{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 93 41 59 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 
93 41 59 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 93 41 59 0 0 
1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 93 41 59 0 0 1 0 0 0 0 0 
0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 40 "Harvesting an Age-Distrib
uted Population" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 31 "Part 1.  Sustainabl
e harvesting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 31 "Build the Leslie growth matrix " }{XPPEDIT 18 0 "L" "6#
%\"LG" }{TEXT -1 78 ", as you did in the Leslie Growth Models module.
\n[Notes:  (1) We first define " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT 
-1 66 " to be a matrix of zeros. (2) Next we insert symbolic birth rat
es " }{XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 21 " in the fir
st row of " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 229 ".  (3) Then we
 insert symbolic survival rates on the first subdiagonal.  (4) The for
-from-do structure is a way to tell Maple to do the same operation ove
r and over.  (5) The odd word \"od\" is \"do\" backwards -- it means \+
\"end do\".]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L:=diag(0,0,0,0,0,0,0,0,0,0,0,0):" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a:='a':for j from 1 to 12 do L[1,
j]:=a[j] od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "b:='b':for j from 1
 to 11 do L[j+1,j]:=b[j] od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "pr
int(L);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Construct the matric
es " }{XPPEDIT 18 0 "H" "6#%\"HG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "I[1
2" "6#&%\"IG6#\"#7" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "(I[12]-H)*L" 
"6#*&,&&%\"IG6#\"#7\"\"\"%\"HG!\"\"F)%\"LGF)" }{TEXT -1 18 " in symbol
ic form:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "h:='h':H:=diag(h[1],h[2
],h[3],h[4],h[5],h[6],h[7],h[8],h[9],h[10],h[11],h[12]):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 45 "I12:=array(identity,1..12,1..12): evalm(I12):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalm((I12-H)&*L);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Reconstruct the characteristic pol
ynomial " }{XPPEDIT 18 0 "p(lambda)" "6#-%\"pG6#%'lambdaG" }{TEXT -1 
1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p(lambda):=charpoly(L,lambd
a);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Reconstruct the auxiliar
y function " }{XPPEDIT 18 0 "q(lambda" "6#-%\"qG6#%'lambdaG" }{TEXT 
-1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "c:='c':for j from 1 to 1
1 do c[j]:=mul(b[i],i=1..j) od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "
j:='j': q(lambda):=a[1]/lambda+sum(a[j]*c[j-1]/lambda^j,j=2..12);\n" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Assign birth and survival rates \+
for the New Zealand sheep population." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 86 "a:=array(1..12,[0,0.045,0.391,0.472,0.484,0.546,0.543,0.502,0.46
8,0.459,0.433,0.421]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "b:=array(
1..11,[0.845,0.975,0.965,0.950,0.926,0.895,0.850,0.786,0.691,0.561,0.3
70]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "for j from 1 to 12 do L[1,
j]:=a[j] od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for j from 1 to 11 \+
do L[j+1,j]:=b[j] od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(L);
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Now let Maple compute the e
igenvalues of L." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(L);\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Recall that we assigned the n
ame " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" }{TEXT -1 79 "
 to the uniqe positive eigenvalue of L.  Fill in that dominant eigenva
lue here:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "lambda1:= ???;\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Find a uniform harvesting policy:
 What fraction of every age group can be harvested to return the popul
ation to its starting state?  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "Part 2.  Harvesting th
e youngest class" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 36 "Calculate the net reproduction rate " }{XPPEDIT 
18 0 "R" "6#%\"RG" }{TEXT -1 34 ".  Recall that we earlier defined " }
{XPPEDIT 18 0 "c[j" "6#&%\"cG6#%\"jG" }{TEXT -1 7 " to be " }{XPPEDIT 
18 0 "b[1]*b[2]...b[j]" "6#;*&&%\"bG6#\"\"\"F(&F&6#\"\"#F(&F&6#%\"jG" 
}{TEXT -1 10 " for each " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 65 ".
 The following line computes the sum of products of appropriate " }
{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 7 "'s and " }{XPPEDIT 18 0 "c" "
6#%\"cG" }{TEXT -1 3 "'s." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "R:=a[1
]: for j from 2 to 12 do R:=R+a[j]*c[j-1] od: R;\n" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 83 "Calculate the fraction of lambs to be harvested i
n a sustainable harvesting policy." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Calculate the stable popul
ation distribution for this policy." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
260 27 "Part 3.  Optimal harvesting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Calculate the age distribution f
or the optimal harvesting policy." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 
16 "Part 4.  Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "36" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }