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654
Special Feature
Ecology,
81(3), 2000, pp. 654–665
q
2000 by the Ecological Society of America
LIFE HISTORIES AND ELASTICITY PATTERNS: PERTURBATION
ANALYSIS FOR SPECIES WITH MINIMAL DEMOGRAPHIC DATA
S
ELINA
S. H
EPPELL
,
1,4
H
AL
C
ASWELL
,
2
AND
L
ARRY
B. C
ROWDER
3
1
Department of Zoology, Duke University Marine Lab, 135 Duke Marine Lab Road,
Beaufort, North Carolina 28516 USA
2
Biology Department, Woods Hole Oceanographic Institute, Woods Hole, Massachusetts 02543 USA
3
Nicholas School of the Environment, Duke University Marine Lab, 135 Duke Marine Lab Road,
Beaufort, North Carolina 28516 USA
Abstract.
Elasticity analysis is a useful tool in conservation biology. The relative
impacts of proportional changes in fertility, juvenile survival, and adult survival on as-
ymptotic population growth
l
(where ln(
l
)
5
r,
the intrinsic rate of increase) are determined
by vital rates (survival, growth, and fertility), which also define the life history character-
istics of a species or population. Because we do not have good demographic information
for most threatened populations, it is useful to categorize species according to their life
history characteristics and related elasticity patterns. To do this, we compared the elasticity
patterns generated by the life tables of 50 mammal populations. In age-classified models,
the sum of the fertility elasticities and the survival elasticity for each juvenile age-class
are equal; thus, age at maturity has a large impact on the contribution of juvenile survival
to
l
. Mammals that mature early and have large litters (‘‘fast’’ mammals, such as rodents
and smaller carnivores) also generally have short lifespans; these populations had relatively
high fertility elasticities and lower adult survival elasticities. ‘‘Slow’’ mammals (those that
mature late), having few offspring and higher adult survival rates (such as ungulates and
marine mammals), had much lower fertility elasticities and high adult or juvenile survival
elasticities. Although certain life history characteristics are phylogenetically constrained,
we found that elasticity patterns within an order or family can be quite diverse, whilesimilar
elasticity patterns can occur in distantly related taxa.
We extended our generalizations by developing a simple age-classified model param-
eterized by juvenile survival, mean adult survival, age at maturity, and mean annual fertility.
The elasticity patterns of this model are determined by age at maturity, mean adult survival,
and
l
, and they compare favorably with the summed elasticities of full Leslie matrices.
Thus, elasticity patterns can be predicted, even when complete life table information is
unavailable. In addition to classifying species for management purposes, the results gen-
erated by this simplified model show how elasticity patterns may change if the vital rate
information is uncertain. Elasticity analysis can be a qualitative guide for research and
management, particularly for poorly known species, and a useful first step in a larger
modeling effort to determine population viability.
Key words: age-based model; conservation; elasticity analysis; life history; life table; mammal;
management; matrix model; population model.
I
NTRODUCTION
Elasticity analysis can be used to categorize popu-
lations according to the response of population growth
l
to perturbations that affect vital rates. An elasticity
pattern is composed of the relative contributions of
matrix entries to population growth that are grouped
in biologically meaningful ways for comparative anal-
ysis. For example, in animal populations, we may want
to compare the relative contributions of fertility, ju-
Manuscript received 9 October 1998; revised 26 May 1999;
accepted 29 May 1999. For reprints of this Special Feature, see
footnote 1, p. 605.
4
Present address: U.S. Environmental Protection Agen-
cy, National Health and Environmental Effects Research
Laboratory (NHEERL), Western Ecology Division, 200
SW 35th Street, Corvallis, Oregon 97333 USA.
E-mail: heppell@mail.cor.epa.gov
venile survival, and adult survival; while, in plants, the
categories are often growth, reproduction, and stasis.
Elasticity patterns are similar for populations or species
that share life history characteristics. Comparisons
have been made across taxa for plants (Silvertown et
al. 1993, 1996), birds (Sæther et al. 1996, Sæther and
Bakke 2000), and turtles (Cunnington and Brooks
1996, Heppell 1998). These elasticity patterns may pro-
vide general rules of thumb for categorizing popula-
tions according to their relative responses to pertur-
bations of particular life stages. Although general pat-
terns exist for certain taxa, such as long-lived fresh-
water turtles, there is often considerable variation in
the elasticity patterns of closely-related species, and
even populations within a species, due to habitat char-
acteristics or disturbance regimes that affect vital rates
(Silvertown et al. 1996, Oostermeijer et al. 1996). Be-
March 2000 655
ELASTICITY ANALYSIS IN POPULATION BIOLOGY
Special Feature
cause of variance and uncertainty in vital rates, it is
important to know what combinations of vital rates and
population growth rates give rise to particular elasticity
patterns.
There is a practical need for a classification of pop-
ulations according to their likely response to pertur-
bation. Conservation management plans cannot be as-
sessed for most species, because demographic data to
construct detailed age- or stage-specific models are un-
available. Obtaining complete estimates of vital rates,
including their temporal variances and covariances, is
time consuming for any species and may be impossible
for those that have long life spans or that cover wide
geographic ranges, such as marine taxa (Boyce 1992,
Caughley 1994, Heppell and Crowder 1998). Green and
Hirons (1991) suggested that
,
2% of threatened bird
populations have enough biological data to parameter-
ize even simple population models. Consequently, de-
mographic models are generally used on a case-by-case
basis for particular, well-studied species, rather than as
a general framework for setting management priorities
(Caughley 1994, Groom and Pascual 1998). Although
quantitative analysis of extinction risk is included as
one criteria for risk categorization used by the Inter-
national Union for the Conservation of Nature (IUCN),
most species are classified by population size, as well
as observed rates of decline or habitat loss (IUCN
1996). While the status of a population and its habitat
is clearly an important factor, it does not provide guid-
ance for optimizing management efforts (Mace and
Hudson 1999). Because demographic information is
incomplete for most threatened taxa, it is clear that a
generalized approach is needed to guide research and
management for poorly known species (Carroll et al.
1996).
Life history strategies are inexorably linked to vital
rates. Thus, the impact of a perturbation to survival,
growth, or fertility should be in part dependent on the
life history strategy of the affected population. For ex-
ample, a decrease in adult survival rate should be more
detrimental to population growth in populations that
are long lived with low annual reproductive success,
than in those that are short lived and highly fecund.
Recently, members of the IUCN fish conservation
group suggested that life history characteristics be in-
cluded as a weighting factor for risk classification, with
early-maturing, highly fecund cod and tuna classified
as less at risk than slow-growing species such as sharks
(IUCN 1997). However, an objective method for as-
sessing risk to threatened species according to their life
history characteristics has yet to be developed.
Correlations among life history characteristics have
been well studied for mammals, birds, reptiles, and
other taxa (see Charnov (1993) and references therein).
Many life history variables are strongly correlated with
adult body size (Western 1979, Stearns 1983, Wooton
1987) and can be constrained by phylogeny (Stearns
1983, Harvey and Pagel 1991). Other relationships are
less intuitive and independent of body size (Harvey
and Zammuto 1985, Gaillard et al.1989, Read and Har-
vey 1989, Charnov 1993). Mammal life tables are com-
mon in the literature and have been used in comparative
studies of life history tactics (Millar and Zammuto
1983, Promislow and Harvey 1990, Purvis and Harvey
1995). Several authors have used their results to cat-
egorize mammals on a ‘‘fast–slow’’ continuum, with
species that reproduce early, have large litters, and ex-
hibit short generation times contrasted with those with
late maturity, one or fewer offspring per year, and long
generation times (Read and Harvey 1989, Promislow
and Harvey 1990). Links between elasticity patterns
and the fast–slow continuum may provide a useful tool
for classifying mammals according to their likely re-
sponse to stage-specific perturbations.
However, elasticity analysis requires a population
model, and the formulation of even a simple deter-
ministic projection matrix requires a large amount of
demographic information for a population, including a
complete birth and death schedule. If data are scarce,
we need a way to predict relative responses to pertur-
bations based on elasticity analysis of approximate
models. We can use elasticity patterns generated from
models of similar, well-studied species. One problem
with this approach is how to determine what constitutes
a ‘‘similar’’ species, that is, should we rely on phy-
logeny, body size, or some suite of life history char-
acteristics? Alternatively, we can develop simple mod-
els based on minimal demographic data, such as partial
life cycle analysis (Caswell 1989). The characteristic
equations of certain kinds of simplified models make
it easy to interpret the source of elasticity patterns.
Life tables are easily converted to age-based projec-
tion matrices for elasticity calculations. We present
here an analysis of 50 published life tables for mam-
mals, many of which were used in past life history
analyses. We compare elasticity patterns across and
within mammalian orders to identify the source of elas-
ticity patterns, i.e., the relative proportional contribu-
tions of reproduction, juvenile survival, and adult sur-
vival to the growth rate of a population. We then derive
a new method to estimate elasticities without a com-
plete life table, and we compare the results with those
of complete age-based models. Our results reveal much
about how mammals and other vertebrates can be cat-
egorized by life history types, and how elasticity pat-
terns may shift with changes or uncertainties in age at
maturity, adult survival, and population growth rate.
We hope to provide a valuable tool for making pre-
dictions about the effects of perturbations for poorly
known species.
M
ETHODS
Analysis of mammalian elasticity patterns
We examined 50 published life tables for 44 mammal
species representing 10 orders and 25 families (Ap-
Special Feature
656
SELINA S. HEPPELL ET AL.
Ecology, Vol. 81, No. 3
pendix A). Order and family classifications are listed
according to Wilson and Reeder (1993). In 32 cases,
the life tables included age-specific reproductive rates
(female newborns per female per year), which gener-
ally varied according to proportion of females who
were mature in a given age class. In the remaining life
tables, survivorship and age at first breeding were the
only age-specific data available, so the mean fertility
provided by the author(s) was prescribed to all mature
age classes. Life tables that were constructed on half-
year intervals were rounded to the next integer year.
We calculated several standard life history measures
for each life table, including the net reproductive rate
for the population
R
0
, mean generation time
T
c
, sur-
vivorship to first maturity
l
a
, and life expectation at
birth
e˚
0
, and life expectation at maturity
e˚
a
(expectation
of further life; Pianka [1978]):
k
R
5
lm
(1)
O
0ii
i
5
0
k
il m
O
ii
i
5
0
T
5
(2)
ck
lm
O
ii
i
5
0
k
e˚
5
l
(3)
O
0i
i
5
0
k
l
O
i
i
5a
e˚
5
(4)
a
l
a
where
l
i
is the survivorship of a cohort to age
i, m
i
is
the number of female offspring produced annually by
a female aged
i, k
is the maximum age, and
a
is the
age at first reproduction (the first age class with
m
±
0).We then converted each life table to a prebreeding,
birth pulse projection matrix (Leslie matrix [Leslie
1945, Caswell 1989]) in which the survival probability
P
i
and fertility
F
i
of age class
i
are given by the fol-
lowing:
l
i
1
1
P
5
(5)
i
l
i
F
5
lm
. (6)
i1i
F
i
thus includes survival to age 1. We calculated the
elasticity matrices Efrom the eigenvectors of each pro-
jection matrix A(Caswell et al. 1984, de Kroon et al.
1986):
avw
ij i j
e
5
(7)
ij
l^
w,v
&
where vand ware the left and right eigenvectors of
the projection matrix A, and
^
w, v
&
is the scalar product
of the two vectors. We summed the elasticities of
l
to
changes in
P
i
and
F
i
across age classes to obtain the
following quantities of interest:
0 1. Fertility elasticity, which is the effect of a
proportional change in reproductive output for all adult
age classes (the sum of elements in the top row of E).
1 2. Juvenile survival elasticity, which is the ef-
fect of a proportional change in all annual survival rates
for age 1 to the year just prior to maturation (the sum
of the subdiagonal elements of Efrom column 1 to
column
a2
1, where
a
represents the first age class
that includes breeding females).
2 3. Adult survival elasticity, which is the effect
of a proportional change in all annual survival rates
for mature individuals (the sum of the subdiagonal el-
ements of Efrom
a
to
k
).
To examine the relationship between mean fertility,
mean adult survival, and elasticity patterns, we cal-
culated weighted means of age-specific adult survival
P
i
and age-specific fertility
F
i
.
We weighted the means
according to the probability of survival to age
i,
so that
k
2a
(
l
)
O
a1
i
lP
1
···
1
lP
i
5
1
aa
k
2
1k
2
1
¯
P
55
(8)
k
2a2
1
l
1
···
1
l
a
k
2
1
(
l
)
O
a1
i
i
5
0
and
k
2a
(
lm
)
O
a1
i
a1
i
lF
1
···
1
lF
i
5
0
aa
kk
¯
F
55
l
. (9)
1k
2a
l
1
···
1
l
a
k
2
1
(
l
)
O
a1
i
i
5
0
We plotted the summed elasticities from each matrix
on a three-way proportional graph after Silvertown et
al. (1993), where maximum elasticities for fertility, ju-
venile survival, and adult survival occur at the apexes
of the triangle. We grouped the species broadly: ‘‘ro-
dents’’ (including rabbits), ‘‘carnivores’’ (including
bats), ‘‘marine’’ (whales, seals, sea lions, and manatee,
all restricted to at most one offspring per year), ‘‘graz-
ers’’ (ungulates, zebra, hippopotamus, and elephant),
and primates. We also plotted the proportional contri-
butions of fertility, juvenile survival, and adult survival
to population growth, in order to show which species
have similar elasticity patterns and how the patterns
are affected by generation time. Although a correlation
analysis of life table results and all of the summed
elasticities was inappropriate, due to the proportional
nature of elasticities (Shea et al. 1994), we calculated
Spearman rank correlation coefficients for adult sur-
vival elasticity and life table results to show how these
results are related. To compare relationships within vs.
between mammalian orders, we also calculated the cor-
relation coefficients for life tables of orders Artiodac-
tyla, Rodentia, and Carnivora. Life table calculations
and matrix analyses were performed using Mathcad
software (MathSoft, Cambridge, Massachusetts, USA).
March 2000 657
ELASTICITY ANALYSIS IN POPULATION BIOLOGY
Special Feature
Spearman rank correlations were calculated according
to Zar (1984).
We note the following useful property of elasticities.
The first row of an age-classified projection matrix con-
tains the fertilities
F
i
,
which adopt the value of zero
for
i
,a
, where
a
is the age at first reproduction. In
the characteristic equation, the survival probabilities
for age classes 1 through
a2
1 appear only as the
product
P
1
P
2
...
P
a2
1
. A proportional change in any
of these probabilities has the same effect on this prod-
uct. Thus, the elasticities
e
,
e
,...,
e
are identical.
PP P
12
a2
1
Because the sum of any row in an elasticity matrix E
equals the sum of the corresponding column (de Kroon
et al. 1986), the sum of the fertility elasticities must
equal
e
, and, by extension,
e
,...,
e
.If
a5
1,
PPP
12
a2
1
there are no prebreeding age classes in the projection
matrix (recall that we are constructing matrices with a
prebreeding census), hence no juvenile survival elas-
ticity. When
a.
1, the relationship between the
summed juvenile survival elasticity and the summed
fertility elasticity is
a2
1k
e
5
(
a2
1)
e
. (10)
OO
PF
ij
i
5
1j
5
1
Elasticity patterns of a simplified demographic model
Because a complete age-classified life table requires
a large amount of data, we consider a simplified model,
in which adult survival is assumed to have the age-
invariant value
P
¯,
and annual fertility is assumed to
have the age-invariant value
F
¯
. Such a model is a good
approximation for long-lived organisms with little se-
nescence, but it can be applied to any species. In at
least some cases where detailed comparisons have been
possible, simplified models have done a good job of
capturing the essentials of full age-classified models
(Brault and Caswell 1993, Levin et al. 1996). The tran-
sition matrix is given by the following:
¯
0 0 ··· 0
F
P
00 0 0
1
B
5
0
P
00
_
(11)
2
00
5
00
¯
000
PP
a2
1
where
P
¯
is the weighted mean of adult annual survival
(Eq. 8). In the prebreeding census models used in this
analysis, survival to age 1 is included in the weighted
mean fertility term
F
¯
(Eq. 9) (Caswell 1989). Adults
that survive remain in the final stage indefinitely. The
characteristic equation for this matrix is given by
a2
1
2
(
a2
1)
¯
PF
l
P
j
12
j
5
1
1
5
. (12)
¯
l2
P
Again, the juvenile survival probabilities appear only
as their product; thus, all prereproductive survival elas-
ticities are equal, as well as being equal to the elasticity
of
l
to changes in
F
¯
. Because the elements of the elas-
ticity matrix sum to unity, the elasticity of
l
to changes
in mean adult survival for matrix Bis
e
5
(1
2a
e
).
¯¯
PF
(13)
This relation also follows directly from loop analysis,
as Bis composed of two loops (the birth-to-maturity
loop and the adult stasis loop), whose elasticities must
sum to unity (van Groenendael et al. 1994). Thus, the
elasticity matrix of a prebreeding census model with a
single adult stage looks like this:
0000
e
¯
F
e
0000
¯
F
E
5
0
e
0 0 0 (14)
¯
BF
_
0
5
0
_
000
ee
¯¯
FP
The eigenvectors of Blead to a simple equation for
e
F
¯
(see Appendix B):
¯
P
2l
e
5
. (15)
¯
F
¯
(
a2
1)
P
2al
Because adult survival is age independent, cohorts
persist indefinitely in this model. If adult survival rate
is high, it can be many years before a cohort becomes
so small that it has no influence on population growth.
To partially compensate for this, we discounted
P
¯
for
maximum life span using an equation given by Caswell
(1989) for calculating the probability of leaving a stage
given that it has a fixed duration:
TT
2
1
ss
2
12 12
ll
g5
(16)
T
s2
1
12
l
where
s
is the annual survival, and
T
is the stage length
in years. We used
s5
P
¯
and
T
5
(
k
2a1
1) to
calculate
g
, from which we computed an adult survival
rate:
ˆ¯
P
5
P
(1
2g
). (17)
When
P
¯
is replaced with
P
ˆ
, cohorts still persist indef-
initely, but survival is reduced to mimic the effect of
senescence.
Using Eqs. 13 and 15, it is possible to calculate the
elasticities of
l
to changes in fertility, juvenile survival,
and adult survival knowing only
P
¯
(or
P
ˆ
),
a
, and
l
.
Juvenile annual survival rates, while necessary to com-
plete matrix Bitself, are not needed to determine the
elasticities of B, if
l
can be estimated by other means.
In the absence of detailed demographic information, a
crude estimate of the elasticities can be obtained by
setting
l
to a value based on observed long-term pop-
Special Feature
658
SELINA S. HEPPELL ET AL.
Ecology, Vol. 81, No. 3
F
IG
. 1. Three-way proportional graph of the survival
elasticities calculated for 50 mammal life tables. Species are
sorted generally into five groups: ‘‘marine’’ (whales,
pinnipeds, manatee), ‘‘primates,’’ ‘‘carnivores’’ (including
bats), ‘‘rodents’’ (including rabbits), and ‘‘ungulates’’
(including elephant and hippopotamus). Because the three
elasticity values sum to unity, the position of each point on
the plot gives the relative contribution of survival in each
stage to population growth. Points close to a corner have high
elasticities for that life stage. Points along the left-hand side
of the triangle are species that mature in a single year and
do not have a juvenile stage.
ulation trends (assuming stable age distribution, Crouse
et al. 1987), or to a default value of 1.0, corresponding
to a stationary population. These estimates can be used
to explore how age at maturity, adult survival, and
population growth rate affect elasticity patterns in
poorly known species.
Comparing matrix elasticities with elasticity
approximations
Because model Breplaces age-specific survival and
fertility with their mean values, we compared the elas-
ticities from the matrices Aand B, which were cal-
culated from the same data, using linear correlation
(Excel 7.0 [Microsoft, Redmond, Washington, USA]).
Because 22 of the life tables did not include a juvenile
stage (age at maturity
5
1 yr) the correlation analysis
for juvenile survival elasticity was restricted to the re-
maining 28 populations.
R
ESULTS
Mammalian elasticity patterns
Plotted together on a three-way proportional graph,
elasticity patterns cluster according to age at matura-
tion (Fig. 1). The distance from a point to a corner of
the triangle represents the inverse of its value; for ex-
ample, points close to the adult survival corner have
high adult elasticities. The line of points along the left
margin of the triangle has a juvenile survival elasticity
of zero, because those points represent populations that
mature at age 1 and, therefore, do not have a juvenile
stage. The vertical line of points in the middle of the
triangle are populations that mature at age 2, where
fertility and age-1 survival elasticities are equal (Eq.
10). The remaining points are from populations that
mature in
$
3 yr. Marine mammals and primates cluster
along the right side of the triangle; with late maturation
and long life spans, these species have low fertility
elasticities. Two ‘‘grazers’’ also fall in this group: the
elephant and hippopotamus. All of the ungulates in the
sample mature by age 2, but generally have higher adult
survival rates and adult survival elasticities than ‘‘car-
nivores’’ or ‘‘rodents.’’ The larger grazers, marine
mammals and primates, all have adult survival elastic-
ities near or in excess of 10
3
the fertility elasticity.
When the elasticities for each population are plotted
according to the three age-at-maturation groups (1, 2,
and 3
1
), the patterns become more discernable (Fig.
2). There is a general decrease in fertility elasticity with
increasing generation time (
T
c
, Eq.2). Adult survival
elasticity increases with generation time within the
a
5
1 and
a5
2 groups (Fig. 2A and B, respectively).
Juvenile survival elasticity is more variable, but also
increases with age at maturity (Fig. 2C). This is in part
due to the fact that, for a given adult survival rate, an
increase in age at first reproduction results in a pro-
portionately larger juvenile stage. Note that the differ-
ences in elasticity patterns among the mammals that
were sampled are also related to body size, with later
age at maturation and smaller litter sizes in larger an-
imals. Mammals with similar elasticity patterns are of-
ten not related; reindeer and yellow-bellied marmots
have different vital rates (see Appendix A), but nearly
identical summed elasticities (Fig. 2b).
To explore the relationship between summed elas-
ticities and life history properties, we calculated rank
correlations between adult survival elasticity and the
life table statistics given in Appendix A. These results
are displayed in Table 1. Recall that juvenile survival
elasticity is fertility elasticity multiplied by (
a2
1)
(Eq. 10); thus, fertility elasticity correlates strongly
with age at maturity and all life table results that are
dependent on age at maturity, while correlations with
the summed prereproductive elasticities (fertility
1
ju-
venile survival) are simply opposite those shown for
adult survival elasticity. For all mammal life tables
combined, adult survival elasticity was negatively cor-
related with fertility, and positively correlated withsur-
vivorship to maturity, as well as with variables asso-
ciated with adult survival and life span. Surprisingly,
adult survival elasticity was not significantly correlated
with generation time. A plot of adult survival elasticity
vs. generation time reveals a hump-shaped correlation
where populations with very long generation times
have lower adult survival elasticities (Fig. 3). Within
the rodents, which generally have short generation
times, generation time is positively correlated with
adult survival elasticity. Some other within-order cor-
relations were stronger than those calculated for all life
tables combined, but most were insignificant due to
March 2000 659
ELASTICITY ANALYSIS IN POPULATION BIOLOGY
Special Feature
F
IG
. 2. Area plots showing the stage-specific elasticities
for each mammal population, grouped by age at maturity and
ordered by increasing generation time. (A) Age at first
maturity
5
1 yr; no juvenile stage. (B) Age at first maturity
5
2 yr; fertility elasticity
5
juvenile survival elasticity (see
Methods
). (C) Age at first maturity
.
2yr.
small sample size (Table 1). Within the Artiodactyla,
there was a strong negative correlation between age at
maturity and adult survival elasticity, which was nei-
ther a significant relationship for the other orders nor
the combined data set. In addition, the signs of several
of the correlations within this order are reversed, al-
though the relationships are not significant. This is in
large part due to the hippopotamus life table, which is
quite different from the other Artiodactyla (see Figs.
1 and 2C). There was no correlation between adult
survival elasticity and net reproductive rate
R
0
or
l
for
any of the three orders, nor for the entire data set com-
bined.
Elasticity patterns generated by a simplified model
The results of the mammal life table analysis suggest
that elasticity patterns, when summed across age clas-
ses, depend on adult survival and other demographic
characteristics that could be measured or estimated
from limited field data. To explore this, we examined
the elasticity patterns generated by a simplified model
without adult age structure (matrix B, Eq. 11). The
elasticities of this matrix are determined by age at ma-
turity, adult annual survival, and
l
(Eqs. 13 and 15;
see also Appendix B).
Fig. 4 shows the elasticity patterns generated by Eqs.
13 and 15 for a complete range of adult survival rates,
and four different ages at maturity, in populations that
are declining (
l5
0.8), stable (
l5
1.0), or increasing
(
l5
1.2). For the declining populations, the plots are
truncated at
P
¯
5
0.8, as a population cannot decline
faster than the survival rate of adults, even if there is
no recruitment to the adult population (
P
¯
#l
). These
results elucidate the source of the patterns from the life
table analysis. For example, fertility elasticity is neg-
atively correlated with generation time (Fig. 2). Long
generation times may be due to late age at maturity,
high adult survival (which increases the mean age of
mothers and, hence,
T
c
; Eq. 2), or both (see Appendix
1). Fig. 4 shows that populations that mature later have
low fertility elasticities, and fertility elasticity decreas-
es with increasing adult survival (moving to the right
of each plot). As
P
¯
approaches
l
, the elasticity of
l
to
changes in adult survival increases rapidly. Changes in
juvenile survival or fertility have little or no effect on
l
in these cases, because these populations are in ‘‘sta-
sis,’’ and adult survival completely determines popu-
lation growth. Thus, long-lived species at low popu-
lation growth rates have high adult survival elasticities
and low fertility elasticities. Summed juvenile survival
elasticities are not predictable from generation time for
populations with age at maturity
.
2 yr (Fig. 2C). For
a given adult survival rate, the relative contribution of
juvenile survival increases as
a
increases.
P
¯
cannot be
greater than unity, so adult survival elasticity is re-
duced, relative to juvenile survival elasticity, as ju-
venile stage length increases. Increasing
l
for a given
age at maturity also increases the relative contribution
of juvenile survival, which has a greater contribution
over all adult survival rates when
l.
1 than when
l
,
1. Over these ranges of parameters, elasticity pat-
terns are more sensitive to adult survival and
a
than
to
l
.
We compared the elasticities estimated from the sim-
plified model (B) with those from the age-classified
mammal projection matrices (A). Parameters for B
were estimated as follows:
a
, earliest age class with a
fertility estimate;
F
¯
, the mean annual fertility rate (Eq.
9);
P
¯
, the mean adult survival rate (Eq. 8), and
l
, ob-
tained from the life table matrix. Fertility elasticities
from the two models were highly correlated (
r
5
Special Feature
660
SELINA S. HEPPELL ET AL.
Ecology, Vol. 81, No. 3
T
ABLE
1. Spearman rank coefficients (
r
s
) for several life table results and adult survival elasticity.
Variable All life tables
(
N
5
50) Artiodactyla
(
N
5
10) Carnivora
(
N
5
10)
Rodentia
(
N
5
16)
[Sciuridae
(
N
5
13)]
Age at earliest maturity (
a
)
Maximum age (
k
)
Fertility (
F,
weighted mean; includes
survival to age 1)
Adult annual survival (
P,
weighted
mean)
l
2
0.192
0.317*
2
0.375**
0.300*
2
0.091
2
0.811**
2
0.352
2
0.067
0.152
2
0.030
0.138
0.480
2
0.709*
0.685*
2
0.394
0.085
[
2
0.167]
0.619*
[0.461]
0.099
[
2
0.324]
0.482
[0.338]
0.471
[0.209]
R
0
Generation time (
T
c
)
Survivorship to maturity
Life expectation at birth (
e˚
0
)
Life expectation at maturity (
e˚
a
)
2
0.056
0.252
0.310*
0.258
0.313*
0.006
2
0.042
0.012
2
0.248
0.115
2
0.455
0.479
2
0.097
0.479
0.624*
0.413
[0.118]
0.519*
[0.338]
0.393
[0.066]
0.309
[0.157]
0.469*
[0.439]
Note:
Coefficients have been corrected for ties (Zar 1984).
*
P
,
0.05, **
P
,
0.01.
F
IG
. 3. Adult survival elasticity vs. generation time for
three mammalian orders examined in the correlation analysis
(Table 1): Artiodactyla, Carnivora, and Rodentia. Order
Cetacea, whose four life tables all have very long generation
times, is shown for comparison. Key to numbers inside
symbols: 1
5
Family Sciuridae, 2
5
Family Bovidae; 3
5
Family Cervidae.
0.958). However, juvenile and adult elasticities were
not as well correlated (
r
5
0.697 and 0.722, respec-
tively;
P
,
0.001). One likely reason for the scatter is
that the two models yield different values of
l
. After
discounting adult survival by maximum age (Eqs. 16
and 17), which reduced
l
, the correlation coefficients
increased to
r
5
0.967 for fertility elasticities,
r
5
0.93
for juvenile survival elasticities, and
r
5
0.857 for adult
survival elasticities. When we calculated estimated
elasticities using
P
ˆ
and
l5
1.0, the correlations be-
tween the two model types were again very high (
r
5
0.958, 0.792, and 0.815 for fertility, juvenile survival,
and adult survival elasticities, respectively). The re-
siduals for each correlation showed that Eq. 15 tended
to slightly overestimate adult survival elasticity (hence
underestimating prereproductive elasticity), as age at
first reproduction increased. For most populations in
this study, though, combining adult age classes into an
adult stage did not dramatically affect the elasticity
pattern; thus, Eq. 15 can be used to predict the summed
elasticities of
l
to changes in fertility, juvenile survival
and adult survival.
D
ISCUSSION
How elasticity patterns relate to life history
characteristics in mammals
Because survival, growth, and fertility rates deter-
mine elasticities, relationships exist between elastici-
ties and basic life history parameters. Both fertility and
juvenile survival elasticities are strongly correlated
with age at maturation, mean fertility, generation time,
and life expectancy. Fertility contributes more to
growth rates of populations with early age at maturation
and short generation times. Populations with high mean
adult survival rates have low fertility elasticities and
higher adult survival elasticities, with juvenile survival
elasticity dependent on the proportion of life that is
spent as a juvenile. Although the parameters in the life
tables of this study are variable within families and
even genera, the qualitative nature of fertility elastic-
ities are predictable from age at maturation and gen-
eration time alone (Figs. 2 and 4). These are the same
variables that have been shown to predict life history
patterns, with age at maturation highly correlated with
adult survival (Millar and Zammuto 1983, Read and
Harvey 1989, Promislow and Harvey 1990).
Further connections can be drawn between demo-
March 2000 661
ELASTICITY ANALYSIS IN POPULATION BIOLOGY
Special Feature
F
IG
. 4. Results of Eqs. 13 and 15, showing the effects of age at maturity (rows),
l
(columns), and adult annual survival
rate (
x
-axes), on summed survival elasticities (
y
-axes). Each area plot is for a range of elasticities with increasing adult
annual survival for a given age at maturity and annual survival rate. In the first column, populations are declining, in the
middle column populations are stable, and in the right column populations are increasing. The maximum adult survival rate
is 0.8 for models with
l5
0.8, as
l
cannot be less than adult survival in models of type B(Eq. 11).
graphic elasticities and the life history patterns ob-
served in mammals. In long-lived species that mature
late and have few offspring (Read and Harvey’s ‘‘slow’’
mammals (1989)), fecundity and early offspring sur-
vival are less critical than juvenile survival to maturity.
Thus, increasing juvenile survival (quality), through
large offspring size at birth, small litter size, and pa-
rental care, has a greater effect on fitness than does
increasing litter size. In contrast, mammals that mature
early and have shorter life spans (‘‘fast’’ mammals)
have much higher fertility elasticities; individuals with
these life history traits will benefit more from an in-
crease in offspring number (quantity). A similar ar-
gument has been made for birds (Sæther et al. 1996,
Sæther and Bakke 2000).
Phylogenetic constraints have confounded some
comparative studies of mammal life history traits, and
numerous techniques have been developed to remove
these effects (Harvey and Pagel 1991). Results from
various analyses have been conflicting, primarily due
to different comparative methodologies. Sæther and
Bakke (2000) found that phylogenetic corrections did
not alter their correlations between elasticities and life
history traits of birds. We found that elasticity patterns
for mammals were dependent on generation time and
its components, age at maturity and adult annual sur-
Special Feature
662
SELINA S. HEPPELL ET AL.
Ecology, Vol. 81, No. 3
vival. Elasticity patterns must be loosely dependent on
phylogeny, to the extent that these variables are cor-
related with body size and constrained by phylogeny.
Although the elasticity patterns of the broad taxonomic
groups we defined cluster somewhat in Fig. 1, the rel-
ative impacts of perturbations to fecundity, juvenile
survival, and adult survival do not appear to be gen-
eralizable by these groups. This makes sense, because
elasticities are determined directly by the survival and
fertility rates, which may vary dramatically even
among populations subjected to different environmen-
tal conditions. The general pattern of decreasing fer-
tility elasticity with generation time that we observe
for all early maturing species (Fig. 2) also occurred
within the ground squirrels (
Spermophilus
spp.). How-
ever, ground squirrels with short generation times, such
as
S. armatus,
have summed elasticities that are more
similar to red fox and black-footed ferret than conge-
neric ground squirrels with delayed maturity. Thus,
phylogeny is often not a reliable indicator of which
vital rates will have the greatest impact on population
growth. Future analyses of elasticity patterns could uti-
lize new techniques for statistically evaluating com-
positional data (e.g., Aitchison 1986, Billheimer et al.
1998) with a larger data set, perhaps one with elastic-
ities generated by Eq. 15.
Elasticity patterns of age-based models
In age-classified matrices, the proportional contri-
butions of each of the prereproductive age classes, as
well as the sum of the fertility elasticities, are equal.
If the adult age classes are combined into a single stage,
with mean annual survival and fertility (matrix B, Eq.
11), the elasticity of
l
to fertility, juvenile survival,
and adult survival can be estimated algebraically. This
generalization only applies to age-based models where
age-specific annual survival and fertility rates are the
only model parameters, as in life tables. However, be-
cause most mammals and birds have age-dependent
vital rates, and many other vertebrates have good age–
length relationships, these predictions of summed age-
class elasticities should hold qualitatively for a wide
range of taxa.
An important part of understanding the limitations
of a method is to know how various parameters will
affect the results. While one of our main objectives is
to seek patterns in elasticities across species for com-
parative analysis, this also serves to help us predict
how the elasticities will change if our parameter esti-
mates change or are incorrect (Caswell 1996). We did
not carefully screen our sample of published life tables
for reliability or consistency of methods. Unreliable
life table parameters may lead to faulty interpretations
of population growth rates and, potentially, elasticity
values (Wisdom et al. 2000). Populations that have
undergone recent changes in vital rates may not be
appropriate for certain life table calculations that de-
pend on a stable age distribution (Caughley 1966). Fur-
thermore, they may show different responses to per-
turbation than are predicted by the elasticities. Our
analysis shows that incorrect adult survival rates or age
at first reproduction may have a substantial effect on
elasticity patterns, but faulty fertility or juvenile sur-
vival rates may be less critical, because the relative
values of summed elasticities change less dramatically
with changes in
l
(Fig. 4). Shifts in elasticity patterns
may be predictable for some changes in vital rates that
result in changes in
l
, such as density-dependent re-
ductions in annual juvenile survival (Grant and Benton
2000).
Applications to conservation biology
For conservation purposes, our analysis of mammal
elasticity patterns suggests that the population growth
rates of ‘‘fast’’ mammals that mature early willrespond
to improved survival of offspring, while the growth
rates of ‘‘slow’’ mammals that mature late and have
few offspring per year will respond better to improved
adult or juvenile survival rates. A similar result has
been found for bird species (Sæther and Bakke, 2000).
In populations with very late age at maturity, or rela-
tively low adult survival, the elasticity of
l
to changes
in juvenile survival may become primary (Fig. 4; Hep-
pell 1998). The decision to concentrate management
efforts on a particular life stage should be influenced
by variables such as age at maturation and mean annual
adult survival, rather than taxonomic relationships, and
will depend on the level of potential increase in sur-
vival rates (Green and Hirons 1991, de Kroon et al.
2000).
The elasticity approximation equation (Eq. 15) pro-
vides a shortcut for life history analysis that may be
very useful for categorizing species for management.
Full life table data are notoriously difficult to collect,
but estimates of age at maturity and average adult sur-
vival may be more accessible. These parameters should
be priorities in initial research efforts. Estimates of
l
may be difficult to obtain (only
;
20% of the Habitat
Conservation Plans reviewed in a recent study (Savage
1998) contained information about population trends)
but Fig. 4 shows that
l
has less effect on predicted
elasticity patterns than age at maturity or adult survival.
Elasticity approximations have at least two important
applications: (1) to make preliminary management pro-
posals that account for life history characteristics of
data-poor populations; and (2) to categorize species or
populations according to their elasticity patterns, as a
first step in various modeling efforts, such as choosing
a ‘‘model’’ species for simulations. The elasticity pat-
terns generated by Eq. 15 cannot substitute for detailed,
long-term studies of wild populations. However, the
patterns are relatively robust and permit qualitative
comparisons of management alternatives.
Conservation biologists should apply elasticity anal-
ysis cautiously. Uncritical prioritization through elas-
ticities for various life stages can result in poor pre-
March 2000 663
ELASTICITY ANALYSIS IN POPULATION BIOLOGY
Special Feature
scriptions for research or conservation efforts (Sæther
et al. 1996, Silvertown et al. 1996, Wisdom et al. 2000).
In particular, interventions targeted at one life stage
may affect other vital rates in surprising ways. Unless
those effects can be included in the calculations, chang-
es in
l
cannot be predicted. Perhaps most importantly,
elasticities give us information about population pro-
jections, given that current conditions (vital rates or
prescribed changes in vital rates) are maintained, lead-
ing to a stable age distribution. Although it may be
tempting to use the quantitative differences in elasticity
values to rank management options, the assumptions
and restrictions of deterministic elasticity analysis
make qualitative comparisons much more prudent (de
Kroon et al. 2000). Elasticity of
l
often does an ex-
cellent job of qualitatively predicting the elasticities of
other indices appropriate for models that include den-
sity dependence, environmental or demographic sto-
chasticity, or spatial structure (Caswell 2000, Grant and
Benton 2000, Neubert and Caswell 2000). In the best
of all worlds, elasticity analysis would be a first step
in a larger framework of population viability analysis
that included stochastic simulations, multiple models,
and detailed sensitivity analysis (Ferrie´re et al. 1996,
Beissinger and Westphal 1998).
A
CKNOWLEDGMENTS
We would like to acknowledge R. Powell, Jim Gilliam,A.
Read, and W. Morris, who provided insightful comments on
this work. C. Pfister, H. de Kroon, N. Schumaker, and two
anonymous reviewers offered many useful suggestions for
the manuscript. This work was part of a dissertation (S. Hep-
pell) at Duke University and was supported by a grant from
the National Marine Fisheries Service and the University of
North Carolina Sea Grant (R/MER-21 and NOAA/NMFS
NA90AA-D-S6847), as well as a postdoctoral appointment
from the U.S. Environmental Protection Agency, National
Health and Environmental Effects Research Laboratory. H.
Caswell acknowledges support from National Science Foun-
dation Grant DEB-9582740, Woods Hole Oceanographic In-
stitute Contribution 9931.
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APPENDIX A
Detailed life tables, including sources, parameters, and results for 50 mammal populations, may be found inESA’s Electronic
Data Archive:
Ecological Archives
E081-006.
