heppell et al. 2000 elasticity analysis

7/21/2019 Heppell Et Al. 2000 Elasticity Analysis

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Life Histories and Elasticity Patterns: Perturbation Analysis for Species with

Minimal Demographic Data

Selina S. Heppell; Hal Caswell; Larry B. Crowder

Ecology, Vol. 81, No. 3. (Mar., 2000), pp. 654-665.

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2/13

Ecolog?, 81(3).

2000,

pp . 654-665

2000 by the Ecological Society of America

LIFE HISTORIES AND ELASTICITY PATTERNS: PERTURBATION

ANALYSIS FOR SPECIES WITH MINIMAL DEMOGRAPHIC DATA

S E L I N A . H E P P E L L , ' ,~

AND

L A R R Y . C R O W D E R ~A L C A S W E L L , ~

'Dep artm ent of Zoology, Duke University Marine Lab, 1 35 Duke Marine Lab Ro ad,

Beallfort, North Carolina 28516 USA

ZBiology Departm ent, W oods Hole Oceanog raphic Institute, W ood s Hole, Massachusetts 02 543 U SA

WNicholas School of the Environ men t, Duke Un iversity Marine Lab, 13 5 Dctke Marine Lab Roa d,

Bea ~lfor t , or th Carolina 28516 USA

Abstract

Elast ici ty analysis is a useful tool in conservat ion biology. The relat ive

impacts of proport ional c hanges in fert i l i ty , juveni le survival , an d adul t survival on as-

ymptot ic popu lat ion gro wth h (wh ere ln(A) r the intrinsic rate of increase) are determine d

by vi tal rates (survival , growth, and fert i l i ty), which also define the l i fe his tory character-

i s t ics o f a species o r popula t ion . Be cause we d o no t have good demog raphic in format ion

for most threatened populat ions, i t i s useful to categorize species according to their l i fe

his tory characteris t ics and related elast ici ty pat terns. To do this , we compared the elast ici ty

pat t erns genera ted by the l i fe t ab les o f 5 0 mam mal popula t ions . In age-c lass if i ed model s ,

the su m of the fert i l i ty elast ici t ies and the sur viva l elast ici ty for each juveni le age-class

are equal ; thus, age at maturi ty has a large impact on the contribut ion of juveni le survival

to h . Mam mals tha t mature ear ly and have l arge l i tt e rs ( fas t mammals , such as rodent s

and sma ller carnivores) also general ly have short l i fespans; these populat ions had relat ively

high fert il i ty elast ici t ies and low er adul t surv ival elast ici t ies . Slow ma mm als (those that

mature l a te) , hav ing few offspr ing and h igher adu l t surv ival ra tes (such as ungula tes and

mar ine m ammals ) , had much lower fer t i li ty e las ti c i ti es and h igh adul t o r juveni le surv ival

elast ici t ies . Although certain l i fe his tory characteris t ics are phylogenet ical ly constrained,

we fou nd that elast ici ty pat terns within an order or family can be qu i te diverse, while s imilar

elast ici ty pat terns can occur in dis tant ly related taxa.

We extended our general izat ions by developing a s imple age-classified m ode l para m-

eterized by juveni le survival , mean adul t survival , age at maturi ty, and mean ann ual fert i l ity .

Th e elast ici ty pat terns of this model are determined by ag e at maturi ty, mean a dul t survival ,

and h , and they compare favorab ly wi th the summed e las t i c i t i es o f fu l l Les l i e mat r i ces .

Thus, elast ici ty pat terns can be predicted, even when complete l i fe table information is

unavai lable. In addi t ion to classifying species for management purposes, the resul ts gen-

erated by this s implified model show how elast ici ty pat terns may change i f the vi tal rate

information is uncertain. Elast ici ty analysis can be a qual i tat ive guide for research and

management , part icularly for poorly known species , and a useful f i rs t s tep in a larger

model ing effort to determine populat ion viabi l i ty .

Key wo rds: age-b ased nzodel; cons ervatio n; elu:i.ticity ana lysis; life lzistory; life table; murnm al,

management; matrix model; population nlodel.

I N T R O D U C T I O N

venile surviva l , and adul t survival ; w hile, in plants , the

Elast ici ty analysis

can be

used to

categorize

p o p u -

categor ies a re o f ten growth , reproduct ion , and s tas is .

lations

accord ing

to

the response

of

populat ion growth

Elast ici ty pat terns are s imilar for populat ions or species

h to perturbat ions that affect vi tal rates . An elast ici ty

that share l i fe his tory characteris t ics . Comparisons

pattern

is

composed of

the relat ive contribut ions

of

have been made across t axa for p lan t s (Si lver town e t

mat r ix en t r i es to popula t ion growth that are g rouped

al . 19 93, 19 96), birds (Ssether et al . 19 96, Saether and

in b io log ically m eaningfu l ways for comparat ive anal -

Bakke 2000) , and tu r t l es (Cunning ton and Brooks

ys is . For exam ple , in an imal popula tions , we m ay w ant

1996 , Heppell 1998) . These e las t ic i ty pat terns may pro-

to compare the relat ive contribut ions of fert i l i ty , ju-

vide general rules of thumb for categorizing popula-

t ions according to their relat ive responses to pertur-

Manuscript received 9 October 1998; revised 26 May 1999; bations

part icular l i fe s tages. genera1 pat-

accepted 29 Map 1999. For reprints of this Special Feature, see

terns exis t for certain taxa, such as long-l ived fresh-

footnote 1. p 605.

water turt les , there is often considerable variat ion in

~ r e s e i t d d r es s : U . S . E n v i r o n m e n ta l P r o te c ti o n A g e n -

the elast ici ty pat terns of closely-related species , and

cy , Na ti on j il Hea l t h and E nvi ron men t a l E f f ec t s Rese a rch

even populat ions within a species , due to habi tat char-

L a b o r a t o r y ( N H E E R L ) , W e s t e r n

E c o l o g y D i v i s i o n . 200

S W 3 5 t h S t r e e t . C o r v a l l is . O r e e o n 9 7 3 3 3 U S A .

acteris t ics or dis turbance regim es that affect vi tal rates

E - m a i l : h e p p e l l @ m a i l . c o r . e p a . g o v

(Si lvertown e t al . 1996, Oostermeijer et al . 199 6). Be-
mailto:[email protected]:[email protected]


3/13

655

arch

2000

ELASTICITY ANALYSIS IN POPULATION BIOLOGY

cause of variance and uncertainty in vi tal rates , i t i s

important to know w hat comb inat ions of vi tal rates and

populat ion growth rates give rise to part icular elast ici ty

pat terns.

There is a pract ical need for a classificat ion of pop-

ulat ions according to their l ikely response to pertur-

bat ion . Conservat ion m anagement p lans cannot be as -

sessed for mos t species , because demographic data to

construct detai led age- o r s tage-specific mo dels are un-

avai lable. Obtaining complete est imates of vi tal rates ,

including their temporal variances and covariances, is

t ime consuming for any species and may be imposs ib le

for those tha t have long l i fe spans or tha t cover wide

geographic ranges , such as mar ine t axa (Boyce 1992 ,

Caughley 1994 , Heppel l and Crow der 1998) . Green and

Hirons (1991) suggested that


4/13

656

SELINA

S.

HEPPELL ET AL.

Ecology, Vol.

81,

No. 3

pendix A). Order and family classificat ions are l is ted

accord ing to Wi l son and Reeder (1993) .

In

32 cases ,

the l i fe tables included age-specific reproduct ive rates

( female newborns per female per year) , which gener-

al ly varied according to proport ion of females who

were mature in a given age class . In the remaining l i fe

tables , survivorship and age at f i rs t breeding were the

only age-specific data avai lable, so the mean fert i l i ty

provided by the author(s) was prescribed to al l mature

age classes . Life tables that were constructed on half-

year intervals were rounded to the next integer year.

We calculated several s tandard l i fe his tory measures

for each l i fe table, including the net reproduct ive rate

for the populat ion

R

mean genera t ion t ime T sur-

vivorship to first maturity

l

and l i fe expectat ion at

(expectat ionnd l i fe expectat ion at maturi tyirth

of further l i fe; Pianka [1978]):

changes in P, an d F across age classes to obtain the

fol lowing quant i t ies of interest :

0 1. Fert i li ty elast ici ty, which is the effect of a

proport ional c han ge in reproduct ive ou tput for al l adul t

age classes (the sum of elements in the top row of

E .

1

2. Juveni le survival elast ici ty, which is the ef-

fect of a proport ional chan ge in al l annual survival rates

for age 1 to the year just prior to maturat ion (the su m

of the subdiagonal elements of

E

f rom co lumn 1 to

co lumn a 1, where a represents the fi rs t age class

that includes breeding females).

2 3. Adult survival elast ici ty, which is the effect

of a proport ional change in al l annual survival rates

for mature individuals ( the sum of the subdiagonal el -

ements of

E

f rom a to k).

To examine the relat ionship between mean fert i l i ty ,

mean adul t surviv al , and elast ici ty pat terns, we cal-

culated weighted means of age-specific adul t survival

P,

and age-specific fertility

F,.

We weighted the means

according to the probabi l ity of survival to age i, so that

(3)

and

where

1

is the survivorship of a cohort to age i

m

is

the number of female offspring produced annual ly by

a female aged

i

k i s the maximum age, and a is the

age at f i rs t reproduct ion (the fi rs t age class with

nz f

0) .

We then converted each l i fe table to a prebreeding,

birth pulse project ion matrix (Lesl ie matrix [Lesl ie

1945, Caswell 19891) in which the surv ival probabi l i ty

P, and fertility

F,

of age class

i

are given by the fol-

lowing:

F,

thus inc ludes surv ival to age 1. We calculated the

elast ici ty matrices E from the eigenvec tors of each pro-

ject ion matrix

A

(Casw ell et al . 198 4, de Kroon e t al .

1986) :

where

v

and w are the left and right eigenvectors of

the project ion matrix

A,

and w,

v )

is the scalar product

of the two vectors . We summed the elast ici t ies of

h

to

We plot ted the summ ed elast ici t ies from each matrix

on a three-way proport ional graph after Si lvertown et

al . (1993 ), where ma ximum elast ici t ies for ferti l ity , ju-

veni le survival , and adul t survival occur at the apexes

of the t r iangle. We grouped the species broadly: ro-

dents ( including rabbi ts) , carnivores (including

bats) , marine (whales , seals , sea l ions, and ma nate e,

al l rest ricted to at most one offspring per yea r), graz-

ers (ungulates , zebra, hippopotam us, and elephant) ,

and primates . We also plot ted the proport ional contri -

but ions of fert i l i ty , juveni le survival , and adul t surviv al

to populat ion growth, in order to show which species

have s imilar elast ici ty pat terns and how the pat terns

are affected by generat ion t ime. Although a correlat ion

analysis of l i fe table resul ts and al l of the summed

elast ici t ies was inappropriate, due to the proport ional

nature of elast ici t ies (Shea et al . 1994), we calculated

Spearman rank correlat ion coefficients for adul t sur-

vival elast ici ty and l i fe table resul ts to show how the se

resul ts are related. To compare relat ionships w ithin v s .

between m ammal ian orders , we a l so ca lcu la ted the cor-

relat ion coefficients for l i fe tables of orders Art iodac-

tyla, Rodent ia, and Carnivora. Life table calculat ions

and mat r ix analyses were performed us ing Mathcad

sof tware (Maths of t , Cambridge , Massachuset t s, USA).


5/13

657

arch 2 ELASTICITY ANALYSIS IN POPULATION BIOLOGY

Spearman rank correlat ions were calculated according

to Zar (1984) .

We note the fol lowing useful prop erty of elast ici t ies .

The fi rs t row of an age-classified project ion ma trix con-

tains the fertilit ies

F

which adopt the value of zero

for

i 2 yr.

small sam ple s ize (Table 1). Within the Art ioda ctyla,

there was a s t rong negat ive correlat ion between age at

matur i ty and adul t surv ival e las t i c ity , which was nei -

ther a s ignificant relat ionship for the other orders nor

the com bined data set . In addi t ion, the s igns of several

of the correlat ions within this order are reversed, al -

though the relat ionships are not s ignificant . This is in

large par t due to the h ippopotamus l i fe t ab le , which i s

qui te different from the other Art iodactyla (see Figs.

1 and 2C) . There was no corre la t ion between adul t

surv ival e las t ic i ty and net reproduct ive ra te R or A for

any of the three ord ers , nor for the ent i re data set com -

bined.

Fig. 4 shows the e las t ic i ty pat terns genera ted by Eqs .

13 and 15 for a complete range of adu l t surv ival ra tes ,

and four different ages at maturi ty, in populat ions that

are dec lining (A

=

0.8), stable (A

=

1.0), or increasing

(A

=

1.2) . For the decl in ing popula t ions , the p lo t s a re

truncated at P = 0.8 , as a popula t ion cannot decl ine

faster than the surviv al rate of adul ts , even i f there is

no recrui tment to the adul t populat ion P A). These

resul ts elucidate the source of the pat terns from the l i fe

table analysis . For example, fert i l i ty elast ici ty is neg-

at ively correlated with generat ion t ime (Fig. 2). Long

genera t ion t imes may be due to l a te age a t matur i ty ,

h igh adul t surv ival (which increases the mean age of

mothers and , hence , T,; Eq. 2 ) , o r bo th (see Appendix

1). Fig. 4 show s that populat ions that mature later have

low fert i l i ty elast ici t ies , and fert i l i ty elast ici ty decreas-

es with increasing adul t survival (moving to the right

of each plot) . As

P

app roac hes A, the elasticity of A to

changes in adu l t surv ival increases rap id ly . Changes in

juveni le survival or fert i l i ty have l i t t le or no effect on

A in these cases, because these populat ions are in s ta-

s i s, and adul t surv ival complete ly determines popu-

la t ion g rowth . Thus , long- l ived species a t low popu-

lat ion growth rates have h igh adul t surv ival elast ici t ies

and low fer t i l i ty e las t i c it i es . Summ ed juveni le surv ival

elast ici t ies are not predictable fro m generat ion t ime for

popula t ions w i th age a t matur i ty > 2 yr (Fig . 2C) . For

a given adul t survival rate, the relat ive contribut ion of

juveni le surv ival increases as increases. P can n o t b e

greater than uni ty, so adul t survival elast ici ty is re-

duced, relat ive to juveni le survival elast ici ty, as ju-

veni le s tage length increases. Increasing

A

fo r a g iven

age at maturi ty also increases the relat ive contribut ion

of juveni le survival , which has a greater contribut ion

over a l l adu l t surv ival ra tes when A > 1 than when A

< 1 Over these ranges of parameters , elast ici ty pat-

terns are mo re sensi t ive to adul t surviv al and than

to A.

We com pared the elast ici t ies est imated fro m the s im-

pl i f ied model (B) with those from the age-classified

mammal pro jec t ion mat r i ces A). Parameters fo r B

were est imated as fol lows: earl iest age class with a

fert i l i ty est imate; E , the mean annual fert i l i ty rate (Eq.

9); P , the mean adul t surv ival ra te (Eq . 8 ) , and A ob-

tained from the l i fe table matrix. Fert i l i ty elast ici t ies

f rom the two model s were h igh ly corre la ted r

=


8/13


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March 2000 ELASTICITY ANALYSIS IN POPULATION BIOLOGY

dult annual survival rate

Fertility Juvenile dult

FIG.4.

Resul t s of Eqs . 13 and 15, showing the ef fects of age at matur i ty ( rows) ,

h

(columns) , and adul t annual survival

rate (x-axes) , on summed survival e las t ic i t ies (y-axes) . Each area plot i s for a range of e las t ic i t ies wi th increas ing adul t

annual survival fo r a given age at matur i ty and annual s urviv al ra te . In the fi rs t column , populat ions a re decl ining, in the

middle colu mn popu lat ions are s table , and in the r ight column populat ions are increas ing. The maxim um a dul t survival ra te

is

0.8

for models wi th

0.8

as

h

cann ot be less than adul t survival in models of type

B

(Eq. 11 .

graphic elast ici t ies and the l i fe his tory pat terns ob-

served in mammals . In long- l ived species tha t mature

la te and have few offspring (Read and Harvey 's s low

mammals (1989)) , fecundi ty and ear ly o f fspr ing sur-

vival are less cri t ical than juveni le surv ival to maturi ty.

Thus , increas ing juveni le surv ival (qual i ty ) , th rough

large offspring s ize at birth, small l i t ter s ize, and pa-

rental care, has a greater effect on fi tness than does

increasing l i t ter s ize. In contrast , mam mals that mature

ear ly and have shor ter l i fe spans ( fas t mam mals )

have m uch high er fert i l i ty elast ici ties; individuals with

these l i fe his tory t rai ts wil l benefi t more from an in-

crea se in offspring numb er (quant i ty). s imilar ar-

gument has been made for b i rds (Ss ther e t a l . 1996 ,

Sre ther and Bak ke 2000) .

Phylogenet i c cons t ra in t s have confounded some

comparat ive s tudies of mammal l i fe his tory t rai ts , and

numerous t echniques have been developed to remove

these ef fec t s (Harvey and Page1 1991) . Resu l t s f rom

various analyses have been confl ict ing, primari ly due

to d i f feren t comparat ive methodolog ies . Ss ther and

Bakk e (2000) found that phylogenet i c correc t ions d id

not al ter their correlat ions between elast ici t ies and l i fe

his tory t rai ts of birds . We fo und that elast ici ty pat terns

for mammals were dependent on genera t ion t ime and

i t s components , age a t matur i ty and adul t annual sur-


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663arch 2000 ELASTICITY ANALYSIS IN POPULATION BIOLOGY

script ions for research or conservat ion e fforts Saether

e t a l . 1996 , Si lver town e t a l . 1996 , Wisdom et a l . 2000) .

In part icular, intervent ions targeted at one l i fe s tage

may affect other vi tal rates in surpris ing ways. Unless

those effects can be included in the calculat ions, chang-

es in cannot be pred ic ted . Perhaps mos t impor tan tly ,

elast ici t ies give us information about populat ion pro-

ject ions, given that current condi t ions vi tal rates or

prescribed c hanges in vi tal rates) are maintained, lead-

ing to a s table age dis t r ibut ion. Although i t may be

tempting to use the qu ant i tat ive differences in elast ici ty

values to rank management op t ions , the assumpt ions

and restrict ions of determinist ic elast ici ty analysis

make qual i ta t ive compar i sons much more prudent de

Kroon e t a l . 20 00) . E las t ic i ty o f of ten does an ex-

cel lent job of qual i tat ively predict ing the elast ici t ies of

other indices appropriate for models that include den-

si ty dependence, environmental or demographic s to-

chas t ic i ty , o r spat ia l s tructure Casw el l2000 , G ran t and

Benton 2000 , N euber t and C aswel l 2000) . In the bes t

of al l worlds, elast ici ty analysis would be a f i rs t s tep

in a larger framework of populat ion viabi l i ty analysis

that included s tochast ic s imulat ions, mult iple models ,

and detai led sensi t ivi ty analysis FerriCre et al . 1996,

Bei ss inger and Wes tphal 1998) .

We would l ike to acknowledge R. Powel l , J im Gi l l iam, A.

Read, and W. Morr i s , who provided ins ight ful comments on

this work. C. Pf i s ter , H. de Kroon, N. Schumaker , and two

anonymous reviewers of fered many useful suggest ions for

the manuscr ipt . This work was par t of a disser ta t ion S . Hep-

pel l ) a t Duke Univers i ty and was suppor ted by a grant f rom

the Nat ional Mar ine F isher ies Service and the Univers i ty of

Nor th Caro l i na S ea Gran t R IM E R-21 and NOAAINM F S

NA90A A-D-S 684 7) , a s we l l a s a pos t doc t ora l appo i n t ment

f rom the U.S . Envi ronmental Protect ion Agency, Nat ional

Heal th and Envi ronmental Effects Research Laboratory. H.

Caswel l acknow ledges suppor t f rom Nat ional Science Foun-

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PPENDIX

Detai led l i fe tables , including sou rces: parameters , and resul t s for 50 mamm al pop ulat ions , may b e found in ESA's Elect ronic

Data A rchive: Ecological Archives E08 1-00 6.


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March 2000

ELASTICITY ANALYSIS IN POPULATION BIOLOGY

PPENDIX B

In this Append ix we der ive the express ions (E q.13 ) and

With this scal ing, the scalar product i s

(Eq.1 5) for the e las tic i t ies of to changes in the s impl if ied

model given by the mat r ix

B

in (E q.11 ) . Th e elas t ic i ty of

to any element i s given by

( w , v ) = a - 1

-

X

(B.4)

x P

h~~

log

x

b

v,tt:

ab

log h , , X

(w,

v)

Using these results, we find that the elast ici ty of X t o changes

in F is

where w and v are the r ight and lef t e igenvectors and (w, v )

i s their scalar product (d e Kroon et a l . 1986) . These can be

wri t ten di rect ly f rom the l i fe cycle graph corresponding to

the mat r ix, us ing the character i s t ic equat ion to s impl i fy som e

of the express ions . The resul t s are:

1

PIA - I

P l P 2 A - 2

Le t e,

:=

e, , , , , be the su mm ed juveni le survival e las t ic i ty.

(B.2)

Because the e las t ic i t ies of a l l the juveni le surviv al probabi l -

i t ies are equal ,

A1S -l ] and we know that

Because e i

e,,,

this implies that

(B.3)

e j

a e ~ (B.lO)

as given by (Eq.

13 .

Thus, the complete e las t ic i ty pat tern

determined by

a , A,

and P

of the matrix B is

heppell et al. 2000 elasticity analysis

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