{VERSION 2 3 "APPLE_68K_MAC" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 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 "" 0 256 1 {CSTYLE "" 
-1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 
0 }{PSTYLE "" 0 257 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 "" 0 258 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 256 "" 0 "" {TEXT -1 43 "Sequences -- Classifying
 Equilibrium Points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 65 "The cell below finds the equilibrium points of the logi
stic model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 
256 30 "p(n + 1) = a (1 - b p(n)) p(n)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 257 15 "Evaluate it now" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "f := p -> a * (1 - b * p) * p:\n\neqpts := solve(f(p) = p, p);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 
"The next cell finds 2-cycles for the same model.  " }{TEXT 258 16 "Ev
aluate it now." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "cycle := solve(f(f(p)) = p,p);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 78 "\nThe next cell saves the two 2-cycle poi
nts that are not equilibrium points.  " }{TEXT 259 16 "Evaluate it now
." }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pt1 :
= cycle[3];\npt2 := cycle[4];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 53 "The next cell computes the derivativ
e of the function" }}{PARA 258 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 260 
14 "g(p) = f(f(p))" }}{PARA 0 "" 0 "" {TEXT 261 1 "\n" }{TEXT -1 1 "\n
" }{TEXT 262 15 "Evaluate it now" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "g := f@f:\ngp := p -> D(g)(p):\ngp(p);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "\nNext we want to solve the equati
on  " }{TEXT 263 12 "gp(pt1) = -1" }{TEXT -1 30 " as shown in the cell
 below.  " }{TEXT 264 23 "Evaluate that cell now." }{TEXT -1 1 "\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(gp(pt1) = -1, a);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "F
inally, we evalaute " }{TEXT 265 7 "gp(pt1)" }{TEXT -1 24 " for specif
ic values of " }{TEXT 266 1 "a" }{TEXT -1 5 " and " }{TEXT 267 1 "b" }
{TEXT -1 37 " as described in the browser window. " }{TEXT 268 24 " Ev
aluate the cell below" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "b := .001:\na := 3.2:\n
evalf(gp(pt1));\n\na := 4:\nevalf(gp(pt1));" }}}}{MARK "0 0 0" 43 }
{VIEWOPTS 1 1 0 1 1 1803 }