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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 20 "Leslie Growth Models" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 258 31 "Part 1.  Age-distributed growth" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "B
uild the Leslie growth matrix " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT 
-1 32 ". \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 rates " }{XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 21 " i
n the first 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 oper
ation over and over.  (5) The odd word \"od\" is \"do\" backwards -- i
t 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 d
o 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 "print(L);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Construct the \+
characteristic polynomial " }{XPPEDIT 18 0 "p(lambda)" "6#-%\"pG6#%'la
mbdaG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p(lambda):
=charpoly(L,lambda);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Constru
ct the auxiliary function " }{XPPEDIT 18 0 "q(lambda" "6#-%\"qG6#%'lam
bdaG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "c:='c':for \+
j from 1 to 11 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 an
d survival rates for the New Zealand sheep population." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 86 "a:=array(1..12,[0,0.045,0.391,0.472,0.484,0.54
6,0.543,0.502,0.468,0.459,0.433,0.421]):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "b:=array(1..11,[0.845,0.975,0.965,0.950,0.926,0.895,0
.850,0.786,0.691,0.561,0.370]):" }}}{EXCHG {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 5 "Plot " }{XPPEDIT 18 0 "q(lambda)" "6#-%\"qG6#%'lambdaG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(lambda)" "6#-%\"pG6#%'lambdaG" }
{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(q(lambda),lam
bda=0..2, y=0..10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(p(lambd
a),lambda=-2..2,y=-2..2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "No
w let Maple compute the eigenvalues directly." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "eigenvals(L);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 259 38 "Part 2.  Properties of Leslie matrices" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Fill in
 the dominant eigenvalue here:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "l
ambda1:= ;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Enter the followi
ng lines, and explain the output." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "k:=10: x0:=[1,1,1,1,1,1,1,1,1,1,1,1]: " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "xk:=evalm(L^k&*x0); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "evalm(L&*xk); evalm(lambda1*xk);\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "By varying k in the preceding step, find an eigenvector f
or the dominant eigenvalue." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 16 "P
art 3.  Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 0" 0 }
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