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THEORETICAL
I’OPULATION
BIOLOGY
14, 215-230 (1978)
A General Formula for the Sensitivity of
Population Growth Rate to Changes in
Life History Parameters
IfAL
Biological
CASWELL
Sciences Group U-42, University
Received
of Connecticut,
Stows, Connecticut 06268
July 15, 1977
This paper considers the sensitivity of population growth to small changes in
birth, growth, survival, and migration probabilities
for an arbitrary population
classification (i.e., age, instar, size, developmental stage, age, and spatial location,
etc.). The stage-specific life history parameters are expressed in a discrete-time
system of linear difference equations, the dominant eigenvalue of which defmes
the population growth rate. The sensitivity of this eigenvalue to production of
class i by class j individuals
is shown to be proportional
to the product of the
reproductive
value of stage i and the abundance of stage j in the stable stage
distribution.
This formula is readily computable,
and several examples are
presented. For the special case of age-structured
populations,
this formula
reduces to those derived by Hamilton,
Emlen, and Goodman.
INTRODUCTION
The problem of the sensitivity of population growth rate to small changes
in life history parameters was first investigated numerically by Lewontin (1965)
using simple artificial life tables. Hamilton (1966), Demetrius (1969), Emlen
(1970), Goodman (1971), Keyfitz (1971), and Mertz (1971) have since tackled
the problem in a more analytical fashion. The purpose of this paper is to present
a new formulation for growth rate sensitivity, which has several advantages over
those derived previously. It is applicable to any linear population growth model,
whether based on age distributions,
on more general developmental
stage
distributions (e.g., size, instars, etc.), or even on multiple classification distributions (e.g., age and size or age and spatial location). This allows existing theory
to be extended to those organisms, such as many plants, for which age is an
inadequate state variable. The formulas are easily computable for real populations, and they allow several results already derived in the literature to be
modified or extended. A brief description of this formula and an application to
a plant population have been presented in Caswell and Werner (1978).
215
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HAL CASWJZLL
DERIVATIONS
Linear Population Models
The derivation
tion growth:
is based on a linear, discrete, time-invariant
qt + 1) = AC(t).
model of popula-
(1)
Here, Z(t) is an n-dimensional vector, the elements of which (xi) give the number
of individuals in the ith stage at time t. A is an tl x n matrix, the elements of
which (aij) are determined by the birth, growth, death, survival, and migration
processes of the population. Such a model is, in essence, the dynamic expression
of an instantaneous life table, and can be regarded either as describing population
growth at the instant for which the life table is valid or as predicting the hypothetical pattern of growth that would result if the life table were to remain in
effect indefinitely.
The assumptions of linearity and time-invariance
exclude
considerations of density effects and temporal fluctuations in the environment.
This formulation was originally developed, using an age structure classification, by Bernardelli (1941), Lewis (1942), and Leslie (1945). If the units of
time and age are the same, the matrix A has the special form commonly known
as a Leslie matrix:
A =
where Fi is the effective fecundity of an i-year-old and Pi is the probability
of
survival from age i to age i + 1.
Most of the analytical work on (I), and all of the work on growth rate sensitivity, has been based on such age-structured models. It is not generally appreciated just how many populations cannot be adequately described using age
distribution
as a state variable (Caswell et al., 1972). Classifications by instar
(Lefkovitch,
1965), size class (Usher, 1966, Hartshorn,
1975), developmental
stage (Werner and Caswell, 1977) or even multiple classifications (Rogers, 1966;
Slobodkin, 1953; Sinko and Streifer, 1969) are more appropriate for a wide
variety of organisms. Any of these classifications can be accommodated within
the framework of Eq. (l), but theory which depends on the special form of (2)
is inappropriate.
The general solution to Eq. (1) is
f(t) = f c,git,
i=l
(3)
GROWTH
RATE
SENSITIVITY
217
where hi and ci are the eigenvalues and right eigenvectors of & respectively.
The constants ci are determined by initial conditions. The long-term behavior
of this solution can be deduced from well-known theorems of matrix algebra
(see the Appendix for a discussion of the non-age-structured
case). In general,
there will be a positive real eigenvalue, &,, , of modulus greater than any other.
Its corresponding eigenvector tiTn will be real and nonnegative. As t gets large,
the contribution
of this maximal eigenvalue and eigenvector dominates the
solution, and the population grows geometrically at a rate X, with a stable age
or stage distribution given by v;, . This growth rate is the focus of most theoretical life history studies, mainly because of its identification
with the average
theorem of
fitness of the population by Fisher (1958), and his fundamental
natural selection, stating that natural selection will favor genotypes which increase
the value of & .
The left eigenvector ii,,, corresponding to X, gives the reproductive value of
the different stages, and is defined by
i&A
= A,ii~ .
(4)
This relation was derived for the age distribution
model by Goodman (1968);
a simple derivation and application to a stage classification model can be found
in Caswell and Werner (1978).
A General Formula for Sensitivity
Within the framework of the linear population model (1) the problem of
growth rate sensitivity reduces to the problem of finding the response of h, to
changes in the entries of d Demetrius (1969), Goodman (1971), Emlen (1970),
and Keyfitz (1971) approached this problem by differentiating
the characteristic
equation of A (or the corresponding equation for the continuous age case) for
age-structured
models. This will not work for arbitrary stage classifications,
since the form of the characteristic equation is not generally known beforehand.
The formula presented here bypasses the characteristic equation in calculating
sensitivities. Originally
derived by Jacobi (1846), it has since been rederived
several times (e.g., Faddeev, 1959; Faddeev and Faddeeva, 1963; Desoer, 1967).
My derivation here is framed in terms of the population growth model (1).
but the formula can be applied to any eigenvalue problem, and will thus be
useful in a number of ecological contexts outside of life history theory. A similar
approach to eigenvector sensitivity will be presented elsewhere (Caswell, 1978).
Let dx be an n x n matrix, the elements of which (da,J are differential
perturbations
of the elements of 2ii. The eigenvectors and eigenvalues of A are
defined by
&
= xiui (
i = I,..., n,
(54
$A
= hiil,’ ,
(zii, F<) = 1,
i = l,..., II,
(5b)
i = l,..., 12,
(5c)
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HAL
CASWELL
where (%,jQ = 3’7 is the scalar product of x and 7. Equation (5~) is a
normalization that may be assumed without loss of generality.
We wish to find the change dh, in the maximum eigenvalue caused by the
change d& in the life history. We start with the equation defining A, :
A-q,, = h,q,, .
(6)
Taking the differential of both sides yields
Next, form the scalar product of both sides of (7) with iim :
Expanding the scalar products and simplifying
dh, = ((di&
(7)
leaves1
, i&J.
(9)
An appealing aspect of this result is that it can deal with simultaneous changes
in several life history stages at once. For most applications, however, and for
comparison with existing theory, the formula can be simplified by assuming that
only one parameter is changing at a time. In this case dif contains only a single
nonzero element, da, , and (9) simplifies to
where u,(i) and w&) are the reproductive value of the ith class and the representation of the jth class in the stable stage distribution, respectively. Since aii
measures the production of class i individuals by a class j individual in a single
time unit, (10) has the reasonable interpretation that A,,, is most sensitive to
changes in life history parameters describing the production of high reproductive
value individuals by members of abundant age classes.
For the age-structured case, only the first row and subdiagonal terms can be
modified, and (10) can be rewritten as
dp,
=
____
daj,,.j
= %n(i + 1) %n,(i),
-d&n = -d&n = urn(l) urn(j).
dFj
da,,
1 If the normalization
(SC) is not assumed, Eq. (9) should be replaced
<@A>%, Em>/<%, G> and Eq. (13) by I dL I < II dAIl * I 4 I * I rS, Ill<%,
(12)
by dA,,, =
%>I.
GROWTH
RATE
219
SENSITIVITY
Thus, the sensitivity of & to changes in fecundity is directly proportional
to the stable age distribution, while the sensitivity to survival changes involves
reproductive values as well.
A potentially useful sensitivity index can be derived from (9). Taking norms of
both sides yields
where the matrix norm i/ dAI/ = (x:i,j du~j)l~L.
The sensitivity index S = 1V, 1 . j ii,,, / relates the absolute change in h,,
to the overall magnitude of the change in the life history parameters. Because
of the normalization (5c), S > 1; values considerably larger than 1 indicate
that the life history involved may be (because the relationship is only an inequality) unstable in the sense that X, is highly sensitive to life history alterations.
Note that S is not limited to changes in only one parameter at a time.
Some Examples
One of the advantages of this sensitivity formula (10) is that it is easily computable, given only a routine for evaluating eigenvectors. Such routines are now
readily available, eliminating the need to work directly with the algebra of the
characteristic equation. Figures 1 through 4 show some examples for populations
classified by criteria other than age. Figure 1 shows the results for a population
of Dipsucus syhestris, a weedy biennial plant [see Caswell and Werner (1978) and
_--
431
-_-
SENSITIVITY
!4xTIX
---
___
--_
___
___
.OOl
___
___
___
_-_
_--
___
_-_
___
_-_
--
--
75
___
--_
__-
___
___
-__
.042
__-
.g,
__-
---
_--
___
___
---
.ooz
___
--_
__-
---
-_-
.Ol
.Ol
.Ol
.13
---
___
_-_
.97
.40
.f?
.Ol
_--
___
--_
.07
.Ol
.oo
.13
.24
---
---
5.2
2.2
1.2
.05
.25
---
---
.OO?
.Ol
.OO
.04
.25
.17
---
17.3
11.4
6.1
.25
1.3
.23
---
--_
___
___
.oo
.o*
.75
--_
(59)
(25)
(13)
.54
2.8
.51
(.25
FIG. 1. The state vector, population
growth matrix, and sensitivity matrix for an
experimental population of teasel (Dipsacur sylvestris Huds.) classified by developmental
stage. The entries in the sensitivity matrix are the sensitivities of A,,, to changes in the
corresponding
entry of the growth matrix. The figures in parentheses are the sensitivities
to changes involving major life history changes: from a biennial to an annual (lower left)
or a perennial (lower right) habit. Model from Werner and Caswell (1977).
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HAL
CASWELL
Werner and Caswell (1977) for details]. Detailed studies of this species showed
that age classification models were inferior to a developmental stage classification
in describing demographic transitions (Werner, 1975) or overall population
growth (Werner and Caswell, 1977). The population and sensitivity matrices for
one of eight experimental populations are shown in Fig. 1. The sensitivities
show a general tendency to increase down rows and to decrease across columns.
The highest sensitivities are to changes in the lower left corner, to parameters
representing the most rapid production of large reproductive individuals by
small individuals.
PENTACLETHRA
+-
U
t
+
MACROLOBA
REPRODUCTION
SURVIVAL (IN SAME SIZE CLASS)
GROWTH (TO NEXT SIZE CLASS)
FIG. 2. Sensitivity
of A,,, to changes in reproduction,
survival,
and growth for
Pentaclethra macroloba, a tropical tree, classified by size. Model from Hartshom (1975).
GROWTH
221
,1&Q
____
_-__
-._.
----
__._
.ljO,
_-__
--_.
--.-
____
____
.2X)6
_.__
-..-
---
.1x5
----
_... ___- __..
.$jjq;
-__.
-1
FIG. 3. The population growth matrix for the human population of California and
the rest of the United States, classified by age and spatial location. The upper half of the
state vector describes the age distribution in California; the lower half that in the rest of
the United States. Population growth within the two regions is described by the upper
left and lower right quadrants of the matrix, respectively. Age-specific migration rates
from the United States to California are in the upper right quadrant; the corresponding
rates in the reverse direction are in the lower left. Model from Rogers (1966).
In Fig. 2, the analysis is applied to Hartshorn’s (1975) data for Pentuckthra
mucroloba, a tropical rainforest tree. The model uses a size classification based
on tree height or diameter. Size classes were such that an individual could not
grow more than one class larger in a single year. Thus, the matrix contained
nonzero entries only in the first row (representing reproduction),
the diagonal
(representing survival in the same class), and the subdiagonal (growth to the
next class). The sensitivities of X, to changes in these parameters are plotted in
Fig. 2 as a function of size.
Several patterns are evident in the figure. The population growth rate is more
sensitive to changes in growth and survival than to changes in fecundity
throughout
the entire life cycle. There is a peak in growth rate sensitivity
in the seedling stage and then another sharp increase in the middle of the size
class spectrum. These results bear on Hartshorn’s (1975) analysis of the impact
of seed and seedling predation on population growth. Although he concluded
that predation would have no significant effect, Fig. 2 suggests that an increase
in seedling predation, at least, would have an important impact on h,,, .
Discrete population models can also be applied to populations classfied by
multiple criteria. Figures 3 and 4 examine a model using an age and spatial
location classification. The model, attributable to Rogers (1966) describes the
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HAL
CASWJZLL
growth of the human populations of California and the rest of the United States,
and age-specific migration between the two regions. Figure 3 shows the population matrix; the analysis could be extended to populations in several regions, the
result being larger partitioned growth matrices.
The sensitivities of A, to changes in survival and fecundity in the two regions
are nearly identical, (Fig. 4) although dh,/dFj falls off faster with age in
California than in the rest of the United States. The sensitivity of A,, to migration
parameters, however, is strikingly different. Migration from the United States
to California at any age is about an order of magnitude more important than
migration in the other direction,
Figure 5 shows sensitivity graphs for several age-classified populations.
Very diverse taxa of organisms have sensitivity graphs of very similar form.
’ lO()CALIFORNIA
USA
MIGRATION
IO’
IO- 2-
lo-36
’
’
20
’
’
40
-&-%k
AGE (YRS)
FIG. 4.
Sensitivity
of A,,, to changes in the matrix
shown in Fig. 3.
GROWTH
RATE
223
SENSITIVITY
It is notable that sensitivity to life history alterations can vary by as much as
six orders of magnitude over the reproductive lifespan of an individual.
Since
dA,,/dFj and dA,/dPj have been identified by Emlen (1970) with the selective
pressures operating on genes with age-specific effects on fecundity and survival,
this extreme variation may be reflected in extreme differences in age-specific
evolutionary changes. This is the basis of the theory of senescence (see below),
but no one seems to have noticed the magnitude of the differences that may be
involved.
AGE
120 wks
b
I
FIG. 5. Sensitivities to changes in reproduction
and
populations. The abcissa has been scaled to the length of
organism. (a) Laboratory
population of the flour beetle,
(b) Laboratory
population of the Orkney vole, Microtus
(c) The human population of the United States in 1965
55 vrs
c
’
-dXm
survival for three age-classified
the reproductive
period of each
Culandra oryzae (Birch, 1948).
orcademis (Leslie et al., 1955).
(Keyfitz and Flieger, 1968).
The sensitivity coefficient S, defined by Eq. (13), is difficult to interpret at
this point. Table I shows the extent of variation present in the models examined
in this paper. The trend shown in the last three lines of the table suggests that
shorter lived and/or more rapidly growing populations have a higher overall
sensitivity to life history alterations. It is tempting to interpret the extremely
high S value for Pentaclethra macroloba, together with the lack of a clear pattern
in the sensitivity graph in Fig. 2, as suggesting that Hartshorn’s choice of a state
variable had somehow missed an essential aspect of population structure. This
conjecture is supported by a comparison of the age structure and stage structure
models for Dipsacus sykwtris; the developmental
stage structure is known to
224
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CASWELL
TABLE I
Sensitivity Coefficients for the Populations Show in Figs. 1 to 4.
Population
s
Llipsacus sylvestris
Pentaclethra
65.79
macroloba
105.41
California/U.S.A.
1.51
Calandra
ory3zae
7.82
Microtus
orcadensis
2.00
U.S.A.
1.17
be more accurate (Werner and Caswell, 1977). Over the seven populations
examined, the ratio of the age-structured to the stage-structured sensitivity
coefficients averages 3.42 (SE = 1.88). M ore extensive comparative studies are
needed before this statistic can be clearly interpreted.
IMPLICATIONS
Other
Sensitivity
AND CONNECTIONS
Formulas
The characteristic equation for the special case of age-structured models can
be written out explicitly, and dh,ldF, and dh,ldPj derived by implicit differentiation. Hamilton (1966) and Emlen (1970) arrived by this means at formulas
equivalent to Eq. (lo), but did not realize that they could be simplified in terms
of cm and iim . Goodman (1971) and Keyfitz (1971) first pointed out the connection, although their derivations were more complex and limited to age-structured
models.
Inequalities
Demetrius (1969) derived several inequalities relating sensitivities at different
ages; these inequalities play an important role in the theory of senescence. His
results can be extended, using Eq. (10) and the well-known formulas for e’,,,
and ts, (Pollard, 1973, p.43):
Z&(l) = 1,
v,(j)
= PoPI ... Pj-IA;;;‘,
(14)
%2(l) = 1,
%n(i) = i
i=j
(PPi+1 . . . pfel) FJki-‘.
(15)
GROWTH
RATE
225
SENSITIVITY
These versions of Us,,and tim are not scaled following Eq. (5c), but the necessary
corrections cancel out in all of the following analyses.
Consider first the relative magnitude of dh,/dFj and dA,,/Fi+l , given by
dA,/dFj = PI .*. Pj_lAi’,
dh,jdFi+l
= PI ... Pj-lPjXf:‘)m
(16)
(17)
Thus,
dh,/dFj
dA,ldFi+l
A,
= pj ’
(18)
so dA,,/dFj > dh,/dFj+l if and only if A,,, > Pj . Demetrius’ (1969) result, that
the sensitivity to changes in fecundity is nonincreasing if and only if A, >
max(Pi), follows immediately. It also follows that if survival is age independent,
the fecundity sensitivity curve will be exactly exponential.
Demetrius [1969, Eq. (ll), although there is a typographical omission of p
in his paper] showed that
dUdPi
dh,ldPj
pi
>P==
(i > j).
(19)
This inequality can be sharpened. From (10) we know that
Substituting (14) and (15) into (20) and rearranging yield
4#pj
dA,ldPj+,
pj,l
= - Pj
+
=- p3%l f
Pi
Pj(Fj+&;
Fj+1
-1 ... + pj+zPi+3 e-eP,,-lF,Xp+l)
Fj+1
Pi%(i + 2)
Sensivity to changes in survival clearly decreases with age if Pj+l > Pi, in
accord with Demetrius’ result (19). H owever, Eq. (20) shows that even if
Pj+l < Pj , the contribution of the second term may still result in a declining
curve.
If sensitivities separated by more than a single age interval were being compared (say, ages j and K) the Fj+l term in (20) would be replaced by a sum of
terms, representing the portion of the reproductive value of a j-year-old lying
between agesj and K. Thus, the age dependence of survival sensitivity is affected
by future reproductive value as well as by survival at the ages in question.
Demetrius (1969) also evaluated the relative importance of fecundity and
226
HAL
CASWELL
survival at a given age but was able to obtain results only by assuming a particular
set of Fj values. Here we can do better, for
dh,,JdP, =
dh,,JdFj
%7di+ l)%(j)
%n(1) @m(i)
=
%n(i+ 1)
%nU)
*
c-w
This result has implications for the theory of life history optimality (Caswell,
1978).
Senescence
Hamilton (1966) and Emlen (1970) studied life history sensitivities in the context
of the problem of the evolution of senescence. This theory, originated by
Medawar (1952, 1955) and Williams (1957), is based on the declining of the
dA,,,/dPj curve with increasing age. [Demetrius (1978) has recently introduced
a theory based instead on entropy maximization.] If a pleiotropic gene has
a beneficial effect on survival at one age and an equal deleterious effect on
survival at a later age, its net effect on h, (and thus the selective pressure favoring
its increase; see Emlen, 1970) will still be positive. Deleterious gene effects will
thus accumulate late in life, because they have little effect on fitness there.
Equation (10) helps to clarify one issue in this area. Fisher (1958) had conjectured that senescence was related to the form of the reproductive value
curve. If selection pressure were directly proportional to reproductive value,
it would lead to senescence beginning at the peak in the reproductive value
curve. When Medawar (1952) examined the problem, however, he based his
analysis directly on the reproductive contribution of each age class to the next
generation. He showed that even in a population where reproductive value did
not vary with age the form of the stable age distribution would lead to a declining
importance of older age classes, and thus to the evolution of senescence. At the
end of his paper, however, he casually mentions Fisher’s theory and suggests
that reproductive value may have something to do with the situation. Hamilton
(1966) and Emlen (1970) both argued that only fitness and not reproductive
value alone was directly related to senescence.
In fact, Medawar was more correct than he or his critics may have realized,
and Fisher was approximately half correct. dh,/dPj is related to both reproductive
value and the stable age distribution. In Medawar’s (1952) hypothetical population of test-tubes, reproductive value was constant; the decline in the age distribution led to the potential for senescence. Hamilton (1966) modeled a hypothetical, immortal VoZvox-like creature, whose reproductive value increased with
age. Again, senescence evolved, under the impact of the declining dhldPj curve
caused by the product of the increasing reproductive value and the declining
age distribution.
GROWTH
RATE
SENSITIVITY
221
Other .Qplications
The general formula (10) f or eigenvalue sensitivity is valid for any matrix,
and thus has potential applications to a number of other areas of ecology.
Competition matrices and linear compartment models (cf. Hett and O’Neill,
1974) spring to mind immediately. In both cases the stability of the system is
determined by the largest eigenvalue, and the rate of return to equilibrium,
if stable, by the modulus of the smallest eigenvalue. The sensitivity structure of
these matrices would be well worth investigating. For population models, the
rate of approach to the stable stage distribution is largely determined by the
second largest eigenvalue of A, and the sensitivity of this eigenvalue to life
history changes would bear investigating as well.
APPENDIX
The assertion that an arbitrary stage classification model ultimately grows
at a rate h, with a stable stage distribution tim is not strictly true, and its assumption in applying Eq. (10) requires some justification. For the assertion to be
true, h,, must be real, nonnegative, greater in modulus than any other eigenvalue,
and have a real, nonnegative eigenvector.
The eigenvalue spectrum of age structured Leslie matrices has been studied
by Sykes (1969), Demetrius (1971), and Cull and Vogt (1973, 1974). The Leslie
matrix is clearly nonnegative, and can be shown to be irreducible. On these
grounds, the Perron-Frobenius theorem (Gantmacher, 1959, p. 53) guarantees
the existence of a positive real eigenvalue X, with an associated real, nonnegative
eigenvector. The modulus of h, is greater than or equal to that of any of the
other eigenvalues. If the greatest common divisor of the ages at which fecundity
in nonzero is one, then A is also primitive (Demetrius, 1971) and 1h, ( is
strictly greater than the modulus of any other eigenvalue.
A stage classification model can reasonably be assumed to be nonnegative.
To assume otherwise is to admit the possibility of negative population sizes.
Irreducibility need not be assumed, for there is a weaker form of the PerronFrobenius theorem, applicable to reducible nonnegative matrices (Gantmacher,
1959, p. 66). It differs from the stronger version only in allowing the possibility
of a maximum eigenvalue of zero. Such an eigenvalue would imply the singularity
of L?i,and hence a redundancy in the classification of the population into stages.
Definition of a new state variable could eliminate this and lead to a population
matrix of full rank.
The existence of two or more eigenvalues of the same modulus cannot be
completely ruled out without a detailed study of the particular matrix involved.
However, I conjecture (and it would be nice to prove) that primitivity is a generic
condition for matrices; i.e., that the existence of roots of the same modulus is
228
HAL
CASWELL
structurally unstable. Hirsch and Smale (1974) p rove that rqeuted eigenvalues
are structurally unstable; it seems that a proof based on the continuity of eigenvalues would do the same for imprimitivity. If my conjecture is true, then only
for very special life histories, not subject to measurement error (e.g., a Leslie
matrix with precisely periodic fecundities) would imprimitivity be a problem.
Cull and Vogt (1973, 1974, 1976) h ave recently analyzed the properties of
imprimitive Leslie matrices. The contribution of the second (complex) eigenvalue, of modulus equal to that of A, , leads to an oscillatory pattern of population growth. The average trend of this growth, however, is given by A,,,, and
so the theory in this paper could be applied to imprimitive matrices by merely
considering the average growth rate over a single cycle.
ACKNOWLEDGMENTS
This researchwas supported by NSF Grants GB 43378 and DEB76-19278, which are
gratefully
acknowledged.
I thank Lloyd Demetrius and J. Merrit Emlen for helpful
comments.
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