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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.