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{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
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{PARA 0 "" 0 "" {TEXT -1 69 "   Lets look at the single species discre
te population dynamics model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "P[t+1]-P[t] = F(P[t
]);" "6#/,&&%\"PG6#,&%\"tG\"\"\"F*F*F*&F&6#F)!\"\"-%\"FG6#&F&6#F)" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "   or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "         " }{XPPEDIT 18 0 "P[t+1] = f(P[t]);" "6#/&%\"PG6#,&%\"t
G\"\"\"F)F)-%\"fG6#&F%6#F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "   where" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{XPPEDIT 18 0 "f(P[t
]) = P[t]+F(P[t]);" "6#/-%\"fG6#&%\"PG6#%\"tG,&&F(6#F*\"\"\"-%\"FG6#&F
(6#F*F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 6 "   for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "f(P) = alpha*P-beta*P^2;" "
6#/-%\"fG6#%\"PG,&*&%&alphaG\"\"\"F'F+F+*&%%betaGF+*$F'\"\"#F+!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
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{PARA 0 "" 0 "" {TEXT -1 15 "   Lets define " }{XPPEDIT 18 0 "f(P);" "
6#-%\"fG6#%\"PG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F(P);" "6#-%\"FG6
#%\"PG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "f := P -> alpha*P - beta*P^2;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
y := alpha*P - beta*P^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F := P -> f(P)-P;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "     Lets calculate the derivative of f.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "yp := d
iff(y,P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fp := unapply(yp,P);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "   Lets determine where our mod
el is valid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "so
lve(f(P)=0,P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "  \+
 Lets find the equilibrium populations for this system." }}{PARA 0 "" 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(y=P,P);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(P)=P,P);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 69 "   Lets determine the two period points
 and three period points of f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "solve(f(f(P))=P,P);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(f
(f(f(P)))=P,P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "  \+
  Lets choose an alpha and a beta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "alpha := 1+r;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "beta := r/K;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(f(f(P))=P,P)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(f(f(f(P)))=P,P);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r := 0.3;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "K := 100;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solv
e(f(f(P))=P,P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(f(
f(f(P)))=P,P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "  \+
 Now lets analyze f and F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 43 "   Lets determine where our model is valid." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(P)=0,P);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "plot(\{y,P\},P=0..alpha/beta,color=[blue,red]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(abs(yp),P=0..alpha/beta);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(abs(yp)<1,P);" }}}
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" {MPLTEXT 1 0 27 "plot(F(P),P=0..alpha/beta);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 49 "  Now lets change our model slightly and \+
consider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
7 "       " }{XPPEDIT 18 0 "P[t+1] = f(P[t])" "6#/&%\"PG6#,&%\"tG\"\"
\"F)F)-%\"fG6#&F%6#F(" }{TEXT -1 7 " + g(t)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "g := t -> 10;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "g(10.);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "P[0] := K/2.;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for t from 1 by 1 to 100 do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "  P[t] := f(P[t-1])+g(t):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "P[100];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
listplot([seq(P[t],t=0..100)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 20 "   Now lets consider" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 
0 "P[t+1]-P[t] = f*P[t-1]-d*P[t];" "6#/,&&%\"PG6#,&%\"tG\"\"\"F*F*F*&F
&6#F)!\"\",&*&%\"fGF*&F&6#,&F)F*F*F-F*F**&%\"dGF*&F&6#F)F*F-" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "
   or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " \+
  " }{XPPEDIT 18 0 "P[t+1] = P[t]+f*P[t-1]-d*P[t];" "6#/&%\"PG6#,&%\"t
G\"\"\"F)F),(&F%6#F(F)*&%\"fGF)&F%6#,&F(F)F)!\"\"F)F)*&%\"dGF)&F%6#F(F
)F2" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(1-d)*P[t]+f*P[t-1];" "6#,&*&,&
\"\"\"F&%\"dG!\"\"F&&%\"PG6#%\"tGF&F&*&%\"fGF&&F*6#,&F,F&F&F(F&F&" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }}}{EXCHG {PARA 0 ">
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{PARA 0 "" 0 "" {TEXT -1 27 "   Determine the stability." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 
0 "lambda = (1-d+sqrt((1-d)^2+4*f))/2;" "6#/%'lambdaG*&,(\"\"\"F'%\"dG
!\"\"-%%sqrtG6#,&*$,&F'F'F(F)\"\"#F'*&\"\"%F'%\"fGF'F'F'F'F0F)" }
{TEXT -1 9 "    or   " }{XPPEDIT 18 0 "lambda = (1-d-sqrt((1-d)^2+4*f)
)/2;" "6#/%'lambdaG*&,(\"\"\"F'%\"dG!\"\"-%%sqrtG6#,&*$,&F'F'F(F)\"\"#
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"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lamb
da[1] := (1-d+sqrt((1-d)^2+4*f))/2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "lambda[2] := (1-d-sqrt((1-d)^2+4*f))/2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "f := 0.02;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "d := 0.009;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda[1
];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda[2];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "P[0] := K/2.;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for t from 1 by 1 to 100 do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "  P[t+1] := (1-d)*P[t]+f*P[t-1]:" }
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{MPLTEXT 1 0 7 "P[100];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
listplot([seq(P[t],t=0..100)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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