Madan K. OIi and Bertram Zinner Matrix population models have

~
OIKOS 93: 376-387. Copenhagen2001
Madan K. OIi and Bertram Zinner
Oli, M. K. and Zinner, B. 2001. Partial life cycle analysis: a model for pre-breeding
cunsus data. -Oikos 93: 376-387.
Matrix population models have become popular tools in researchareasas diverse as
population dynamics, life history theory, wildlife management, and conservation
biology. Two classesof matrix models are commonly used for demographic analysis
of age-structuredpopulations: age-structured (Leslie) matrix models, which require
age-specificdemographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easierto parameterize becausedata needed tp estimate parameters for thesemodels are collected much
more easily than those needed to estimate age-specificdemographic parameters.
Partial life cycle models also allow evaluation of the sensitivity of population growth
rate to changes in ages at first and last reproduction, which cannot be done with
age-structuredmodels. Timing of censusesrelative to the birth-pulse is an important
consideration in discrete-time population models but most existing partial life cycle
models do not addressthis issue,nor do they allow fractional values of variables such
as agesat first and last reproduction. Here, we fully developa partial life cycle model
appropriate for situations in which demographic data are collected immediately
before the birth-pulse (pre-breedingcensus).Our pre-breeding censuspartial life cycle
model can be fully parameterized with five variables (age at maturity, age at last
reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some
important applications even when age-specificdemographic data are available (e.g.,
perturbation analysisinvolving agesat first and last reproduction). We have extended
the model to allow non-integer values of ages at first and last reproduction, derived
formulae for sensitivity analyses,and presentedmethods for estimating parameters
for our pre-breeding census partial life cycle model. We applied the age-structured
Leslie matrix model and our pre-breeding censuspartial life cycle model to demographic data for several species of mammals. Our results suggest that dynamicalproperties
of the age-structured model are generallyretained in our partial life cycle
model, and that our pre-breedingcensuspartial life cycle model is an excellent proxy
for the age-structured Leslie matrix model.
M. K. Oli, Dept of Wildlife Ecology and Conservation,Univ. of Florida, 303 Newins-Ziegler
Hall, Gainesville,FL 32611-0430,USA ([email protected]).-B. Zinner, Dept
of Discrete and Statistical Sciences,Math Annex, Auburn Univ., Auburn, AL 36849,USA.
Matrix population models have become popular tools
in researchareas as diverse as population dynamics,life
history theory, wildlife management,and conservation
biology (van Groenendael et aI. 1988, Caswell 1989a,
Tuljapurkar and Caswell 1997). Two classesof matrix
models are commonly used for demographic analysis of
age-structured populations: age-structured (Leslie) matrix models, and partial life cycle models (Leslie 1945,
1948, Caswell 1989a,Oli and Zinner 2001). When age-
specific demographic data are available, age-structured
models are the models of choice because they adequately incorporate age-specific differences in demographic parameters. However, age-structured models
are difficult to fully parameterize becausethey require
age-specific demographic data, which are difficult to
collect in the field, and investigators often must rely on
incomplete demographic information. Also, age-structured matrix models do not allow the evaluation of the
Accepted 13 February 2001
Copyright @ OIKOS 2001
ISSN 0030-1299
Printed in Ireland -all rights reserved
376
OIKOS 93:3 (2001)
P.
"'{
sensitivity of population growth rate to changes in
some life history variables, such as ages at first and last
reproduction (Oli and Zinner 2001). Partial life cycle
models, on the other hand, can be parameterized with
partial demographic data (which are collected much
more easily), and also allow the evaluation of the
sensitivity of population growth rate to changes in
demographic variables including ages at first and last
reproduction. Partial life cycle models are particularly
useful in situations where age-specific demographic
data are not available, and when researchersare interested in perturbation analysesinvolving time variables
such as ages at first and last reproduction (Cole 1954,
Caswell 1989a,Levin et al. 1996,Slade et al. 1998, Oli
and Zinner 2001).
An important issue in modelling birth-pulse populations is the timing of the census relative to the birth
pulse. In most studies of vertebrate populations, demographic data are collected either just before (hereafter
pre-breeding census)or just after (hereafterpost-breeding census) the birth-pulse (Caughley 1977, Caswell
1989a, Oli and Zinner 2001). In general, pre-breeding
censusmodels assumethat mortality of adults between
the census and the following birth pulse is negligible,
and the post-breeding censusmodels assumethat mortality of neonates betweenthe birth pulse and the next
census is negligible (Caswell 1989a, Oli and Zinner
2001). If data are collected immediately before the birth
pulse, neonates are nearly one time unit old when they
are counted for the first time. Becausenot all animals
born during a birth-pulse survive to be counted in the
next census,one must consider mortality of neonates
from birth until one time unit of age when estimating
fertility rates. If demographic data are collected immediately after the birth pulse (post-breeding census)such
that neonatal mortality between the birth and the following census is negligible, animals are counted soon
after they are born, and no adjustment in neonatal
mortality is necessaryin estimating fertility rates. Because of these differences, matrix parameters are estimated differently for pre- and post-breeding census
situations (Caswell 1989a). Consequently, partial life
cycle models appropriate for post-breeding censussituations may not be appropriate for the analysis of
demographic data collected from pre-breeding censuses
(Oli and Zinner 2001).
Oli and Zinner (2001)have derived a partial life cycle
model appropriate for post-breeding censussituations.
Fig. 1. An age-structured life
cycle graph. Parametersare:
Fi = age-specificfertility rates,
Pi = age-specificsurvival rates,
cx= age at maturity, co= age at
last reproduction, and
~ = longevity. Multiple
transitions are representeq by
dotted lines.
01'-:08 9J:J (200t)
Here, we fully develop a partial life cycle model appropriate for the analysis of demographic data collected
from pre-breeding censuses.We extend the model to
allow non-integer values of ages at first and last reproduction, derive formulae for estimating the sensitivity
of population growth rate to changesin model parameters, and presentmethods for estimating parameters for
our pre-breeding census partial life cycle model. Using
data from the literature, we show that results of our
partial life cycle model resemble those obtained from
corresponding age-structured matrix models.
The model
We consider a bjrth-pulse population in which births
occur at discrete time t, t = 0, 1, 2, ..., k. Demographic
data are assumedto be collected just before the birth
pulse such that mortality of adult females between the
census and the birth pulse that follows is negligible.
Becausethe census is taken just before birth occurs,
juveniles will not be counted until the next censuswhen
they are almost one time unit old.
We denote age at first reproduction by IX,age at last
reproduction by <0,and maximum longevity by ~. We
define age classesas follows: organismsof age x belong
to ageclassi(i= 1,2, ...) if i-I <x~i. We denote the
fertility of organisms belonging to age class i by Fi.
Then, Fi is zero for i < IXand i > <0,by the definition of
IXand <0.Organisms belonging to age class i survive
with age-specificsurvival probability Pi" A life cycle of
this kind can be graphically representedas an age-structured life cycle graph (Fig. 1). The population projection matrix (sensuCaswell1989a) corresponding to the
age-structured life cycle depicted in Fig. 1 is (dotted
lines representmultiple transitions):
~OO".FF
P.I
0
...F "
o...or
...0
0
...0
0
...0
(
P.2 ...0
0
...0
0
...0
(
0
.A-I
0
0 p..,
A=I
0
0
0
0
0
0
(1)
0
0
0
0
0
0
0 P.
0
0
0
0
0
0 po.'
0
0
0
0
0
0
"'{
0
0
0
~-) oJ
0+J
~IA)
<~):-"Ev-Po
0
j Pa+l
-
~)-:-.~~ (~~:.(:v
PIA)
P(Jj+l
Pp.l
(
Aij=
-
Pj, tor i = 1 and (X:S;j:S; 0)
Pj, for i=j+0, 1 and l:S;j:S;~elsewhere.
BecauseFi = 0 for all i > m, age classesbeyond m make
no contribution to the long-term dynamics of the population, and thus can be ignored in our density-independent setting. The matrix parameters Fi (i= 1,2, ...,m)
and Pi (i=1,2,...,m-l)
can be estimated from the
life table data using the pre-breeding census formulation of Caswell (1989a):
~
0
0~
0
0
0 ~
0
0
P-
0
o
0
0
A(a,ro,~,p',F) = I
(4)
(2)
4
An exampleof the projection matrix corresponding to a
partial life cycle graph with Cl= 2, and ffi = 5 (age
classesbeyond ffi ignored) is:
and
(3)
F.=IJm;,
,
~O
F
jcii'
jcii'
j,ii'
Pj
0
l:)
I)
I)
Ii is the survivorship (probability at birth ofsurviving
0 P a 'J
I)
I)
to age i), II is the probability at birth of A(cx,ro,Pj,Pa,F)=
0
0
]
I)
?a
D
surviving to one time unit of age, and mi is the fecundity (the average number of daughters born to a female
0
l();> ] u ().
l
" 0
of age i). Caswell (1989a)describes in detail the construction and analysis of age-structured matrix models. The characteristic equation for the life cycle of the type
Here, we are concerned with the derivation of a model shown in Fig. 2 and the corresponding projection matrix A(Cl,0>,Pj, Pa' F) is obtained by setting the determithat can be parameterized with partial demographicdata
collected from pre-breeding censuses,and in which nant of the matrix A( Cl,0>,P j, Pa' F) -AI to zero, where
age at first reproduction (cx)and age at last reproduc- A( Cl,0>,Pj' Pa' F) is the projection matrix corresponding
to a pre-breeding censuspartial life cycle graph, and I
tion (ro) appear explicitly as model parameters.
We assume that the age-specificfertilities F", F,,+ I' is the identity matrix. The general characteristic equa..., Fro-I' Fro can be adequately approximated by an tion is:
age-independentaverage fertility parameterF (averaged
J
J
a )..",-cx-1
over age class cx,cx+ I, ..., ro), that age-specificsurvival O=).."'-Fpcx-IJ."'-CX-Fpcx-lp
probabilities until the first birth event Guvenile or pre-FPcx-lp2)..",-cx-2
-Fpcx-lp3).."'-cx~3
J
a
J
a
reproductive survival probabilities) PI, P2, ..., P,,-I
(5)
-FPj-IP:-cx-l)..
-FPj-1 p:-cx.
can be adequately approximated by a juvenile survival
parameter Pj' and that age-specificadult survival rates
Eq. 5 by ,-coand rearranging yields ('- * 0):
P ", P ,,+ I' ..., P ro- I can be adequatelyapproximated by
an adult survival parameter P a. If we replace age-spe-
)
e(7')
p. J
(~)
3
-+
A
Pj
Pa
~t)-:-{v
p
a
2. A pre-breeding censuspartial life cycle graph. Parame-terswhich is equivalent to:
are: F = age-independentaverage fertility rate, Pj = juve-nile
survival rate, Pa = adult survival rate, (X= age at maturity,
(J)= age at last reproduction. Age classesbeyond (J)are ig-nored.
378
~)!)!
F
000
Pj=
where
Dividing
Pa
Fig.
c;ificparameters in Fig. I by these approximations, the
result is a prebreeding census partial life cycle graph
(Fig. 2; age classesbeyond 0) ignored). The projection
matrix corresponding to the partial life cycle graph
(Fig. 2) is a function of five variables (cx,0), PpPa' and
F):
In general, A is a f:\ by f:\ matrix with
(6)
OIKOS 94:4 (2001)
In general, A is a ~ by ~ matrix with
cific parameters in Fig. I by these approximations, the
result is a prebreeding census partial life cycle graph
(Fig. 2; age classesbeyond 0) ignored). The projection
matrix corresponding to the partial life cycle graph
(!ig. 2) is a function of five variables (Cl,0), Pi' Pa' and
F):
~, for i = 1 and Cl ~j ~ <0
P ft for i = j + 1 and 1 ~j ~ 13-0,
A..=
, IJ
elsewhere.
Fi = 0 for all i > 0), age classesbeyond 0)makeno
0
contribution to the long-term dynamics of the population, and thus can be ignored in our density-indepenPi
dent setting. The matrix parameters Fi (i= I, 2, ...,0)and
0
Pi (i= 1,2, ...,0)-1) can be estimated from thelife
A(a.ro.~.p'.F)
table data using the pre-breeding census formulation of Caswell (1989a):
= 1;+ 1
,-
.p-
0
.0
P..
J
.0
(4)
0 ..
0 ..
0 ..
P.
0
0
(2)
I
.j',
An example of the projection matrix corresponding to a
partial life cycle graph with tX= 2, and Q)= 5 (age
classesbeyond Q)ignored) is:
.
F.=l\m;,
(3)
where /i is the survivorship (probability at birth of
surviving to age i), /1 is the probability at birth of
surviving to one time unit of age, and mi is the fecundity (the average number of daughters born to a female
of age i). Caswell (1989a)describes in detail the construction and analysis of age-structured matrix models.
Here, we are concerned with the derivation of a model
that can be parameterized with partial demographic
data collected from pre-breeding censuses,and in which
age at first reproduction «(1)and age at last reproduction (0) appear explicitly as model parameters.
We assume that the age-specificfertilities Fa, Fa+l,
..., F",-I, F", can be adequately approximated by an
age-independentaveragefertility parameterF ~averaged
over age class (1,(1+ I, ..., 0), that age-specificsurvival
probabilities until the first birth event Guvenile or prereproductive survival probabilities) PI, P2, ..., Pa-1
can be adequately approximated by a juvenile survival
parameter PJ' and that age-specificadult survival rates
P a' P a+ I, ..., P",-I can be adequatelyapproximated by
an adult survival parameter P a. If we replace age-spe-
Because
000
p.
nd
Dividing
Fig.
0
~O
F
j..'
j..."
j.1;',
Pj
0
l:)
I:>
I:>
0
Pa
/J
I:>
I:>
0
0
]':>a
[)
:>
0
0
l');>
A(tX, 00,Pj, P a' F) =
~
]
I
u
lr)-,
The characteristic equation for the life cycle of the type
shown in Fig. 2 and the corresponding projection matrix A(Cl,ffi, Pj, Pa' Fjis obtained bJ setting the determil!,antof the ma~ix A( Cl,ffi, Pj' Pa' F) -AI to zero, where
A( Cl,ffi, Pj' Pa' F) is the projection matrix corresponding
to a pre-breeding census partial life cycle graph, and I
is the identity matrix. The general characteristic equa-
tion is:
0 = Am -FPtx-l).m-tx
J
_Fptx-lp
J
-Fp~-lp2Am-tx~2
J
a
-Fptx-lJ
pm-tx-l'l
a
a
Am-tx-l
-Fp~-lp3Am~tx-3
J
a
II.
-
P-;ntx-l
r J
pa.m-tx
(5)
Eq. 5 by Acoand rearranging yields (A # 0):
+FPJ-lj..
(~ )
-CX
2
j..
+FPJ-lj..
t~~~~
p.
~--;~-0-::-G-Pj
Pa
'Pa
-cx(f
+"
)"'-CX
)3
-+
,A
a
J
2. A pre-breeding censuspartial life cycle graph. Parame-terswhich is equivalent to:
are: F = age-independentaverage fertility rate, P j = juve-nile
survival rate, P a = adult survival rate, cx= age at maturity,
rof"
= FPj-1A. -,,
ro = age at last reproduction. Age classes beyond ro are ig-nored.
[
(~ )
k~O
378
k
A.
-
(6)
OIKOS 944 (2001)
m
+
o=p",
Somelong-lived organismssurvive and reproduce for
a long time, but exact age at last reproduction is
unknown. We have extended the model to adequately
addresssuch situations (Appendix I).
For A.# P a' Eq. 6 can be written as:
-(f)OO-CX+
1 =FPj-lj.-OX
(7)
Sensitivityanalysis
Finally, multiplication of Eq. 7 by (Pal). -1) and rear-
The evaluation of the sensitivity of population growth
rate (A) to absolute or proportional changes in demographic variables is an important aspect of demographic
analyses (Caswell 1989a, Horvitz et al. 1997, de Kroon et
FP~-l1..
-~-FP~-lpco-~+lA.
J
J
a
a
(8) al. 2000, Heppell et al. 2000). The sensitivity quantifies
changes in A in response to small absolute changes in a
demographic variable. Likewise, the elasticity or proporThe asymptotic population growth rate (A) is the largest
tional sensitivity quantifies changes in A in response to
real root of Eq. 8 and can be obtained numerically.
small, proportional changes in a demographic variable
Equivalently, A can be estimated as the dominant eigen(de Kroon et al. 1986, Caswell 1989a, Horvitz et al. 1997,
value of the projection matrix corresponding to the
de Kroon et al. 2000). The sensitivity of A to small
partial life cycle graph. Using different approaches,
changes in demographic parameters is quantified by the
models similar to Eq. 8 have been derived by Caswell
partial derivative of A with respect to a model parameter,
(1989a)and Slade et al. (1998).
p (i.e., aA/ap, where p is IX,CO,Pp PQ' or F). Using implicit
differentiation of Eq. 8, we find that:
rangementyields:
01..
-=-
(9)
OCl
FP;p")'ln(~)
oj.
(10)
am
oJ..
-0
oP.J
-
01..
(11)
'II
"'\
=
oPa
-Fpap"'
ja
-Fpap"'O}
Ja
+
-
+ 10)-FP~P"'+
Ja
(
Fpap"'cx
Ja
I+
-FP~P"'
FP~P"'
Ja
)
Jap
+
p.paA"'
Ja
)
A -a+",
+
a
p
'
pi
(12)
+aA"'
ja
and
oJ.
-=-
(13)
OIKOS 93:3 (2001)
379
of
~-I
Pa~
Elasticities or proportional sensitivities are calculated
by multiplying sensitivities by (PIA), wherep is Cl,OO,
Pi'
Pa' or F. Notice that the sensitivity of A to changes in
Cl will always be negative becausea delayed maturity
causes a decline in A, and that the elasticity of A to
changes in Pi' Pa' and F sum to 1 as they should (de
Kroon et aI. 1986,Mesterton-Gibbons 1993).
P
j~
IejPj
iI-I
,
(15)
«-I
00-1
L
j-~
ejPj
;-1
.
(16)
Parameter estimation
Our pre-breeding census partial life cycle model is
appropriate for the analysis of partial demographic
data collected from pre-breeding censuses,and can be
fully parameterized if estimates of cx,00,Pi' Pa' and F
are available. However, researcherswill find our model
Thus, F, Pj' and P a may be estimated as weighted
averages,weighted according to the contribution of
each age class to the stable age distribution.N3he approximations in Eqs 14-16 will determine whether and
to what extent dynamical properties of the age-strucuseful even when age-specificdemographicdata are tured matrix A are retained in the partial cycle matrix
available for such purposes as the calculation of the A(IX,00,Pj, Pa' F). If F, f;, and Pa are chosen such that
sensitivity of A to changes in cxand 00,and the estima- ~ = A and e = e (where A and A are dominant eigenvaltion of life table response experiment (L TRE; sensu ues of the matrix A and A(IX,00,Pj, Pa' F), respectively,
Caswell 1989b)contributions of cxand 00(e.g., Levin et and e and e are the corresponding right eigenvectors),
al. 1996, Oli and Zinner 2001, Oli et al. 2001). When results from the two models will be identical.
age-specificdemographic data are available, parameters
for the partial life cycle model should be estimated
from life table data or from the age-structured projection matrix such that dynamical properties of the origi- Extending the model to incorporate fractional
nal age-structuredmodel are retained as much as
possible. We proceed with the assumption that the
age-structured Leslie matrix has been parameterized
using the pre-breeding census formulation of Caswell
(1989a), and presentmethods of estimating parameters
for our pre-breeding census partial life cycle models
from the age-structured Leslie matrix. Moreover, in a
Leslie matrix, cxand 00correspond to the first and last
age class with non-zero fertility, respectively, and
statistical estimation of these variables may not be
a; and ro
Not all individuals in a population begin or terminate
their reproductive career at the same age. When onset
or termination of reproduction is spread over multiple
age classes,it might be preferable to use the population's average ages at first or last reproduction, and
these values can be fractional. The application of Eq.
8 to analyze such data may not be appropriate, becausethis model was derived on the assumption that IX
and (J)are integers. Here, we extend the partial life
necessary.
Our goal is to approximate age-specific fertilitiesF",F"+I'
cycle model of Eq. 8 such that the resulting model will
...,F"'-I,F",
by an age-independentaverage allow fractional values of ages at first and last reprofertility parameter F, age-specific survival probabilities duction. To keep the problem mathematically
until the first reproductive event PI, ...,P,,-I by ajuvenile
tractable, however, we consider a birth-pulse populasurvival parameter Pi' and age-specific tion of the type represented in Fig. 2 in which onset
adult survival probabilities P ", ..., P'" -I by an adult and termination of reproduction are spread over two
survival parameter P a such that dynamical properties age classes.
of the age-structured matrix A are retained in the
Let Xi be the proportion of organisms in age class i
matrix A(cx,00,Pi' Pa' F') corresponding to a pre-breed- that become primiparous before they advance to the
ing censuspartial life cycle graph. This goal is achieved next age class,and let Yj be the proportion of organif F, Pi' and P a are chosen such that >:~ A and e = e, isms in age class i that reproduces in age class i but not
or when A(cx,00,Pi' Pa' F')e ~ Ae (Oli and Zinner 2001). beyond (Oli and Zinner 2001). Next, we define Xj and
This result leads to the following approximations:
Yj as follows:
w
F-i~tt~-
-;;;--,
Lei
i=~
380
-X,
IeiFi
(14)
X-
i-
for i = CXo
X, for i = 1%0
+ 1,V,
otherwise,
OIKOS 93:3 (2001)
D..=
Yi=
+
+
FY
where
CXo
isbegin
the earliest
age class
in0which
or more
organisms
reproduction,
and
~ X <one
1, and
/
-Y,
0 is the column vector of zerosof length <00'and D isa <00
by <00diagonal matrix with:
for i = roo
J,ifi<ao
Y, for i = roo+ 1
1,1
'-0, otherwise,
XPJ+(l-X)Pa,
if i=~
l>a, if i> tXo'
where roois the earliestageclass in which reproduction is As above, the general characteristic equation of the
terminated in one or more organisms,and 0 :$;Y < I (Oli matrix A(cxo,roo,PpPa' F, X, Y) is obtained by setting
and Zinner 2001). Then, the average age at t.!te determinant of the matrix [A(CXo,
roo,PpPa'
which reproduction begins (cx)is given by: cx= CIa(I F, X, Y) -AI] to zero:
X) + (CIa+ 1) X = CIa+ X, and the average age at last
0 = 1..roo+
I-F(IX)p;o-I1..roo-",,+ 1
reproduction (ro)is given by: ro= roo+ Y.
We have assumedthat the earliest age class at which
-FP;o-I(XPj
+ (1 -X)P
a)1..roo-""
reproduction beginsis cxo,but only the fraction (1 -X) of
-FP;o
-I(Xp j + (1 -X)P a)P a1..roo-~!f;
organisms becomes primiparous at that age and the
fraction X does not; thus, fertility for age class CXo
is
-FP-",,-I
j
(XPj + (1 -X)P a)P a1..
(I -X)F. Similarly, the fraction of organisms in which
reproduction is terminated in the age class roois (I -Y)while
-Fyp",,-I J
( XpJ+ ( 1- X) p a )proo-""
a
(18)
the fraction Y will reproduce for the last time in ageclass
roo+ I. Consequently,fertility rate for the ageclass which, for A.# 0, can be written as:
(roo+ I) is YF. Becausethe age class CXo
consistsof bothnulliparous
and primiparous organisms,the survival rate
= F(1 -X)p~-l
A-ao
for that ageclassis [XPj + (I -X)P a]' If we incorporate
these changes into the pre-breeding census partial lifecycle
FP;o-I(XPj+ (1-X)P a)A.
-ao-1 ""':~O-I (~)k+
graph and the corresponding projection matrix
(Eq. 4), the result is a projection matrix with parameters
( 1- X) p )A.-"",-I p"",-ao
Fypao-I
(Xp.+
J
J
a
a'
cxo,roo,Pp Pa' F, X, and Y. Denote the projection matrixcorresponding
(19)
to a partial life cycle with parameters cxo,
roo, PpPa' F, X, and Yby A(CIa,roo,PpPa' F, X, Y). For For P a* A, one obtains
example, for CIa= 3 and roo= 5, we have:
0
A(ao,Q)o,~.p',F,X.Y)
=1
0
F
F
Fylr.
0
F(I-X)
0
0
0
0
0
Pj
0
0
0
0
0
0
0
0
0
xp;+(I-X)p'
00
0
p.0
0
00
0
Op.
0
(17)
= FpCXO-l I
J
r
1- ( -p O)OIO-ao.
A
x
1-
When all females begin reproduction at age CXo
andreproduce
for the last time at age COo,
then X = 0 and
Multiplication
Y = 0, and projection matrices in Eqs 17 and 4 areidentical.
yields:
In general, A(cxo,COo,
Pi' Pa' F, X, Y) is an
(coo+ 1) by (coo+ 1) matrix of the form:
0 = Pal. -I +
where f denotes a row vector of length 000with:
U, if i < tXo
.1;=
F(l -X),
if i = tXo
.., otherwise,
OIKOS 93:3 (2001)
a)A.-roo-I p:;",-a.,
(20)
by (1 -P a/A) and rearranging
FP;o-I(1
+ FP;o-I(XPj
0 ]'
Y(XP AI -X)P
(~A)""'".
-X)(I.
+ (1-
-"0 -Pal.
finally
-tJO-1
X)P a)
X [I. -"0 -1 + (Y -1 )P~
-"01. -0'0 -1
-nO'O-2p~-"O+I]-I.
(21)
Let the right hand side ofEq. 21 be denoted by G(CXo,
roo,
X, Y, Pi' Pa' F, A). Then, the asymptotic population
growth rate A is the largest real root of Eq. 21. When
X = 0 and Y = 0 Eq. 21 is identical to Eq. 8 and A
calculated from the two equations will match exactly.
381
~
~i
,..,
./
~=~
"3
~
1.0041
~.
382
-.G
~
In general, results obtained from the two models will
resembleclosely when X and Yare close to zero or one
(Fig. 3).
Eqs 14-16 may be used to estimate model parameters from an age-structured Leslie matrix, but these
equations should be adjusted to reflect redefinition of (X
and rooAlternative methods for estimating parameters
from an age-classifiedmatrix may be considered (e.g.,
arithmetic or geometric average),but adequacyof such
methods should be investigated. When age-classified
data are not available, however, parameters for our
model may be estimated as simple or weighted
averages.
As above, the sensitivity of A to changes in a model
parameter p is given by aA/ap,and the elasticity is given
by (aA/ap)(pIA). The sensitivity of A to changes ih P p
Pa' and ft can be calculated directly through implicit
differentiation of G«Xo,roo,X, Y, PpPa' ft, A). However,
the sensitivity of A to changes in (X and ro requires
additional consideration of the fact that thesevariables
are defined piecewise,(Xin terms of (xoand X, and ro in
1.010
1-- Eq.8 !1-
iu(x
a:x( ao, X)
for cx=cxo+X, CXo=1,2, ..., and O~X< 1. Since I- is
implicitly defined in Eq. 21, we may use implicit differIentiation to calculate aI-lax. Using implicit differentiation of G(CXo'
0)0' X, Y, Pp Pa' ft, 1-),we find that:
(
(~cx
I
(5G loG
ax I ef'
(22)
ind
(
(~ /~)(~
=-
vX 01..
(
I..
).J(XI..
(23)
Eq...21!
1.000
~
a~'
01rl(X) =
r ..
Likewise,the sensitivity and elasticity of A to changesin:0
( (where 00=000+ Y, 000= 1,2...,
and O~ Y> 1) areround
j
to be:
A
1.005
terms of 000and Y. In Eq. 21, A is an implicit function
of CXo
and X (as well as of other variables). Becausewe
have defined (1=(10+ X (where (10= 1, 2, ..., and 0 ~
X < 1), A may also be"viewed as a function of (1. We
note that A depends continuously on (1,and that
oj.
~
laG
§ffi= -W!"ai'
i
0.995
(24)md
0.990
,
0.985
...
0.980
0.975
i
6
=-(~/~)( ~).
(25)
-
7
a
1.0044
B
;1
...r--
1.0043
.-<
The sensitivity as well as elasticity of A to changesin cx
and 00are piece-wise continuous, with discontinuities
located at cx= I, 2, 3, ..., and 00= 1, 2, 3,
Piece-wise
continuity in the sensitivity and elasticity of A to
changesin cxand 00is due to the piece-wisedefinition of
A in terms of CXo
and X, and of 00in terms of 000and Y
(Oli and Zinner 2001). The elasticity of A to changesin
F, Pp and P a sum to unity (de Kroon et al. 1986,2000,
Mesterton-Gibbons 1993).
1.0042
Examples
We applied the age-structuredLeslie matrix model and
our partial life cycle model to life table data for 12
populations of mammals to evaluate the adequacy of
co
our partial life cycle model. For each life table, we
Fig. 3. Changes in population growth rate (A) in responseto
changes in: (A) age at maturity (cx) and (B) age at last estimatedage-specificsurvival probabilities Pi using Eq.
reproduction (00)calculated using Eq. 8 and Eq. 21. Values of 2 and fertility rates Fi using Eq. 3. Population growth
t!1e parameters were: cxo=3, 000=31, Pj=0.776, Pa = 0.872,
rate, and sensitivity and elasticity matrices were calcuF = 0.225, and (A) X was allowed to vary but Y was held
constant at Y = 0, and (B) Y was allowed to vary but X was lated using the Leslie matrix model following Caswell
held constant at X = O. See text for details.
(1989a). The sensitivity of population growth rate to
31.0
31.2
31.4
31.6
31.8
OIKOS 93:3 (2001)
where
1.6
for Tibetan monkeys,where the growth rate differed by
5.8%. Similarly, elasticities calculated from the twomethods
were very similar (Table 1). These results
suggest that our pre-breeding census partial life cycle
model adequatelycaptures the dynamical properties ofthe
corresponding age-structured matrix model, and
that approximations in Eqs 14-16 are adequate forestimating
parameters for our partial life cycle model.
Our results also indicate that, in populations included
in our analysis, population growth rate is insensitive tochanges
in age at last reproduction, but very sensitiveto
changesin survival rates (Table 1).
1.8
-;-
u
>u
:: 1.41.2
0)
m
:e
co
e::.
~
1.0
0.8
0.8
0.9 1.0 1.1
1.2 1.3
1.4 1.5
1.6
I.. (Age-specific)
Fig. 4. Population growth rate (I.) calculated using the agespecific Leslie matrix model, and the pre-breeding census
partial life cycle model (Eq. 8) for severalmammalian populations. Values of the partial life cycle model parameters are
given in Table 1. Censusseswere assumed to be taken just
before the birth-pulse. Population growth rates calculated
from the two methods were highly correlated (r = 0.990, P =
0.0001).
Discussion
Although age-structured Leslie matrix models make
maximum use of age-specificdemographic data, such
data are difficult to gather in the field. Consequently,
investigators frequently have to rely on incomplete
demographic data which are easier to collect. Partial
life cycle models, on the other hand, can be parameterized with partial demographic data. To fully parameterchanges in F, Pj' and P a was calculated from the ize an age-structuredmatrix model, (2n -<x) parameters
sensitivity matrix (see Caswell 1989a for details in cal- (where n = number of age classes)are required. For
culation of the sensitivity matrix) as follows:
example, one must estimate 20 fertility terms and 19
survival terms to parameterize an age-structured model
0>
OA
for a population with 20 age classesand <x= 1; a partial
~, = ):::mj,
(26) life cycle model, on the other hand, can be fully
of j~cx
parameterized with five parameters (ages at first and
,,-I
01last reproduction, juvenile and adult survival rates, and
- = L OJ,
(27)
oP.J j~1
age-independentaverage fertility parameter) regardless
of the number of age classes.In our example calculations, we have used age-structured data to estimate
",-1
01..oPa
parameters
for the pre-breedingcensuspartial life cycle
= L °i'
(28) model. However, our partial life cycle model can also
i="
be used when estimatesof ages at first and last reprom is a vector consisting of the first row of the duction, juvenile and adult survival rates, and average
sensitivity matrix, and n is a vector consisting of the fertility rates are available; data needed for estimating
lower sub-diagonal entries of the sensitivity matrix. these parameters can be collected much more easily
Elasticities were calculated by multiplying sensitivities than those neededto estimate age-specificdemographic
by piA, where p is F, Pi, or P a (de Kroon et al. 1986, parameters. An additional desirable property of partial
2000, Caswell 1989a). From each Leslie matrix, we life cycle models is that variables such as ages at first
estimated parameters for the partial )ife cycle modelusing
and last reproduction appear as explicit model parameEqs 14-16. For the partial life cycle model, ters, and the sensitivity of population growth rate to
sensitivitieswere calculated using Eqs 9-13, and elastic- changesin thesevariables can be estimated directly (Oli
ities were calculated as above.
and Zinner 2001). Becauseof these advantages,partial
If our partial life cycle model were a reasonable life cycle models have received wide applications in
approximation to the age-structured Leslie matrix population biology (e.g., Cole 1954,Lande 1988, Levin
model, we would expect the population growth rate as et al. 1996, Oli et al. 2001).
well as sensitivities and elasticities calculated from the
Assuming that mortality does not occur until age at
two models to resemble closely. Population growth last reproduction, Cole (1954) derived a partial life
rates calculated from the two models compared fa- cycle model. This model, however, does not provide
vourably (Fig. 4). In 8 of 12 populations we analyzed, realizable estimatesof population growth rate, because
population growth rates calculated from the two mod- organismsof any age can die. Using different methods,
els differed by < 2%. The largest difference in popula- Caswell (1989a) and Slade et al. (1998) il1dependently
tion growth rate calculated from the two models was derived analogous partial life cycle models. Two other
OIKOS 93:3 (2001)
partial life cycle models that ignore age at last repro- tioned partial life cycle models have not explicitly
duction have been presented by Lande (1988) and considered the timing of census relative to the birthLevin et al. (1996; also seeCaswell 1989a).As discussed pulse in deriving their models, although models of
earlier, an important consideration in modelling birthpulse
Caswell (1989a) and Slade et al. (1998) are similar to
populations is the timing of censusrelative to the our pre-breeding census partial life cycle model. Rebirth pulse. For matrix population models, age-struc- cently, we (Oli and Zinner 2001) have derived a
tured or not, estimation of model parameters depends partial life cycle model appropriate for the analysis of
on whether the demographic data were collected frompre-demographic data collected from post-breeding cenor post-breeding censuses(Caswell 1989a).Conse- suses.The partial life cycle model presented here was
quently, the application of a model derived for pre-breeding
specifically derived for the analysis of demographic
censussituations may not be appropriate forthe data, age-structured or partial, collected from preanalysis of demographic data collected from post-breeding
breeding censuses,and also allows fractional v!ilues of
censuses.However, authors of the aforemen- ages at first and last reproduction, cxand co, respec1. Comparison of population growth rate (A) and elasticities calculated from the age classified Leslie matrix model andthe
pre-breeding census partial life cycle moQel (Eq. 8) for several populations of mammals. For the Leslie matrix model,the
elasticity of ). to changes in P, fa and F was calculated from the elasticity matrix using Eqs 26-28. Parameters for the
partial life cycle model (P, P a and F) were estimated using Eqs 14-16, and elasticities were calculated as described in the text.Values
of demographic v~riables used to parameterize the partial life cycle model also are given. Censuseswere assumed tobe
taken just before the birth-pulse.
Species/model
Parameter
values
ro
p.J
1. Spiny pocket mouse (Liomys adspersus)1
Age-structured.
Partial life cycle
4
12
0.935
A
Pa
i!
0.9700.966-
0.897
0.156
2. Blue sheep (Pseudoisnayaur)2
Age-structured.
-Partial life cycle
2
16 0.950
3. African buffalo (Synceruscaffer)3
Age-structured.
Partial life cycle
3
18 0.743
4. Caribou (Rangifer tarandus)4
Age-structured.
Partial life cycle
2
15 0.714
5. Warthogs (Phachochoerusaethiopicus)5
Age-structured.
-Partial life cycle
2
12 0.309
6. Feral horse (Equus caballus)6
Age-structured.
-Partial life cycle
3
19
0.972
7. Lions (Pantheraleo)7
Age-structured.
Partial life cycle
3
17 0.432
-0.059
0.337
1.053
1.053
-
0.751
0.285
1.063
1.045
-
0.920
0.304
1.117
1.107
0.715
0.633
0.903
0.915
0.975
0.452
0.669
0.534
0.963
-0.338
-
0.943
0.910
-
0.395
0.245
1.013
1.014
0.927
0.891
8. Giant panda (Ailuropoda melanoleuca)8
Age-structured.
Partial life cycle
3
9
0.566
0.867
9. North American black bears {Ursus americanus)9
Age-structured.
-Partial life cycle
5
19 0.663
0.824
10. Northern sea lions (Eumetopiasjubatus)IO
Age-structured.
--Partial life cycle
3
31
0.776
0.872
11. Tibetan monkeys (Macaca thibetanus)11
Age-structured.
-Partial life cycle
3
7
0.802
0.868
12. Olive baboons (Papio cynocephalus)7
Age-structured.
--Partial life cycle
4
25
0.846
0.967
-0.048
-0.136
0.154
PJ
-
0.3190.3990.574
0.125
Po
0.468
0.107
0.133
0.593
0.545
0.204
0.228
0.008
0.228
0.043
0.2580.251
0.613
0.623
0.041
-
0.1680.170
0.663
0.660
-0.460
0.044
0.209
0.657
0.583
1.246
1.246
-0.119
0.011
0;316
0.549
0.526
0.158
1.047
-
-
0.993
-0.319
0.056
0.267
0.248
0.600
0.628
0.133
0.124
0.156
0.380
0.358
0.429
0.463
0.179
-0.169
0.054
0.392
0.392
0.510
0.510
0.098
0.098
0.191
0.713
-0.092
0.007
0.228
0.659
0.096
0.114
0.053
0.524
0.509
0.214
-0.554
-
0.011
0.332
0.343
0.557
-0.143
1.5201.608
3.124
1.153
0.172
-
1.017
0.480
m
0.237
0.542
0.262
0.254
0.111
0.115
* Partial life crcle parameters (cx,0), Pp PQ' and F) do not explicitly appear in the age-structuredmodel:
Data sources: Flemming (1971); 2Wegge(1979); 3 Sinclair (l977); 4 Messier et al. (1988); 5Rodgers (1984); 6 Garrot and Taylor
(1990); 7 Packer et aI. (1998); 8Wei et al. (1989); 9Yozdis and Kolenosky (1986); 1O
Calkins and Pitcher (1982); II Li et al. (1995).
384
Table
Elasticities
F
0.203
0.1290.126
0.1690.170
0.1710.208
0.301
0.150
0.190
1.166
GIKaS 93:3 (2001)
tively. This model can be parameterized with partial
demographic data, but it is also useful even when
age-specificdemographic data are available. For example, the sensitivity of population growth rate to changes
in age at first reproduction, an important life history
variable with substantial potential for influencing population dynamics (Cole 1954, Lewontin 1965, Oli and
Dobson 1999), cannot be estimated using the standard
Leslie matrix models; our partial life cycle model can be
used for this purpose.
In a seminal study that investigated population consequencesof life history patterns, Cole (1954) found
that age at maturity generally had a greater relative
influence on population growth rate than other life
history variables. In a similar study, Lewontin (1965)
found that, among life history variables he considered,
age at maturity had the largest relative influence on
population growth rate. Test of ideas such as Cole's
and Lewontin's requires estimation of the sensitivity or
elasticity of population growth rate to changes in life
history variables, becauseabsolute or proportional sensitivity of population growth rate to changes in life
history traits are interpreted as selectiongradients (e.g.,
Caswell 1989a, Roff 1992, Stearns 1992). However,
variables such as ages at first and last reproduction do
not appear as explicit parameters in the Leslie matrix
model, and the sensitivity or elasticity of population
growth rate to changes in these variables cannot be
estimated using standard techniques.
One of the most important uses of our partial life
cycle model is that the sensitivity and elasticity of
population growth rate to changes in ages at first and
last reproduction can be estimated directly because
these variables appear as explicit parameters in our
model. Since elasticities are scaled,dimensionlessquantities, they are directly comparable among life history
variables and across populations or species(Horvitz et
al. 1997)and thus are appropriate for testing theoretical
predictions regarding the relative importance of life
history variables to population growth rate. Using our
partial life cycle model, we estimated the elasticity of
population growth rate to changes in ages at first and
last reproduction, juvenile survival, adult survival, and
fertility (Table 1). Our results show that adult survival
rate had the largest potential influence on growth rates
of most populations included in our analysis. Population growth rate was insensitive to changes in age at
last reproduction, but moderately sensitive to changes
in juvenile survival and fertility rate. Age at maturity
had the highest elasticity only in one population (Tibetan monkeys). In general, our results indicate that
age at maturity may have large influence on growth
rates of populations with high fertility rates, but that
adult or juvenile survival rates are more influential in
populations characterized by low fertility rates. These
findings are consistent with the suggestion that the
relative importance of life history variables to populaOIKOS 93:3 (2001)
tion growth rate may vary, depending on the pattern of
life history (Stearns 1992). Disagreement between our
results and those of Cole (1954)and Lewontin (1965)
may have beenbecauseof the fact that Cole and
Lewontin used in their analysesvalues of fertility rates
much larger than those observed in populations included in our analyses.
Our model (Eq. 8) was derived on the assumption
that agesat first and last reproduction only take integer
values,and the sensitivity of population growth rate to
changesin thesevariables is difficult to interpret. However, the extended model (Eq. 21) addressest~is concern, and partial derivatives of ). with respectto IXand
(J)calculated using this model have natural interpretations. Despite some differences in the formulation, our
calculations (Figs 3 and 4, Table 1) indicate that these
two models yield very similar results, particularly when
IXand (J)have near-integervalues. Theseresults suggest
that the use of the simpler model (Eq. 8) will not
generally compromise precision of the analyses nor
conclusions of an investigation.
Although age-specificinformation is lost in a partial
life cycle model, we have found that dynamical properties of the age-structured matrix model are generally
retained in our partial life cycle model (Fig. 3, Table 1).
For example, population growth rate calculated from
our partial life cycle model differed only by < 2% in 8
out of 12 populations from those obtained from the
age-structured model. Elasticities calculated from the
two methods differed to some extent for some populations. However, the relative magnitudes, rather than
actual values,of elasticitiesare of primary interest (e.g.,
Horvitz et al. 1997), and in no case did the relative
magnitudes of elasticities differed between the two
methods (Table 1). These results suggestthat our prebreeding census partial life cycle model is an excellent
proxy for the full age-classifiedmatrix model, and that
Eqs 14-16 are adequate for estimating parameters for
our partial life cycle model. Our model, although
derived specificallyfor birth-pulse populations, can also
be used to model birth-flow populations if age-structured matrix parameters are estimated using the birthflow formulation of Caswell (1989a: 9-12), and the
partial life cycle model is parameterized from the agestructured projection matrix using Eqs 14-16.
Acknowledgements-We
thank N. B. Frazer and M. P.
Moulton for helpful comments on the manuscript. A computer
program used to perform analysespresented in this paper is
available from M. K. Oli on request. Tliis research was
supported by the Florida Agricultural Experiment Station, and
approved for publication as Journal SeriesNo. R-O8036.
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Appendix I
Consider a birth-pulse population in which organismssurvive
with a juvenile survival probability P j per time
unit until reproduction begins at age CX.
Once reproduction is achieved,adult individuals survive with an adult
survival probability of P a and contribute to the population with an averagefertility rate of F per time unit. Weassume
that some individuals in the population surviveand
reproduce for a long time, but that exact age at last
reproduction (00)is not known. The pre-breedingcensus
projection matrix corresponding to such life histories isa
function of CX,Pi' Pa' and F:
0
p.J
0
0
(11)
J
0
characteristic equation for the life cycle of the typerepresented
by the projection matrix A (Eq. II) can beobtained
by setting the determinant of the matrix A AI to zero as describedearlier. The generalcharacteris-tic
equation can be written as:
I.,OX
-l.,ox-IPa-Pj-IF=O.
asymptotic population growth rate, A, is the largest
real root ofEq. 12.The sensitivity of population growthrate
to changes in a model parameter p is the partialderivative
of A with respectto p (i.e., oA/op),and can beobtained
through implicit differentiation of Eq. 12:
OA
A( -A
In(A)
+ A -a+
lPJa-1In(Pj)F+'
-=OCI
P a In(A))
,
CIA -P
aCI + P a
OIKOS 93:3 (2001)
OA
--p~-2PA2-CX(IX-1)
-J
aP; -IX).
oJ..
-P alX + P a'
CIA -P
OIKOS 93:3 (2001)
acx+
P a'
(14) --='
of
pa-l).
I
-A."CX+A.IX-lp
a
CX-A"-lp
(16)
a
The proportional sensitivity (elasticity) of population
growth rate to changesin a model parameterp is given
(15)
by [(aA/ap)(pIA)].
387