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{SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 262 40 "Pod-specific demography
of killer-whales" }{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 79 "by \+
Prof. Matt Miller, Department of Mathematics, University of South Caro
lina \n" }{TEXT 263 6 "email:" }{TEXT -1 21 " miller@math.sc.edu \n" }
}{PARA 259 "" 0 "" {TEXT -1 54 "Maple worksheet to accompany Brault a
nd Caswell, 1993" }}{PARA 262 "" 0 "" {TEXT 258 40 "Pod-specific demog
raphy of killer-whales" }{TEXT -1 10 ", Ecology " }{TEXT 257 2 "74" }
{TEXT -1 12 ": 1444-1454." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 438 "This paper is an analysis of a stage-based model \+
of the dynamics of killer whales, a modification of the Leslie model. \+
In this case, the assumption is that the stage of an organism (yearli
ng, juvenile, mature female, post-reproductive female) is a better ind
icator of reproductive performance and survival than age is. This ass
umption works well for many plants, and any organism in which size is \+
correlated with survival and fecundity." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 335 "The general form of the model treat
s the stages as elements of a vector. The population projection matri
x A (sometimes called a Lefkovich matrix, or, among mathematicians, tr
ansition matrix) is used to calculate the number of individuals in eac
h stage class at time t+1, based on the numbers at time t. This is eq
uation 1 on p. 1445:" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 259 2 "n(" }
{TEXT -1 3 "t+1" }{TEXT 260 9 ") = A n(" }{TEXT -1 1 "t" }{TEXT 261
2 ") " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}
{PARA 0 "" 0 "" {TEXT -1 44 "The (non-zero) elements of the matrix A a
re:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Pi
the probabilities of surviving and remaining in the same stage." }}
{PARA 0 "" 0 "" {TEXT -1 63 "Gi the probabilities of surviving and mov
ing to the next stage." }}{PARA 0 "" 0 "" {TEXT -1 90 "Fi the fertili
ty, or number of female offspring at time t+1, per adult female at ti
me t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(lin
alg): " }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n
orm 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,P2,0,0]
, [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%#P3GF*7&F
*F*%#G3G%#P4G" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Stable age dist
ribution and reproductive value" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 64 "We will now define the matrix with th
e values listed on p. 1447:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "A:=
matrix(4, 4,[ [0,0.0043,0.1132,0], [0.9775,0.9111,0,0],\n[0,0.0736,0.9
534,0], [0,0,0.0452,0.9804] ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG-%'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-" }}}{EXCHG {PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 622 "This is equation 2 on
p. 1447. The elements on the top row are the reproductive outputs. \+
The subdiagonal elements represent the probability of transition to th
e next stage. The diagonal elements represent the probability of remai
ning in a stage. ***Of 1000 yearlings, how many survive to become juve
niles? If there are currently 1000 juveniles (stage 2) and 1000 repro
ductive adults (stage 3), how many yearlings (stage 1) will be present
after 1 year? Suppose an individual is in the highest stage (non-rep
roductive adult). 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
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
236 "Out of the complete list of eigenvalues we find the maximum eigen
value (lambda = population growth rate), and the index to the correspo
nding eigenvector (stable age distribution). This distribution is vec
tor w in equation 5 on p. 1448." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "
EVec:= seq( eigsys[i][3], i = 1 .. 4); # eigenvectors" }}{PARA 12 ""
1 "" {XPPMATH 20 "6#>%%EVecG6&<#-%'vectorG6#7&$\"+aPo=5!\"*$!+Hgv)4\"F
-$\"+K=ND&)!#6$!+?&z*\\R!#7<#-F(6#7&$\"+lI'4Y%!#5$\"+Z=m8QF-$\"+IN<'*Q
F-$\"+t\"**)4RF-<#-F(6#7&$!+JL@a7F<$\"+]ju%f\"F-$!+aVl[)*F<$\"+tHMXIF<
<#-F(6#7&\"\"!FSFS\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52
"EVal:= seq( eigsys[i][1], i = 1 .. 4); # eigenvalues" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 53 "EM:= seq( eigsys[i][2], i = 1 .. 4); # multiplic
ities" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "mags:=op( map( abs, [EVal]
)); # magnitudes of eigenvalues" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%EValG6&$\"+H$)pN[!#7$\"+E8WD5!\"*$\"+](HAM)!#5$\"%/)*!\"%" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%#EMG6&\"\"\"F&F&F&" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%%magsG6&$\"+H$)pN[!#7$\"+E8WD5!\"*$\"+](HAM)!#5$\"%/)
*!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 85 "Next we identify the dominant eigenvalue (lambda), and th
e eigenvector it belongs to." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "lam
bda:=max(mags);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "member( lambda, \+
[mags], 'position'); location:= position;" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 24 "V:=op( EVec[location] );" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%'lambdaG$\"+E8WD5!\"*" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)locationG\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7&$\"+lI'4Y%!#5$\"+Z
=m8Q!\"*$\"+IN<'*QF.$\"+t\"**)4RF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 345 "This is the stable age distrib
ution vector, calculated from the dominant eigenvector of the original
matrix A. Note that in equation 5, vector w is scaled so that the su
m of the elements is 1.0. To scale the stable age vector, we have to \+
do some fiddling. Add up the elements in the vector; then divide all \+
elements in the vector by this sum:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0
37 "total:= sum( V['j'] , 'j' = 1 .. 4 );" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%&totalG$\"+'3$e17!\")" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We now have the stable ag
e distribution vector as printed on p. 1448:" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "w:=scalarmul(V, 1/total);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%\"wG-%'vectorG6#7&$\"+0o=(p$!#6$\"+s@rgJ!#5$\"+dn4HKF
.$\"+(Qs/C$F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 33 "Let's check; is A w = (lambda) w?" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "evalm( A &* w ); scalarmul( w , lambda);" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&$\"+_\"[7z$!#6$\"+q[7TK!#5$\"+
k#\\7J$F,$\"+,V\"HK$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7&$\"+S\"[7z$!#6$\"+\")[7TK!#5$\"+g#\\7J$F,$\"+,V\"HK$F," }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 148 "\nTranspose the A matrix in order to set
up calculation of the reproductive value vector, which is the dominan
t eigenvector of the transposed matrix." }}{PARA 0 "> " 0 "" {MPLTEXT
1 0 17 "AT:=transpose(A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eigsys
T:=eigenvects(AT): # (output printing suppressed)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#ATG-%'matrixG6#7&7&\"\"!$\"%v(*!\"%F*F*7&$\"#VF-$
\"%6\"*F-$\"$O(F-F*7&$\"%K6F-F*$\"%M&*F-$\"$_%F-7&F*F*F*$\"%/)*F-" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "F
ind maximum eigenvalue and index to corresponding eigenvector; this ei
genvector is the reproductive value distribution, i.e., the vector \"
v\" in equation 5 on p. 1448." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "EV
ecT:= seq( eigsysT[i][3], i = 1 .. 4 ); # eigenvectors" }}{PARA 12 ""
1 "" {XPPMATH 20 "6#>%&EVecTG6&<#-%'vectorG6#7&$!*8Se))*!\"*$!+45`!*[!
#7$\"+A)e(z6!#5\"\"!<#-F(6#7&$!+n%>O'QF3$!+o+6`SF3$!+%*Q)42'F3F4<#-F(6
#7&$!+P'=v8\"F3$!+?*oyq*!#6$\"+\\#p/3\"F3F4<#-F(6#7&$!+sSvi]F3$!+KSxx]
F3$!+(oC`[%F3\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "EVal
T:= seq( eigsysT[i][1], i = 1 .. 4 ); # eigenvalues" }}{PARA 0 "> " 0
"" {MPLTEXT 1 0 55 "EMT:= seq( eigsysT[i][2], i = 1 ..4 ); # multiplic
ities" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "magsT:= op( map( abs , [E
ValT] )); # magnitudes of eigenvalues (Note: [ ] turns a sequence into
a list; op turns a list into a sequence.)" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%&EValTG6&$\"+O$)pN[!#7$\"+E8WD5!\"*$\"+j(HAM)!#5$\"%/
)*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EMTG6&\"\"\"F&F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&magsTG6&$\"+O$)pN[!#7$\"+E8WD5!\"*$
\"+j(HAM)!#5$\"%/)*!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "
mu:= max( magsT); # get dominant eigenvalue" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 53 "member(mu, [magsT], 'position'); indexT:= position;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG$\"+E8WD5!\"*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'in
dexTG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 76 "Convert eigenvector lists to actual vectors for use in se
nsitivity analysis." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "VT:= op( EVe
cT[indexT] ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VTG-%'vectorG6#7&
$!+n%>O'Q!#5$!+o+6`SF+$!+%*Q)42'F+\"\"!" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 388 "This is the reproductive
value distribution. Note that in equation 5, vector v is scaled so t
he first element is 1.0. We do a scalar multiplication of all elemen
ts in the vector by the reciprocal of the first element. (Recall that
a scalar multiple of an eigenvector is again an eigenvector with the \+
same eigenvalue.) We now have the reproductive value vector as print
ed on p. 1448:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v:= scalarmul(VT,
1/VT[1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7&$\"
+**********!#5$\"+T[/\\5!\"*$\"+].Kr:F.$\"\"!F2" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Let's just check tha
t AT v = (mu) v. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalm( AT &* \+
v ); scalarmul( v , mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vecto
rG6#7&$\"+K8WD5!\"*$\"+KRtv5F)$\"+AoH6;F)$\"\"!F/" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#-%'vectorG6#7&$\"+E8WD5!\"*$\"+LRtv5F)$\"+BoH6;F)$\"\"!
F/" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Sensitivity analysis (opt
ional for first go around)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 940 "Now we will calculate the sensitivity of
the system to changes in fecundity. This is the analysis proposed on
p. 1447 in equation 3. We are analytically determining the partial d
erivative of lambda (population growth rate) with respect to each elem
ent in the projection matrix. Thus we are asking: given a change in a
particular element in the projection matrix, what is the correspondin
g change in population growth rate? This allows us to determine which
components of the life history have the biggest effects on population
growth, and which may be most sensitive to natural selection. The fi
rst parameter we need is the inner product of the two eigenvectors (st
able age distribution and reproductive value distribution). This is s
ymbolized as in equation 3 on p. 1447. This value is used as \+
the denominator on the right hand size of equation 3. Observe that it
is immaterial whether we use the normalized vecotrs or not." }}{PARA
0 "> " 0 "" {MPLTEXT 1 0 53 "IP:= innerprod(VT , V); # dot, or scal
ar, product " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sensitivity_to_fecu
ndity:= [seq( V[i]*VT[1] / IP , i = 1 .. 4 )]; " }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%#IPG$!+NWV$3%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%9sensitivity_to_fecundityG7&$\"+!RD3A%!#6$\"+\")*o$3O!#5$\"+')*Qko$F
+$\"+ldU*p$F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 111 "Observe that it is immaterial whether we use the norm
alized vectors or not, as long as we do this consistently." }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "[seq( w[i]*v[1] /innerprod( w, v) ,
i = 1 .. 4 )]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"+*QD3A%!#6$\"
+\")*o$3O!#5$\"+')*Qko$F)$\"+ldU*p$F)" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "In terms of the paper th
is corresponds to the top row of matrix S on p. 1448, equation 7, exce
pt that the first and last elements have been suppressed because the o
riginal matrix had zero entries in these locations." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We do a similar calcuat
ion to obtain the sensitivity to survival: " }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 66 "sensitivity_to_survival:= [seq( V[i]*VT[i+1] / IP , i
= 1 .. 3 )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8sensitivity_to_sur
vivalG7%$\"+)4NyU%!#6$\"+6O!*pc!#5$!\"!\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 401 "This vector correspond
s to the subdiagonal of matrix S. The entire sensitivity matrix can \+
be calcuated by generalizing the above operations. Note there are som
e entries that can be taken to be zero, or otherwise ignored, because \+
the A matrix 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 t
he corresponding entry of A)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "S:
=matrix(4,4, [ seq( seq( V[i] * VT[j] / IP, i = 1.. 4), j = 1.. 4 ) ]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'matrixG6#7&7&$\"+!RD3A%
!#6$\"+\")*o$3O!#5$\"+')*Qko$F/$\"+ldU*p$F/7&$\"+)4NyU%F,$\"+j3M&y$F/$
\"+)zRs'QF/$\"+6N'3)QF/7&$\"+H)oAj'F,$\"+6O!*pcF/$\"+*fwDz&F/$\"+!*H)H
\"eF/7&$!\"!\"\"!FGFGFG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 420 "The elasticity values are calculated by \+
scaling the sensitivities by aij / lambda (equation 4 on p. 1447). Th
e resulting matrix is reported as equation 8 on p. 1448. The elastici
ties are a bit easier to interpret than the sensitivities, because the
y represent the proportional contributions of matrix elements to lambd
a. Recall the we can't use E in Maple because this means the base of \+
natural logarithms (2.718...)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e
:=matrix(4,4,[ seq( seq( S[j,i] * A[j,i] / lambda, i = 1 .. 4), j = 1.
. 4)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG-%'matrixG6#7&7&$\"\"
!F+$\"+2L58:!#7$\"+s]^pS!#6F*7&$\"+8a#3A%F1$\"+4$eKO$!#5F*F*7&F*$\"+&>
:&pSF1$\"+&3DcQ&F7F*7&$!\"!F+F>F>F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 401 "***Try seeing if you believe t
hat the sensitivities or elasticities are correct, following the patte
rn illustrated below.*** We will change one of the entries in the ori
ginal \"A\" matrix by 10% and see what percent change there is in lamb
da. Compare this change with the corresponding value in the elasticit
y or sensitivity matrix. Let's first recall the original matrix, eig
envalues, eigenvectors:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A:= eval
m(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"!$
\"#V!\"%$\"%K6F-F*7&$\"%v(*F-$\"%6\"*F-F*F*7&F*$\"$O(F-$\"%M&*F-F*7&F*
F*$\"$_%F-$\"%/)*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lamb
da[old]:= lambda; mu[old]:= mu;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%'lambdaG6#%$oldG$\"+E8WD5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
#muG6#%$oldG$\"+E8WD5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
31 "RV:= evalm(v); SAD:= evalm(w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#RVG-%'vectorG6#7&$\"+**********!#5$\"+T[/\\5!\"*$\"+].Kr:F.$\"\"!F
2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SADG-%'vectorG6#7&$\"+0o=(p$!#
6$\"+s@rgJ!#5$\"+dn4HKF.$\"+(Qs/C$F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 416 "Let's change the F2 value from
0.0043 to 0.00473 (a 10% change) and see what percent change we get i
n lambda. This is the so-called [1,2] entry of the matrix A, that is,
the number that appears in the first row (from top to bottom) and sec
ond column (from left to right). If Caswell's description of elastic
ity is correct, then we should get a change in lambda that is proporti
onal to the elasticity value for F2." }}{PARA 0 "> " 0 "" {MPLTEXT 1
0 57 "A[1,2]:=.00473; # make the change in the affected entry" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6$\"\"\"\"\"#$\"$t%!\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=evalm(A); # confirm the
matrix is OK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7
&\"\"!$\"$t%!\"&$\"%K6!\"%F*7&$\"%v(*F0$\"%6\"*F0F*F*7&F*$\"$O(F0$\"%M
&*F0F*7&F*F*$\"$_%F0$\"%/)*F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 77 "Now let's calculate the eigenvalues a
nd 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 54 "EVec:= seq( eigsys[i][
3], i = 1 .. 4); # eigenvectors " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
%EVecG6&<#-%'vectorG6#7&$!+HmPY7!#5$\"+M4J\"f\"!\"*$!+kt^`)*F-$\"+daV`
IF-<#-F(6#7&$\"+-#H2[%F-$\"+&\\i`#QF0$\"+k%=(**QF0$\"+cf++RF0<#-F(6#7&
$\"+1Gb<5F0$!+fj'p4\"F0$\"+9NC2&)!#6$!+D%)oRR!#7<#-F(6#7&\"\"!FSFS\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "EVal:= seq( eigsys[i]
[1], i = 1 .. 4); # eigenvalues" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "
mags:= op( map( abs, [EVal] )); lambda[new]:=max(mags);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%EValG6&$\"+!*RQX$)!#5$\"+omfD5!\"*$\"+^>$\\O%
!#7$\"%/)*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%magsG6&$\"+!*RQX$
)!#5$\"+omfD5!\"*$\"+^>$\\O%!#7$\"%/)*!\"%" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>&%'lambdaG6#%$newG$\"+omfD5!\"*" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Compare this new va
lue for lambda with the \"estimate\" of lambda at bottom of p 1447. T
his is the old value of lambda:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 12 "lambda[old];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E8WD5!\"*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "This is the ratio of the new
value to the old; the fractional change in lambda is the portion afte
r the decimal. Did you expect lthe new lambda to be larger? Why?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ratio:= lambda[new]/lambda[o
ld];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG$\"+)[^,+\"!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Here is the elasticity value. Com
pare it to the fractional change in lambda." }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 14 "elas:= e[1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%elasG$\"+2L58:!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 390 "Note tha
t we made a 10 percent change in fecundity (a fractional change of 0.1
). If you multiply this elasticity value by the fractional change mad
e in the fecundity, you should get a number close to the fractional c
hange in lambda. If you made a 100% change (a fractional change of 1.
0) in the fecundity value, you should see a fractional change in lambd
a equal to the elasticity value." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 44 "(elas * 0.1) - (ratio - 1); # close to zero?" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$!$y\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 63 "member( lambda[new], [mags], 'position'); location
:= position;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)locationG\"\"#" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 57 "evalm(SAD); # here's the original stable age distribu
tion" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&$\"+0o=(p$!#6$\"
+s@rgJ!#5$\"+dn4HKF,$\"+(Qs/C$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
47 "\nCompare with the new stable age distribution. " }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 116 "V := op( EVec[location] ); total:= sum( \+
V['k'] ,' k' = 1 .. 4 );\nw:= scalarmul(V, 1/total); # compare with SA
D above" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7&$\"+-#H
2[%!#5$\"+&\\i`#Q!\"*$\"+k%=(**QF.$\"+cf++RF." }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%&totalG$\"+%)fJ27!\")" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%\"wG-%'vectorG6#7&$\"+/WJ6P!#6$\"+D\\[oJ!#5$\"+`F2IKF.$\"+y3JI
KF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "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&\"\"!$\"%v(*!\"%
F*F*7&$\"$t%!\"&$\"%6\"*F-$\"$O(F-F*7&$\"%K6F-F*$\"%M&*F-$\"$_%F-7&F*F
*F*$\"%/)*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "EVecT:= seq
( eigsysT[i][3], i = 1 .. 4 ); # eigenvectors" }}{PARA 12 "" 1 ""
{XPPMATH 20 "6#>%&EVecTG6&<#-%'vectorG6#7&$!*n*='))*!\"*$!+FEe9W!#7$\"
+)H:#z6!#5\"\"!<#-F(6#7&$!+!))4'yQF3$!+Y?XpSF3$!+o_U\"3'F3F4<#-F(6#7&$
!+THEK6F3$!+T*omm*!#6$\"+#>J$y5F3F4<#-F(6#7&$!+lR#Q0&F3$!+%Q<)o]F3$!+U
m)yW%F3\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "EValT:= se
q( eigsysT[i][1], i = 1 .. 4 ); # eigenvalues" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 63 "magsT:= op( map( abs , [EValT] )); # magnitudes of ei
genvalues " }}{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&$\"+W>$\\O%!#7$\"+umfD5!\"*
$\"+>RQX$)!#5$\"%/)*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&magsTG6
&$\"+W>$\\O%!#7$\"+umfD5!\"*$\"+>RQX$)!#5$\"%/)*!\"%" }}{PARA 11 "" 1
"" {XPPMATH 20 "6#>&%#muG6#%$newG$\"+umfD5!\"*" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'indexTG\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "VT:= op( EVecT[index
T] ); v:= scalarmul(VT, 1/VT[1] ); # compare with RV above" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%#VTG-%'vectorG6#7&$!+!))4'yQ!#5$!+Y?XpSF+$
!+o_U\"3'F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#
7&$\"+)*********!#5$\"+aP?\\5!\"*$\"+0%Rzc\"F.$\"\"!F2" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(RV);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#-%'vectorG6#7&$\"+**********!#5$\"+T[/\\5!\"*$\"+].Kr:F
,$\"\"!F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 372 "***Now, how did v a
nd 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% from 0.0452
to 0.05198. (Don't forget to reset A[1,2] back to its original value
of 0.0043!) What effect do you expect to see? What happens in fact?
***" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Individual pod data" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "I
f you want to do a similar analysis on the individual pod data in the \+
appendix, you need to set up a symbolic description of matrix A, and t
hen substitute in the values from the table:" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 54 "Ag:=[[0,F2,F3,0],[G1,P2,0,0],[0,G2,P3,0],[0,0,G3,P4]]
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A:=matrix(4,4,subs(\{G1=0.953
5,G2=0.0803,G3=0.0414,P2=0.8827,P3=0.9586,P4=0.9752,F2=0.0067,F3=0.163
2\},Ag));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AgG7&7&\"\"!%#F2G%#F3G
F'7&%#G1G%#P2GF'F'7&F'%#G2G%#P3GF'7&F'F'%#G3G%#P4G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"!$\"#n!\"%$\"%K;F-F*7&$\"%N
&*F-$\"%F))F-F*F*7&F*$\"$.)F-$\"%'e*F-F*7&F*F*$\"$9%F-$\"%_(*F-" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 552 "N
ow move your cursor back to the line that does the first eigenvalue ca
lculation at the top of the worksheet. This is the line that defines \+
\"eigsys\" just after A is first defined. Then go back through the wor
ksheet to get new eigenvalues, eigenvectors, sensitivities and elastic
ities. Alternatively, replace the lines defining A at the top of the \+
worksheet with these lines, and use what has been a closely guarded s
ecret up til now. In the menu bar on top look for Execute Worksheet (o
n some systems it's under View, but this varies). BAM!! DONE!!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 236 "NOTE - if you use new data in the worksheet, you may fin
d that some of the procedures don't work. Usually the problems arise \+
from specific matrices having repeated eigenvalues. We can show you h
ow to work around this problem in class." }}}}}}{MARK "0 2 0" 20 }
{VIEWOPTS 0 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }