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{SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT -1 2 " " }{TEXT 261 39 "Demogr
aphy of the vegetable ivory palm " }{TEXT 262 11 "Pytelephas " }}
{PARA 260 "" 0 "" {TEXT 264 8 "seemanii" }{TEXT 263 47 " in Colombia, \+
and the impact of seed harvesting" }}{PARA 263 "" 0 "" {TEXT -1 77 "by
Prof. Matt Miller, Department of Mathematics, University of South Car
olina" }}{PARA 264 "" 0 "" {TEXT 265 6 "email:" }{TEXT -1 20 " miller
@math.sc.edu" }}{PARA 261 "" 0 "" {TEXT -1 79 "Maple worksheet to acco
mpany Rodrigo Bernal,1998 ; Journal of Applied Ecology " }{TEXT 260
2 "35" }{TEXT -1 7 ":64-74." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 402 "This paper is an analysis of a stage-based mod
el of the dynamics of a palm tree species, using, a modification of th
e Leslie model. In this case, the assumption is that the stage of an \+
organism (size class) is a better indicator of reproductive performanc
e and survival than age is. This assumption works well for many plant
s, and any organism in which size is correlated with survival and fecu
ndity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
304 "The general form of the model treats the stages as elements of a \+
vector. The population projection matrix A (sometimes called a Lefkov
ich matrix, or, among mathematicians, transition matrix) is used to ca
lculate the number of individuals in each stage class at time t+1, bas
ed on the numbers at time t. " }}{PARA 0 "" 0 "" {TEXT -1 21 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 257
2 "n(" }{TEXT -1 3 "t+1" }{TEXT 258 9 ") = A n(" }{TEXT -1 1 "t" }
{TEXT 259 2 ") " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 44 "The (non-zero) elements of the matri
x A are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
68 "Pi the probabilities of surviving and remaining in the same stag
e." }}{PARA 0 "" 0 "" {TEXT -1 63 "Gi the probabilities of surviving a
nd moving to the next stage." }}{PARA 0 "" 0 "" {TEXT -1 90 "Fi the f
ertility, or number of female offspring at time t+1, per adult female
at time t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: wi
th(linalg): " }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected n
ames norm and trace have been redefined and unprotected\n" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A:=matrix(4,4, [ [0,F2,F3,0], [G1,P
2,0,0], [0,G2,P3,0], [0,0,G3,P4] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG-%'matrixG6#7&7&\"\"!%#F2G%#F3GF*7&%#G1G%#P2GF*F*7&F*%#G2G%#P3
GF*7&F*F*%#G3G%#P4G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "For test \+
purposes, try Brault & Caswell 1993 data. Confirm that all results ar
e the same as published in that paper." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 110 "Caswell:=matrix(4, 4,[ [0,0.0043,0.1132,0], [0.9775,
0.9111,0,0],\n[0,0.0736,0.9534,0], [0,0,0.0452,0.9804] ] );" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%(CaswellG-%'matrixG6#7&7&\"\"!$\"#V!\"%$\"
%K6F-F*7&$\"%v(*F-$\"%6\"*F-F*F*7&F*$\"$O(F-$\"%M&*F-F*7&F*F*$\"$_%F-$
\"%/)*F-" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Stable age distribut
ion and reproductive value" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The \+
transition matrix is defined in Table 3 on p 70." }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 179 "A:=matrix(6,6,[[0.7052,0,0,14.8173,14.9770,1
4.7732],\n[0.1548,0.6339,0,0,0,0],\n[0,0.0216,0.9100,0,0,0],\n[0,0,0.0
330,0.9614,0,0],\n[0,0,0,0.0386,0.9220,0],\n[0,0,0,0,0.0188,0.9535]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7(7($\"%_q!\"%\"
\"!F-$\"'t\"[\"F,$\"'q(\\\"F,$\"'Kx9F,7($\"%[:F,$\"%RjF,F-F-F-F-7(F-$
\"$;#F,$\"%+\"*F,F-F-F-7(F-F-$\"$I$F,$\"%9'*F,F-F-7(F-F-F-$\"$'QF,$\"%
?#*F,F-7(F-F-F-F-$\"$)=F,$\"%N&*F," }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 217 " The elements on the top row are the reproductive
outputs. The subdiagonal elements represent the probability of trans
ition to the next stage. The diagonal elements represent the probabili
ty of remaining in a stage. " }}{PARA 0 "" 0 "" {TEXT -1 63 "***What f
raction of seeds remain in the 'seed bank' each year? " }}{PARA 0 ""
0 "" {TEXT -1 329 "***Of 1000 stage 1 individuals (seeds), how many su
rvive to become stage 2? If there are currently 1000 stage 2 and 1000
stage 3, how many seeds (stage 1) will be present after 1 year? Supp
ose an individual is in the highest stage ). What is the probablility
that this individual will still be alive 14 time steps from now?***"
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eigsys:=eigenvects(A); # (output
printing suppressed)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'eigsysG6(7
%^$$\"+rv`_\")!#5$\"+6_j=9F*\"\"\"<#-%'vectorG6#7(^$$!+E^x!\\$!\"*$!+w
2@naF6^$$!+$*zB8TF6$!+@M8\\9F6^$$\"+,Qnm8F*$\"+ym,]`F*^$$\"+i%p'[W!#6$
!+R*)4ixFF^$$!+pY/I>FF$\"+>mE=C!#7^$$\"+a@\"GW\"FN$\"+#y0<:\"FN7%$\"+d
M\"=T*F*F-<#-F06#7($!+qg*H&yFF$!*!f7cRF*$!*%>\\SFF*$\"+g4\"HZ%FF$\"+o$
f6+*FF$!+dOqt8F*7%$\"+&\\5*e5F6F-<#-F06#7($!+&Qb-d'F6$!+.)fIR#F6$!+#*)
=7Z$F*$!+Iwuu6F*$!+Xn.7LFF$!+3&Hq!fFN7%^$F($!+6_j=9F*F-<#-F06#7(^$F4$
\"+w2@naF6^$F:$\"+@M8\\9F6^$F?$!+ym,]`F*^$FD$\"+R*)4ixFF^$FJ$!+>mE=CFN
^$FP$!+#y0<:\"FN7%$\"+!4D1h&F*F-<#-F06#7($\"+[-tUMF6$!+sJw;tF6$\"+ixBH
XF*$!+]6ZLPFF$\"+#=8F*RFN$!+jzt7>!#87%$\"+L8QV*)F*F-<#-F06#7($!+IVsC?F
*$!+!zgM?\"F*$\"+Syuf;F*$!+neLn\")FF$\"+2()oR6F*$!+rYh@OFF" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 269 "Out of t
he complete list of eigenvalues we find the maximum eigenvalue (lambda
= population growth rate), and the index to the corresponding eigenve
ctor (stable age distribution). This method for finding the populatio
n growth rate is described at the bottom of p 280.." }}{PARA 0 "> " 0
"" {MPLTEXT 1 0 66 "EVec:= seq( eigsys[i][3], i = 1 .. nops([eigsys]))
; # eigenvectors" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%EVecG6(<#-%'vec
torG6#7(^$$!+E^x!\\$!\"*$!+w2@naF.^$$!+$*zB8TF.$!+@M8\\9F.^$$\"+,Qnm8!
#5$\"+ym,]`F9^$$\"+i%p'[W!#6$!+R*)4ixF?^$$!+pY/I>F?$\"+>mE=C!#7^$$\"+a
@\"GW\"FG$\"+#y0<:\"FG<#-F(6#7($!+qg*H&yF?$!*!f7cRF9$!*%>\\SFF9$\"+g4
\"HZ%F?$\"+o$f6+*F?$!+dOqt8F9<#-F(6#7($!+&Qb-d'F.$!+.)fIR#F.$!+#*)=7Z$
F9$!+Iwuu6F9$!+Xn.7LF?$!+3&Hq!fFG<#-F(6#7(^$F,$\"+w2@naF.^$F2$\"+@M8\\
9F.^$F7$!+ym,]`F9^$F=$\"+R*)4ixF?^$FC$!+>mE=CFG^$FI$!+#y0<:\"FG<#-F(6#
7($\"+[-tUMF.$!+sJw;tF.$\"+ixBHXF9$!+]6ZLPF?$\"+#=8F*RFG$!+jzt7>!#8<#-
F(6#7($!+IVsC?F9$!+!zgM?\"F9$\"+Syuf;F9$!+neLn\")F?$\"+2()oR6F9$!+rYh@
OF?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "EVal:= seq( eigsys[i
][1], i = 1 .. nops([eigsys])); # eigenvalues" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 66 "EM:= seq( eigsys[i][2], i = 1 .. nops([eigsys])); # m
ultiplicities" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "mags:=op( map( abs
, [EVal] )); # magnitudes of eigenvalues" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%%EValG6(^$$\"+rv`_\")!#5$\"+6_j=9F)$\"+dM\"=T*F)$\"+&\\5*e5!\"
*^$F'$!+6_j=9F)$\"+!4D1h&F)$\"+L8QV*)F)" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%#EMG6(\"\"\"F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%m
agsG6($\"+3l/v#)!#5$\"+dM\"=T*F($\"+&\\5*e5!\"*F&$\"+!4D1h&F($\"+L8QV*
)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Next we identify t
he dominant eigenvalue (lambda), and the eigenvector it belongs to." }
}{PARA 0 "" 0 "" {TEXT -1 95 "The dominant eigenvalue is called the fi
nite rate of increase in the last paragraph of p 282. " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "lambda:=max(mags);" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 54 "member( lambda, [mags], 'position'); indx:= position
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "V:=op( EVec[indx] );" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG$\"+&\\5*e5!\"*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%indxG\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7($!+&Qb-d'!
\"*$!+.)fIR#F+$!+#*)=7Z$!#5$!+Iwuu6F0$!+Xn.7L!#6$!+3&Hq!f!#7" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "The reported estimate of lambda i
n the paper is in Figure 3. The original transition matrix is for a p
opulation with no harvesting. How does the Maple estimate compare to \+
the value in the Figure?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "If the population grows for \+
many generations with constant transition probabilities, the populatio
n stage vector will approach the eigenvector associated with the domin
ant eigenvector of the original matrix A." }}{PARA 0 "" 0 "" {TEXT -1
351 "This is the stable stage distribution vector, calculated from the
dominant eigenvector of the original matrix A. Typically the stable \+
stage vector w is scaled so that the sum of the elements is 1.0. To s
cale the stable stage vector, we have to do some fiddling. Add up the
elements in the vector; then divide all elements in the vector by thi
s sum:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "total:= sum( V['j'] , 'j'
= 1 .. rowdim(A) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&totalG$!+P#
RpY*!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We now ha
ve the stable stage distribution vector:" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "w:=scalarmul(V, 1/total);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%\"wG-%'vectorG6#7($\"+*y5-%p!#5$\"+\"H2y_#F+$\"+X\\nm
O!#6$\"+m[*3C\"F0$\"+K'H&)\\$!#7$\"+X-kRi!#8" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "The sta
ble stage distribution is reported as column SSD in Table 5 on p 70. \+
How does it compare to our estimates?" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 34 "Let's check; is A w = (lambda) w? " }}{PARA 0 ""
0 "" {TEXT -1 337 " Remember this is our definition of an eigenvalue a
nd eigenvector - that the eigenvector multiplied by the matrix is the \+
same as the eigenvector multiplied by the eigenvalue. If we got the c
orrect eigenvalues and eigenvectors, we should obtain the same answer \+
by multiplying the stable age distribution by matrix A or by the eigen
value:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalm( A &* w ); scalarm
ul( w , lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7($\"+
M?1\\t!#5$\"+s;swEF)$\"+v0o#)Q!#6$\"+(f'*RJ\"F.$\"+R(HYq$!#7$\"+X0A2m!
#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7($\"+U?1\\t!#5$\"+q
;swEF)$\"+\"e!o#)Q!#6$\"+(f'*RJ\"F.$\"+W(HYq$!#7$\"+S0A2m!#8" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "Here we transpose the A matrix in
order to set up calculation of the reproductive value vector, which i
s the dominant eigenvector of the transposed matrix." }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 17 "AT:=transpose(A);" }}{PARA 0 "> " 0 "" {MPLTEXT
1 0 57 "eigsysT:=eigenvects(AT): # (output printing suppressed)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ATG-%'matrixG6#7(7($\"%_q!\"%$\"%[:
F,\"\"!F/F/F/7(F/$\"%RjF,$\"$;#F,F/F/F/7(F/F/$\"%+\"*F,$\"$I$F,F/F/7($
\"'t\"[\"F,F/F/$\"%9'*F,$\"$'QF,F/7($\"'q(\\\"F,F/F/F/$\"%?#*F,$\"$)=F
,7($\"'Kx9F,F/F/F/F/$\"%N&*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 188 "Find maximum eigenvalue and index to
corresponding eigenvector; this eigenvector is the reproductive value
distribution, i.e., the left eigenvectors of matrix A ( last paragra
ph on p68).." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "EVecT:= seq( eigsys
T[i][3], i = 1 .. nops([eigsysT]) ); # eigenvectors" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%&EVecTG6(<#-%'vectorG6#7($!+]d\"\\'o!#6$\"+5_1#R'F-
$!+1@Zb@!#5$\"+-n;zA!\"*$\"+_8(Rr#F5$\"+Q%yUe#F5<#-F(6#7(^$$!+-D\"4W%F
-$\"+ " 0 "" {MPLTEXT 1 0 69 "EValT:= seq( eigsysT[i][1], i = 1 .. nops([
eigsysT]) ); # eigenvalues" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "EMT:=
seq( eigsysT[i][2], i = 1 ..nops([eigsysT]) ); # multiplicities" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "magsT:= op( map( abs , [EValT] ));
# magnitudes of eigenvalues (Note: [ ] turns a sequence into a list; \+
op turns a list into a sequence.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&EValTG6($\"++^i5c!#5^$$\"+`v`_\")F($!+._j=9F($\"+fM\"=T*F(^$F*$\"+._
j=9F($\"+$\\5*e5!\"*$\"+N8QV*)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$EMTG6(\"\"\"F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&magsTG6(
$\"++^i5c!#5$\"+*[Y]F)F($\"+fM\"=T*F(F)$\"+$\\5*e5!\"*$\"+N8QV*)F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mu:= max( magsT); # get do
minant eigenvalue" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "member(mu, [m
agsT], 'position'); indexT:= position;" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%#muG$\"+$\\5*e5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'indexTG\"\"&" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Convert eigenve
ctor lists to actual vectors for use in sensitivity analysis." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "VT:= op( EVecT[indexT] ); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VTG-%'vectorG6#7($!+K:d2J!#6$!+A.l+
rF+$!+*=`rR\"!\"*$!+a)oXI'F0$!+,#)\\(*RF0$!+fwBbVF0" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 485 "This is the re
productive value distribution. The reproductive values are reported as
columns of Table 5 on p 70. Note that vector v is scaled so the fir
st element is 1.0. We do a scalar multiplication of all elements in \+
the vector by the reciprocal of the first element. (Recall that a sca
lar multiple of an eigenvector is again an eigenvector with the same e
igenvalue.) We now have the reproductive value vector corresponding \+
the to the column \"v\", printed in Table 5 on p 70:" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 28 "v:= scalarmul(VT, 1/VT[1] );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7($\"+++++5!\"*$\"+H:&\\G#F+$\"+%pk
f\\%!\")$\"+WnxG?!\"($\"+sOP'G\"F3$\"+GB\\,9F3" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "How d
o these values compare to those printed in the table? Remember once a
gain that the authors of the paper probably used a different numerical
accuracy in their calculations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
75 "Let's confirm that we correctly identified the eigenvalue/eigenvec
tor pair." }}{PARA 0 "" 0 "" {TEXT -1 38 "Let's just check that AT v =
(mu) v. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalm( AT &* v ); s
calarmul( v , mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7($
\"+(\\5*e5!\"*$\"+[\"f&>CF)$\"+)>C3w%!\")$\"+')HH[@!\"($\"+#ea@O\"F1$
\"+N\\0%[\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7($\"+$\\
5*e5!\"*$\"+]\"f&>CF)$\"+\">C3w%!\")$\"+#)HH[@!\"($\"+zX:i8F1$\"+K\\0%
[\"F1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0
{PARA 3 "" 0 "" {TEXT -1 51 "Sensitivity analysis (optional for first \+
go around)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 831 "Now we will calculate the sensitivity of the system to c
hanges in fecundity. This is the analysis proposed in Caswell. We ar
e analytically determining the partial derivative of lambda (populatio
n growth rate) with respect to each element in the projection matrix. \+
Thus we are asking: given a change in a particular element in the pro
jection matrix, what is the corresponding change in population growth \+
rate? This allows us to determine which components of the life histor
y have the biggest effects on population growth, and which may be most
sensitive to natural selection. The first parameter we need is the i
nner product of the two eigenvectors (stable age distribution and repr
oductive value distribution). This is symbolized as in the Ca
swell handout. This value is used as the denominator of the calculati
on. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "IP:= innerprod(VT , V); #
dot, or scalar, product " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sens
itivity_to_fecundity:= [seq( V[i]*VT[1] / IP , i = 1 .. rowdim(A) )]; \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IPG$\"+2N$yv\"!\"*" }}{PARA 11
"" 1 "" {XPPMATH 20 "6#>%9sensitivity_to_fecundityG7($\"+.s^h6!#5$\"+C
6bIU!#6$\"+;ecOh!#7$\"+_zww?F.$\"+kp:be!#8$\"+i$pU/\"F3" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 111 "Observe that it is immaterial whether we use the normalized ve
ctors or not, as long as we do this consistently." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 59 "[seq( w[i]*v[1] /innerprod( w, v) , i = 1 ..
rowdim(A) )]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7($\"+.s^h6!#5$\"+B
6bIU!#6$\"+8ecOh!#7$\"+^zww?F,$\"+jp:be!#8$\"+h$pU/\"F1" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 215 "In terms of the paper this corresponds t
o the top row of matrix S, equation 7, in Caswell except that the firs
t and last elements have been suppressed because the original matrix h
ad zero entries in these locations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 67 "We do a similar calcuation to obtain the
sensitivity to survival: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "sens
itivity_to_survival:= [seq( V[i]*VT[i+1] / IP , i = 1 .. rowdim(A)-1 )
];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8sensitivity_to_survivalG7'$\"
+50,aE!#5$\"+[3/->!\"*$\"+,A(\\C\"F+$\"+:'*\\rEF($\"+iv&f?)!#6" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 401 "This vector corresponds to the su
bdiagonal of matrix S. The entire sensitivity matrix can be calcuate
d by generalizing the above operations. Note there are some entries t
hat can be taken to be zero, or otherwise ignored, because the A matri
x had zero entries (but this isn't really necessary, since as we shall
see in a moment we end up multiplying each entry of S by the correspo
nding entry of A)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "S:=matrix(ro
wdim(A),rowdim(A), [ seq( seq( V[i] * VT[j] / IP, i = 1.. rowdim(A)), \+
j = 1.. rowdim(A) ) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'m
atrixG6#7(7($\"+.s^h6!#5$\"+C6bIU!#6$\"+;ecOh!#7$\"+_zww?F2$\"+kp:be!#
8$\"+i$pU/\"F77($\"+50,aEF,$\"+cUgm'*F/$\"+Wb<-9F/$\"+1TJXZF2$\"+')\\(
yL\"F2$\"+u[5'Q#F77($\"+Q.9A_!\"*$\"+[3/->FJ$\"+D$y*eFF,$\"+\"RvqL*F/$
\"+**yXKEF/$\"+z\")*\\p%F27($\"+!4fkN#!\")$\"+LP%Ge)FJ$\"+,A(\\C\"FJ$
\"+B&)H8UF,$\"+G1)y=\"F,$\"+&R*e=@F/7($\"+\\^9%\\\"FX$\"+\"ep?W&FJ$\"+
-n\"R*yF,$\"+:'*\\rEF,$\"+i(>>`(F/$\"+71KV8F/7($\"+\\u&yi\"FX$\"+Q\\3H
fFJ$\"+3*\\.g)F,$\"+^Vd5HF,$\"+iv&f?)F/$\"+)\\NNY\"F/" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 418 "The elasticity values are calculated by \+
scaling the sensitivities by aij / lambda (Caswell handout). The resu
lting matrix is reported as Table 4 on p 70. The elasticities are a b
it easier to interpret than the sensitivities, because they represent \+
the proportional contributions of matrix elements to lambda. Recall t
he we can't use the label E in Maple because this means the base of na
tural logarithms (2.718...)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e:
=matrix(rowdim(A),rowdim(A),[ seq( seq( S[j,i] * A[j,i] / lambda, i = \+
1 .. rowdim(A)), j = 1.. rowdim(A)\\)]);" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%\"eG-%'matrixG6#7(7($\"+`tKNx!#6$\"\"!F.F-$\"+=Y,1HF,$\"+5oS\"
G)!#7$\"+'o$*oX\"F37($\"+(pW)zQF,$\"+n&fny&F,F-F-F-F-7(F-$\"+iY%)zQF,$
\"+aQ*4P#!#5F-F-F-7(F-F-$\"+#pW)zQF,$\"+bSJDQF@F-F-7(F-F-F-$\"+7/IQ(*F
3$\"+5(*3elF,F-7(F-F-F-F-$\"+$p$*oX\"F3$\"+Hh%yJ\"F," }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 401 "***Try seeing \+
if you believe that the sensitivities or elasticities are correct, fol
lowing the pattern illustrated below.*** We will change one of the en
tries in the original \"A\" matrix by 10% and see what percent change \+
there is in lambda. Compare this change with the corresponding value \+
in the elasticity or sensitivity matrix. Let's first recall the orig
inal matrix, eigenvalues, eigenvectors:" }}{PARA 0 "> " 0 "" {MPLTEXT
1 0 13 "A:= evalm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matr
ixG6#7(7($\"%_q!\"%\"\"!F-$\"'t\"[\"F,$\"'q(\\\"F,$\"'Kx9F,7($\"%[:F,$
\"%RjF,F-F-F-F-7(F-$\"$;#F,$\"%+\"*F,F-F-F-7(F-F-$\"$I$F,$\"%9'*F,F-F-
7(F-F-F-$\"$'QF,$\"%?#*F,F-7(F-F-F-F-$\"$)=F,$\"%N&*F," }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lambda[old]:= lambda; mu[old]:= m
u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'lambdaG6#%$oldG$\"+&\\5*e5!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#muG6#%$oldG$\"+$\\5*e5!\"*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "RV:= evalm(v); SAD:= e
valm(w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RVG-%'vectorG6#7($\"+++
++5!\"*$\"+H:&\\G#F+$\"+%pkf\\%!\")$\"+WnxG?!\"($\"+sOP'G\"F3$\"+GB\\,
9F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SADG-%'vectorG6#7($\"+*y5-%p
!#5$\"+\"H2y_#F+$\"+X\\nmO!#6$\"+m[*3C\"F0$\"+K'H&)\\$!#7$\"+X-kRi!#8
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 428 "Let's change the P2 value by
10% (we'll do this by multiplying P2 by 1.1) and see what percent chan
ge we get in lambda. This is the so-called [2,2] entry of the matrix \+
A, that is, the number that appears in the first row (from top to bott
om) and second column (from left to right). If the author's descript
ion of elasticity is correct, then we should get a change in lambda th
at is proportional to the elasticity value for P2." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 61 "A[2,2]:=1.1*A[2,2]; # make the change in the affe
cted entry" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6$\"\"#F'$\"&H(p!
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=evalm(A); # con
firm the matrix is OK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matr
ixG6#7(7($\"%_q!\"%\"\"!F-$\"'t\"[\"F,$\"'q(\\\"F,$\"'Kx9F,7($\"%[:F,$
\"&H(p!\"&F-F-F-F-7(F-$\"$;#F,$\"%+\"*F,F-F-F-7(F-F-$\"$I$F,$\"%9'*F,F
-F-7(F-F-F-$\"$'QF,$\"%?#*F,F-7(F-F-F-F-$\"$)=F,$\"%N&*F," }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Now let's
calculate the eigenvalues and eigenvectors for this revised matrix."
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eigsys:=eigenvects(A): # (output
printing suppressed)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "EV
ec:= seq( eigsys[i][3], i = 1 .. nops([eigsys])); # eigenvectors " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%EVecG6(<#-%'vectorG6#7($!+o4Q*='!\"
*$!+L`J,EF-$!+[3&3h$!#5$!+nIWV6F2$!+!y)QtI!#6$!+Kf%Q:&!#7<#-F(6#7($\"+
rd6]XF-$!+W\\>%f'F-$\"+D1qdWF2$!++S(e'RF7$\"+JV`S
U\"F2$!+IL.<6F2$\"+%)>2q:F2$!+\\48gxF7$\"+uy^%4\"F2$!+sBZ&\\$F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "EVal:= seq( eigsys[i][1], i \+
= 1 .. nops([eigsys])); # eigenvalues" }}{PARA 0 "> " 0 "" {MPLTEXT 1
0 56 "mags:= op( map( abs, [EVal] )); lambda[new]:=max(mags);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EValG6($\"+*))4c1\"!\"*$\"+D:v/f!#5
$\"+tu^6%*F+^$$\"+>Ig(G)F+$\"+]@&Gc\"F+^$F/$!+]@&Gc\"F+$\"+>gKY*)F+" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%magsG6($\"+*))4c1\"!\"*$\"+D:v/f!#
5$\"+tu^6%*F+$\"+,[nL%)F+F.$\"+>gKY*)F+" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>&%'lambdaG6#%$newG$\"+*))4c1\"!\"*" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Compare this new value fo
r lambda with the old value of lambda:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "lambda[old];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&
\\5*e5!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "This is the ratio \+
of the new value to the old; the fractional change in lambda is the po
rtion after the decimal. Did you expect lthe new lambda to be larger?
Why?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ratio:= lambda[new
]/lambda[old];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$\"+'oEj+\"
!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Here is the elasticity va
lue. Compare it to the fractional change in lambda." }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 14 "elas:= e[2,2];" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%%elasG$\"+n&fny&!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 410 "Note that we made a 10 percent change in one element of the ma
trix (a fractional change of 0.1). If you multiply this elasticity va
lue by the fractional change made in the matrix element, you should g
et a number close to the fractional change in lambda. If you made a 1
00% change (a fractional change of 1.0) in the matrix element, you sho
uld see a fractional change in lambda equal to the elasticity value."
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "(elas * 0.1) - (ratio - 1
); # close to zero?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!'E*R&!\"*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "member( lambda[new], [mags]
, 'position'); indx:= position;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%
trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%indxG\"\"\"" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "evalm(SAD); # here's the original s
table age distribution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7($\"+*y5-%p!#5$\"+\"H2y_#F)$\"+X\\nmO!#6$\"+m[*3C\"F.$\"+K'H&)\\$!#7$
\"+X-kRi!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "\nCompare with the
new stable age distribution. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 120 "V := op( EVec[indx] ); total:= sum( V['k'] ,' k' = 1 .. rowdi
m(A) );\nw:= scalarmul(V, 1/total); # compare with SAD above" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7($!+o4Q*='!\"*$!+L`J,EF+$
!+[3&3h$!#5$!+nIWV6F0$!+!y)QtI!#6$!+Kf%Q:&!#7" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%&totalG$!+FM,-$*!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20
"6#>%\"wG-%'vectorG6#7($\"+By!Ql'!#5$\"+Dy]'z#F+$\"+B\\z\")Q!#6$\"+ZDC
H7F0$\"+6Q+/L!#7$\"+-.dSb!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "
Transpose the A matrix in order to set up calculation of the reproduct
ive value vector, which is the dominant eigenvector of the transposed \+
matrix." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AT:=transpose(A);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eigsysT:=eigenvects(AT): # (outpu
t printing suppressed)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ATG-%'mat
rixG6#7(7($\"%_q!\"%$\"%[:F,\"\"!F/F/F/7(F/$\"&H(p!\"&$\"$;#F,F/F/F/7(
F/F/$\"%+\"*F,$\"$I$F,F/F/7($\"'t\"[\"F,F/F/$\"%9'*F,$\"$'QF,F/7($\"'q
(\\\"F,F/F/F/$\"%?#*F,$\"$)=F,7($\"'Kx9F,F/F/F/F/$\"%N&*F," }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "EVecT:= seq( eigsysT[i][3], i = 1 .
. rowdim(A) ); # eigenvectors" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&EV
ecTG6(<#-%'vectorG6#7(^$$\"+6\"G`y#!#6$\"+nen8jF.^$$!+D>-^TF.$\"++Ye^y
F.^$$!+w#F?
$!+i*>6g%F?^$$\"+\"p$)>O#F?$!+W\">\"=XF?<#-F(6#7($!+!R\")yJ(F.$\"+^KSB
aF.$!+jV%>o#F9$\"+7<\"of#F?$\"+1&eq8$F?$\"+hC*z(HF?<#-F(6#7(^$F,$!+nen
8jF.^$F2$!++Ye^yF.^$F7$!+)f%\\v6g%F?^$F
H$\"+W\">\"=XF?<#-F(6#7($\"+\"enw7\"F.$\"+m:$)=F9$\"+Sx&=
$=F9$!+)fY[U%F?$!+&o=\"\\8!\")<#-F(6#7($!+k]'\\D$F.$!+c[IyvF.$!+M4C#H
\"F?$!+3#*\\$4'F?$!+lo3cRF?$!+9n?*G%F?<#-F(6#7($!+LVjlFF.$!+r_P%Q$F.$!
+>O/#4$F9$\"+![)*)R9F9$\"+y=tO5F\\q$\"+E)f0%pF?" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 63 "EValT:= seq( eigsysT[i][1], i = 1 .. rowdim(A)
); # eigenvalues" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "magsT:= op( ma
p( abs , [EValT] )); # magnitudes of eigenvalues " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 82 "mu[new]:= max( magsT); member(mu[new], [magsT], '
position'); indexT:= position;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
EValTG6(^$$\"+6Ig(G)!#5$\"+a@&Gc\"F)$\"+a:v/fF)^$F'$!+a@&Gc\"F)$\"+uu^
6%*F)$\"+#*)4c1\"!\"*$\"+SgKY*)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&magsTG6($\"+%zuOV)!#5$\"+a:v/fF(F&$\"+uu^6%*F($\"+#*)4c1\"!\"*$\"+Sg
KY*)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#muG6#%$newG$\"+#*)4c1\"!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%'indexTG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 78 "VT:= op( EVecT[indexT] ); v:= scalarmul(VT, 1/VT[1] ); # compa
re with RV above" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VTG-%'vectorG6#
7($!+k]'\\D$!#6$!+c[IyvF+$!+M4C#H\"!\"*$!+3#*\\$4'F0$!+lo3cRF0$!+9n?*G
%F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7($\"+++++5!
\"*$\"+Y#H#GBF+$\"+T31qR!\")$\"+gG1s=!\"($\"+:2S:7F3$\"+\"yUxJ\"F3" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(RV);" }}{PARA 11 ""
1 "" {XPPMATH 20 "6#-%'vectorG6#7($\"+++++5!\"*$\"+H:&\\G#F)$\"+%pkf\\
%!\")$\"+WnxG?!\"($\"+sOP'G\"F1$\"+GB\\,9F1" }}}{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 -1 365 "***Now, how did v and w change, given that 10% change in
a single entry of the matrix A? Considering the change that was made
can you give an explanation for the result? Now change the survival \+
probability G3 by 15% (multiply it by 1.5). (Don't forget to reset A[
1,2] back to its original value first) What effect do you expect to s
ee? What happens in fact?***" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "The author states on p7
1 that \"an 86% reduction in fecundity of all classes is required to r
educe lambda from 1.0589 to 1.0\". Try making reductions in fecundity
and see if your observations confirm this: First we will get an orig
inal copy of matrix A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "A
:=matrix(6,6,[[0.7052,0,0,14.8173,14.9770,14.7732],\n[0.1548,0.6339,0,
0,0,0],\n[0,0.0216,0.9100,0,0,0],\n[0,0,0.0330,0.9614,0,0],\n[0,0,0,0.
0386,0.9220,0],\n[0,0,0,0,0.0188,0.9535]]);A := matrix([[.7052, 0, 0, \+
14.8173, 14.9770, 14.7732], [.1548, .6339, 0, 0, 0, 0], [0, .216e-1, .
9100, 0, 0, 0], [0, 0, .330e-1, .9614, 0, 0], [0, 0, 0, .386e-1, .9220
, 0], [0, 0, 0, 0, .188e-1, .9535]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7(7($\"%_q!
\"%\"\"!F-$\"'t\"[\"F,$\"'q(\\\"F,$\"'Kx9F,7($\"%[:F,$\"%RjF,F-F-F-F-7
(F-$\"$;#F,$\"%+\"*F,F-F-F-7(F-F-$\"$I$F,$\"%9'*F,F-F-7(F-F-F-$\"$'QF,
$\"%?#*F,F-7(F-F-F-F-$\"$)=F,$\"%N&*F," }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%\"AG-%'matrixG6#7(7($\"%_q!\"%\"\"!F-$\"'t\"[\"F,$\"'q(\\\"F,$
\"'Kx9F,7($\"%[:F,$\"%RjF,F-F-F-F-7(F-$\"$;#F,$\"%+\"*F,F-F-F-7(F-F-$
\"$I$F,$\"%9'*F,F-F-7(F-F-F-$\"$'QF,$\"%?#*F,F-7(F-F-F-F-$\"$)=F,$\"%N
&*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Then we define our fracti
onal reduction in fecundity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 26 "fecundity_reduction:=0.86;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
4fecundity_reductionG$\"#')!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
271 "We will now reduce the seed production entries by this amount. W
e will not change A[1,1], because that element of the matrix defines t
he rate of retention of seeds in the seed bank, not a fecundity. Ther
efore we change only elements 2 to 6 in the top row of the matrix:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for i from 2 to rowdim(A) d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " A[1,i]:=A[1,i]*(1-fecundity
_reduction); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7($\"%_q!\"%$\"\"!F
,F+$\"(AW2#!\"'$\"(!y'4#F/$\"([#o?F/7($\"%[:F*$\"%RjF*F,F,F,F,7(F,$\"$
;#F*$\"%+\"*F*F,F,F,7(F,F,$\"$I$F*$\"%9'*F*F,F,7(F,F,F,$\"$'QF*$\"%?#*
F*F,7(F,F,F,F,$\"$)=F*$\"%N&*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
222 "Now move back to the top of the worksheet, and rerun from a point
below the original definition of matrix A. See if the values for lam
bda, stable stage distribution, and reproductive value distribution ch
ange as expected." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
{MARK "7 44 1" 0 }{VIEWOPTS 0 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1
2 33 1 1 }