%\"PGF%" }{TEXT -1 7 " \+
 and  " }{XPPEDIT 18 0 "Delta*Q;" "6#*&%&DeltaG\"\"\"%\"QGF%" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "-u+v*P;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r*P-r/K*P^2-s*P*Q;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "50-0.007*50^2-0.05*50*10;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "P[0] := 20; Q[0] := 15;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "num \+
:= 50;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "for t from 0 by 1 to num do" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 " P[t+1] := f(P[t],Q[t]);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 " Q[t+1] := g(P[t],Q[t]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "listplot([seq(P[t],t=0..num
)],color=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "listplot
([seq(Q[t],t=0..num)],color=red);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "g1 := l
istplot([seq(P[t],t=0..num)],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "g2 := listplot([seq(Q[t],t=0..num)],color=red):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(g1,g2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "listplot([seq([P[t],Q[t]],t=0..num)
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for t from 0 by 1 to num do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " print(P[t],Q[t]);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P[0] := 20; Q[0] :
= 15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "num := 50;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for t f
rom 0 by 1 to num do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " P[t+1] := \+
f(P[t],Q[t])+5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " Q[t+1] := g(P[t
],Q[t]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "listplot([seq(P[t],t=0..num)],color=blue);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "listplot([seq(Q[t],t=0..num)],color=red);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "g1 := listplot([seq(P[t],t=0..num)],color=blue):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g2 := listplot([seq(Q[t],t
=0..num)],color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "di
splay(g1,g2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "listplot([
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for t f
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