Journal of Ecology 2010, 98, 324–333
doi: 10.1111/j.1365-2745.2009.01627.x
SPECIAL FEATURE
ADVANCES IN PLANT DEMOGRAPHY USING MATRIX MODELS
Life table response experiment analysis of the
stochastic growth rate
Hal Caswell*
Biology Department MS-34, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA; and Max Planck
Institute for Demographic Research, Rostock, Germany
Summary
1. Life table response experiment (LTRE) analyses decompose treatment effects on a dependent
variable (usually, but not necessarily, population growth rate) into contributions from differences
in the parameters that determine that variable.
2. Fixed, random and regression LTRE designs have been applied to plant populations in many
contexts. These designs all make use of the derivative of the dependent variable with respect to the
parameters, and describe differences as sums of linear approximations.
3. Here, I extend LTRE methods to analyse treatment effects on the stochastic growth rate log ks.
The problem is challenging because a stochastic model contains two layers of dynamics: the stochastic dynamics of the environment and the response of the vital rates to the state of the environment. I
consider the widely used case where the environment is described by a Markov chain.
4. As the parameters describing the environmental Markov chain do not appear explicitly in the
calculation of log ks, derivatives cannot be calculated. The solution presented here combines derivatives for the vital rates with an alternative (and older) approach, due to Kitagawa and Keyfitz, that
calculates contributions in a way analogous to the calculation of main effects in statistical models.
5. The resulting LTRE analysis decomposes log ks into contributions from differences in: (i) the
stationary distribution of environmental states, (ii) the autocorrelation pattern of the environment,
and (iii) the stage-specific vital rate responses within each environmental state.
6. As an example, the methods are applied to a stage-classified model of the prairie plant Lomatium
bradshawii in a stochastic fire environment.
7. Synthesis. The stochastic growth rate is an important parameter describing the effects of
environmental fluctuations on population viability. Like any growth rate, it responds to differences
in environmental factors. Without a decomposition analysis there is no way to attribute differences
in the stochastic growth rate to particular parts of the life cycle or particular aspects of the stochastic
environment. The methods presented here provide such an analysis, extending the LTRE analyses
already available for deterministic environments.
Key-words: autocorrelation, fire, Lomatium bradshawii, LTRE analysis, Markovian environments, matrix population models, sensitivity analysis, stochastic growth rate, stochastic
models
Introduction
A life table response experiment (or LTRE; the term was
introduced by Caswell 1989) is a study that compares a
complete set of vital rates under two or more conditions.
What counts as a complete set of vital rates depends on
the situation; it might be a life table, a population projec*Correspondence address. E-mail: [email protected]
tion matrix or a set of parameters (survival, growth, reproduction, etc.) from which a projection matrix may be
calculated. The conditions among which the populations
are compared will be called ‘treatments’ here, although
they may be experimental manipulations or comparative
observations. In analogy with genuine experiments, LTRE
analyses have been classified into fixed (Caswell 1989),
random (Brault & Caswell 1993; Horvitz, Schemske &
Caswell 1997) and regression (Caswell 1996a; Knight,
! 2010 The Author. Journal compilation ! 2010 British Ecological Society
Stochastic LTRE analysis 325
Caswell & Kalisz 2009) designs (see Caswell 2001, Chapter
10).
Life table response experiment analysis focuses on some
demographic summary statistic that is calculated from the vital
rates. LTRE analyses in ecology1 originally focused on population growth rate (k or r ¼ log k), but the approach has also
been applied to such summary statistics as the net reproductive
rate R0 (Levin et al. 1996), the invasion wave speed (Caswell,
Lensink & Neubert 2003) and the operational sex ratio (Veran
& Beissinger 2009). In principle, it can be applied to any dependent variable.
An LTRE study documents differences among treatments
in the dependent variable. The goal of LTRE analysis is
to decompose those differences into contributions from the
differences in each of the vital rates. As the effects of most
treatments are diverse and stage specific, this decomposition is
critical to understanding how a treatment affects population
growth (Caswell 1996a,b).
Because of their particular interest in comparative studies,
plant demographers have applied LTRE analysis in many contexts. To cite just a selection of recent studies, it has been
applied to effects of management (Davis, Dixon & Liebman
2003; Brys et al. 2004; Van der Voort & McGraw 2006; Zuidema, de Kroon & Werger 2007), invasive species (Burns
2008), pathogens (Davelos & Jarosz 2004), floods (Smith,
Caswell & Mettler-Cherry 2005; Elderd & Doak 2006), disturbance (Endels et al. 2007), fire (Kesler et al. 2008), rainfall
(Smith, Caswell & Mettler-Cherry 2005; Lucas, Forseth &
Casper 2008), herbivory (Knight, Caswell & Kalisz 2009),
interspecific interactions (Miriti, Wright & Howe 2001; Freville
& Silvertown 2005; Griffith & Forseth 2005) and effects on
spread speed (Jacquemyn, Brys & Neubert 2005; Jongejans
et al. 2008). Two of the papers in this Special Feature use
LTRE calculations as one of their tools for comparative analysis (Jongejans et al. 2010; Davison et al. 2010).
My goal here is to extend LTRE analysis to stochastic models, by showing how to decompose differences in the stochastic
growth rate, log ks. As stochastic models include both environmental fluctuations and the vital rate responses to those
fluctuations, their structure is richer than that of time-invariant
models. Stochastic LTRE analysis thus requires a new
approach to decomposing these differences. The pay-offs, in
terms of biological understanding, are great.
TWO APPROACHES TO DECOMPOSITION
The familiar LTRE analysis uses derivatives to approximate
the contributions of the vital rates. Suppose that k depends on
a vector h of vital rates, and that observations are available
under two treatments, with
kð1Þ ¼ k½hð1Þ %
eqn 1
kð2Þ ¼ k½hð2Þ %:
eqn 2
To first order,
!
X ð2Þ
ð1Þ @k !
ðhi & hi Þ !! :
kð2Þ & kð1Þ '
@hi !h
i
eqn 3
kð1Þ ¼ k½a; b%
eqn 4
The ith term in the summation is the contribution of differences in hi to the differences in k. The contribution can be
small either because the treatment has little effect on hi or
because k does not respond much to changes in hi. The
derivative in (eqn 3) is evaluated at the mean of h(1) and h(2).
In decomposing differences in the stochastic growth rate, we
encounter variables for which the derivatives in (eqn 3) cannot
be calculated. Fortunately, an alternative method for decomposition is available that does not rely on derivatives. It was
introduced by Kitagawa (1955) to explore the effects of agespecific death rates and of age distribution on crude death
rates. The method was later used by Keyfitz (1968, Section 7.4
cf. Keyfitz & Caswell 2005, Section 10.1) for more general calculations, decomposing differences in age distributions, dependency ratios and population growth rates into contributions
from the entire mortality and fertility schedules. Canudas
Romo (2003) summarizes more recent extensions of the
approach in demography.
Suppose that k depends on two variables, with values (a, b)
in treatment 1 and (A,B) in treatment 2. Thus,
ð2Þ
k
¼ k½A; B%:
The goal of the LTRE analysis is to decompose the treatment effect k[A,B] ) k[a,b] into contributions from A ) a
and B ) b. The Kitagawa–Keyfitz decomposition proceeds
by exchanging variables between the two treatments and
calculating k for all possible combinations. The effect of
A ) a, against the background of B, is k[A,B] ) k[a,B]. The
effect of A ) a, against the background of b, is k[A,b] )
k[a,b]. The overall contribution of A ) a is obtained by
averaging its effect against the two backgrounds:
CðA & aÞ ¼ ð1=2Þðk½A; B% & k½a; B%Þ
þð1=2Þðk½A; b% & k½a; b%Þ:
Similar decomposition analyses have been developed independently
in ecology and human demography. For example, Pollard’s (1988)
study of life expectancy used methods very similar to those I introduced for use on k. Horiuchi, Wilmoth & Pletcher (2008) develop a
method for continuous variables that is essentially identical to that
used by ecologists for regression LTRE calculations (Caswell 1996a,
2000). Canudas Romo (2003) reviews the human demographic
literature.
eqn 6
eqn 7
Similarly, the contribution of B ) b is
CðB & bÞ ¼ ð1=2Þðk½A; B% & k½A; b%Þ
þð1=2Þðk½a; B% & k½a; b%Þ:
1
eqn 5
eqn 8
eqn 9
If this appears familiar, it may be because this process of
averaging differences across different backgrounds is precisely analogous to the calculation of main effects in a
two-way ANOVA (e.g., Steel & Torrie 1960, Section 11.2,
p. 197).
In the next section, I will discuss stochastic population
growth and the components of a stochastic model, and how
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
326 H. Caswell
those are influenced by treatments. This will include a concept
of environment-specific sensitivities of the stochastic growth
rate; these sensitivities are the tool for the component of the
decomposition that uses derivatives. Then I will combine these
results to provide complete LTRE analyses for three cases,
depending on whether the treatments affect the stochastic
dynamics of the environment, the vital rate response or both. I
conclude with an example using data on a plant in a stochastic
fire environment.
STOCHASTIC POPULATION GROWTH
The familiar growth rate k in a constant environment depends
on the vital rates, which I write as a (p · 1) parameter vector h.
The population is projected according to
nðt þ 1Þ ¼ A½h%nðtÞ:
eqn 10
The growth rate is the dominant eigenvalue k of A[h], and
the sensitivity of k to the parameter vector h is
dk
d vec A
¼ ðwT ) vT Þ
:
T
dh
dhT
eqn 11
where the superscript ‘T’ denotes the transpose.
The difference in k between two treatments, k(2) ) k(1), is
determined by the difference between the parameters,
h(2) ) h(1). To first order,
! #
"
dk !!
ðhð2Þ & hð1Þ Þ:
eqn 12
kð2Þ & kð1Þ '
dhT !!h
Note that the dimensions of the terms on the right-hand
side of eqn 12, which are (1 · p) and (p · 1), imply that
k(2) ) k(1) is a scalar, as it should be.
The contributions of the parameters can be written as a
(p · 1) vector C(h), as
! #
"
dk !! T ð2Þ
CðhÞ ¼
*ðh & hð1Þ Þ;
eqn 13
dhT !!h
where s denotes the Hadamard, or element-by-element,
product.
A stochastic model contains two components: a model for
the dynamics of the environment and a model for the response
of the vital rates to the environment (Cohen 1979; Tuljapurkar
1990; Caswell 2001). By contrast, growth in a time-invariant
environment is completely determined by a single set of vital
rates. This difference (Fig. 1) adds an extra layer of complexity
to stochastic LTRE calculations. The differences in log ks
between two treatments is partly due to differences in the environmental dynamics and partly to differences in the vital rates
within each environmental state.
In this paper, I consider the most common case in the ecological literature, in which the environment is described by a
finite-state Markov chain. Examples include years with or
without fire (Silva et al. 1991), years since fire (Caswell & Kaye
2001), years with early or late floods, or with high or low precipitation (Smith, Caswell & Mettler-Cherry 2005) and years
with good or poor sea ice conditions (Hunter et al. 2007;
Jenouvrier et al. 2009a). The Markovian environment case
also includes the situation where the environment is modelled
implicitly by selecting randomly from a set of empirically measured matrices (e.g., Bierzychudek 1982; Cohen, Christensen &
Goodyear 1983; Jenouvrier et al. 2009b). Let u(t) be the state
of the environment at time t. The environmental dynamics are
determined by the Markov chain transition matrix P, where pij
P [u(t+1) ¼ i|u(t) ¼ j ].
To make this more concrete, consider an environment
defined by the time since the last fire (0,1,…, K ) 1 years). The
matrix P contains the probabilities of transition among these
states; it reflects the processes determining fire as a function of
the time since the last fire.
The second part of the model is the response of the vital rates
to the environment. Let h be a vector of parameters that determine the projection matrix A. These could be the matrix entries
or some set of lower level parameters. The vectors h1,…, hK
correspond to environmental states 1,…, K. I will write the
entire set of vital rates as
H ¼ fh1 ; . . . ; hK g:
eqn 14
In our example, h1 would contain the vital rates in
the year of a fire, h2 the vital rates in the year after a fire,
etc.
Given the usual assumptions about the ergodicity of the
environmental dynamics (Tuljapurkar 1990) and of the matrices generated by H, the stochastic growth rate is
logks ½P;H% ¼ lim
1
T!1 T
logkA½hðT & 1Þ%+++A½hð0Þ%n0 k
eqn 15
(a)
(b)
Fig. 1. The determination of the stochastic
growth rate log ks in time-invariant (a) and
stochastic (b) models. The deterministic
growth rate is defined by a set of vital rates.
The stochastic growth rate requires a model
for the stochastic dynamics of the environment and a function giving the response of
the vital rates to the state of the environment.
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
Stochastic LTRE analysis 327
I have written log ks as an explicit function of P and H to
emphasize that it depends on both the environment and the
vital rate responses.
ENVIRONMENT-SPECIFIC SENSITIVITIES
The sensitivity of log ks to the vital rates was given by Tuljapurkar (1990). For the LTRE analysis, we require the derivatives of log ks with respect to the parameters in each state of
the environment; i.e. to each of the vectors hi in H. These environment-specific sensitivities were given by Caswell (2005),
and independently by Horvitz, Tuljapurkar & Pascarella
(2005), and have been applied by Gervais, Hunter &
Anthony (2006); Aberg et al. (2009) and Svensson, Pavia &
Aberg (2009). The derivative of log ks with respect to the vital
rate vector in environment i is
!
T&1
d log ks !!
1X
Jt ½wðtÞT ) vðt þ 1ÞT % d vec A½hðtÞ%
¼ lim
:
T !
T!1 T
Rt vT ðt þ 1Þwðt þ 1Þ
dhT
dh
u¼i
t¼0
eqn 16
This is the stochastic analogue of the expression (eqn 11);
the vectors w(t) and v(t) are the stochastic analogues of the
right and left eigenvectors of a deterministic model, and Rt
is the growth of total population size from t to t + 1. For
a step-by-step algorithm for the calculation, see Caswell
(2001, Section 14.4). To make the sensitivity dependent on
the environment , Jt is an indicator variable, defined as
$
1 uðtÞ ¼ i;
eqn 17
Jt ¼
0 uðtÞ 6¼ i:
If the parameters h consist of the elements of A, then
d vec A/dhT ¼ I, where I is the identity matrix. If h contains
lower level parameters, then d vec A/dhT contains the derivatives of A with respect to these parameters. Equation 16
uses matrix calculus notation (Magnus & Neudecker 1988)
in which ) denotes the Kronecker product, and the ‘vec’
operator turns a matrix into a vector by stacking its
columns one above the other. In this notation, the derivative d log ks/dhT is a (1·p) vector whose jth entry is d log
ks/dhj|u¼i. For ecological introductions to this notation, see
Caswell (2007, 2008, 2009a,b).
and P(2)) and the differences in the vital rate responses (captured in H(1) and H(2)). As log ks is calculated from eqn 15 by
simulation, it cannot be differentiated with respect to P; so, we
will use the Kitagawa–Keyfitz decomposition for the environmental dynamics contribution, and environment-specific
derivatives (eqn 16) for the vital rate response contributions.
I will develop the decomposition analysis in three cases: the
case where only the vital rate responses differ, the case where
only the environmental dynamics differ, and finally the case
where both differ.
CASE 1: DIFFERENT VITAL RATE RESPONSES,
IDENTICAL ENVIRONMENTAL DYNAMICS
Consider two treatments that affect the vital rate responses but
not the environmental dynamics. For example, one might want
to compare low and high fertility sites subjected to a common
fire frequency. The transition matrix P is identical in the two
sites, but the vital rates differ. The stochastic growth rates are
ð1Þ
log kð1Þ
s ¼ log ks ½P; H %
eqn 19
log kð2Þ
s
eqn 20
ð2Þ
¼ log ks ½P; H %:
The difference in log ks is composed of contributions
from vital rate differences in each state of the environment. To first order,
! #
K "
X
@ log ks !!
ð2Þ
ð1Þ
ð1Þ
log kð2Þ
ðhi & hi Þ
&
log
k
'
eqn 21
s
s
@hT !u¼i
i¼1
where the derivatives are environment-specific sensitivities
(eqn 16), and are evaluated at the mean of H(1) and H(2).
The ith term of the summation in eqn 21 is the contribution of differences in the ith environment. These can be
written as contribution vectors
! #T
"
@ log ks !!
ð2Þ
ð1Þ
Cðhi Þ ¼
* ðhi & hi Þ; i ¼ 1; . . . ; K eqn 22
@hT !u¼i
where s denotes the element-by-element, or Hadamard,
product.
CASE 2: IDENTICAL VITAL RATE RESPONSES,
DIFFERING ENVIRONMENTAL DYNAMICS
LTRE analysis for log ks
Suppose now that we have two treatments, and want to
decompose the difference,
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
log kð2Þ
s & log ks ¼ log ks ½P ; H % & log ks ½P ; H %
eqn 18
into contributions. This difference compares growth in
treatment 2 with growth in treatment 1. Treatment 1, the
reference treatment, could be a control in a manipulative
experiment, or some other specific condition of interest
(as in the example to be considered below), or an average
over treatments in a factorial experiment.
The treatment effect on log ks in eqn 18 depends on both
the differences in environmental dynamics (captured in P(1)
Now consider two treatments that affect the environmental
dynamics (given by P(1) and P(2)) but not the vital rate
responses. For example, a comparison of population growth
before and after implementing a fire control strategy that
changes the frequency of fire, but has no effect on the vital rates
in each environmental state. The stochastic growth rates are
ð1Þ
log kð1Þ
s ¼ log ks ½P ; H%
eqn 23
ð2Þ
log kð2Þ
s ¼ log ks ½P ; H%:
eqn 24
The matrices P(1) and P(2) may differ in their long-term frequencies of environmental states. Those long-term frequencies
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
328 H. Caswell
are given by the stationary distributions, i.e. the right eigenvector p corresponding to the dominant eigenvalue of P (which
always equals 1), scaled so that p sums to 1. The same frequency of environmental states, however, can be obtained
from processes with different autocorrelation patterns, from
negative autocorrelation (where states tend to alternate) to
positive autocorrelation (characterized by long runs of the
same state; for an example, see Caswell & Kaye 2001, fig. 2).
So, P(1) and P(2) may differ in their stationary distributions,
autocorrelation patterns or both. To separate these, we construct a Markov chain with the same stationary distribution p
as P, but in which successive environmental states are independent, and hence there is no autocorrelation. This chain has the
transition matrix
Q ¼ peT
eqn 25
where e is a vector of ones. As the next state is independent of the previous state, and the same matrix is applied
at each time, this process is called ‘independent and identically distributed’ (i.i.d.).
The contribution to log kð2Þ
& log kð1Þ
s
s of differences in P is
ð2Þ
ð1Þ
CðPÞ ¼ log ks ½P ; H% & log ks ½P ; H%:
eqn 26
The contribution of the difference in the i.i.d. part of the
environment is
CðQÞ ¼ log ks ½Qð2Þ ; H% & log ks ½Qð1Þ ; H%:
eqn 27
The contribution of differences in environmental autocorrelation, denoted by C(R), is obtained by subtraction:
CðRÞ ¼ CðPÞ & CðQÞ:
eqn 28
CðRÞ ¼ CðPÞ & CðQÞ
Each of C(P), C(Q) andC(R) is a scalar.
2. Write the contributions of the vital rate differences using
the Kitagawa–Keyfitz method
CðHÞ ¼ ð1=2Þflog ks ½Pð2Þ ; Hð2Þ % & log ks ½Pð2Þ ; Hð1Þ %g
þ ð1=2Þflog ks ½Pð1Þ ; Hð2Þ % & log ks ½Pð1Þ ; Hð1Þ %g
eqn 34
C(H) is a scalar, integrating the effects of differences in
all of the vital rate responses. It is decomposed further in
the next step.
3. Use the environment-specific derivatives of log ks to
decompose each term in eqn 34 into contributions from the
vital rates in each environment, using eqn 22
! !T
! !!
@ logks ½Pð2Þ ; H%
ð2Þ
ð1Þ
*ðhi &hi Þ
Cðhi Þ¼ð1=2Þ
!
!
@hT
u¼i
! !T
eqn 35
ð1Þ ! !
@ logks ½P ; H%!
ð2Þ
ð1Þ
*ðhi &hi Þ;
þð1=2Þ
!
T
!
@h
u¼i
i¼1;...;K
! the mean of the
where the derivatives are evaluated at H,
vital rates in the two treatments being compared. C(hi) is
a (p · 1) vector, whose entries give the contributions to
the differences in log ks from each of the vital rates in
environment i.
These calculations are easily implemented by writing subroutines to calculate log ks and the environment-specific sensitivities given a transition matrix and a set of parameters. The
accuracy of the approximations involved can be checked by
comparing
CASE 3: DIFFERENCES IN BOTH ENVIRONMENTAL
Finally, consider two treatments that differ in both the environmental dynamics (P(1) and P(2)) and the vital rate responses
(H(1) and H(2)). The stochastic growth rates are
ð1Þ
ð1Þ
log kð1Þ
s ¼ log ks ½P ; H %
eqn 29
ð2Þ
ð2Þ
log kð2Þ
s ¼ log ks ½P ; H %:
eqn 30
& log kð1Þ
Our goal is to decompose log kð2Þ
s
s into contributions from the differences in the stationary environmental
frequencies (C(Q)), in the autocorrelation pattern (C(R)),
and in the vital rates in each environmental state
(C(h1),…, C(hK)). The decomposition analysis proceeds in
three steps.
1. Write the contributions of the environmental differences
using the Kitagawa–Keyfitz method
CðPÞ ¼ 0:5ðlog ks ½Pð2Þ ; Hð2Þ % & log ks ½Pð1Þ ; Hð2Þ %
eqn 31
CðQÞ ¼ 0:5ðlog ks ½Qð2Þ ; Hð2Þ % & log ks ½Qð1Þ ; Hð2Þ %
eqn 32
þ log ks ½Qð2Þ ; Hð1Þ % & log ks ½Qð1Þ ; Hð1Þ %Þ
?
ð1Þ
log kð2Þ
s & log ks ' CðQÞ þ CðRÞ þ
DYNAMICS AND VITAL RATES
þ log ks ½Pð2Þ ; Hð1Þ % & log ks ½Pð1Þ ; Hð1Þ %Þ
eqn 33
K
X
i¼1
Cðhi Þ:
eqn 36
An example
In this section, I present a stochastic LTRE analysis for an
endangered plant, Lomatium bradshawii, in a stochastic fire
environment (Caswell & Kaye 2001). Lomatium bradshawii
(Apiaceae) is a polycarpic herbaceous perennial plant. It exists
in only a few isolated populations in prairies of Oregon and
Washington. These habitats were, until recent times, subject to
natural and anthropogenic fires, to which L. bradshawii seems
to have adapted. Fires increase plant size and seedling recruitment, but the effect fades within a few years. Populations in
recently burned areas have higher growth rates and lower
probabilities of extinction than unburned populations. For
more information, see Pendergrass et al. (1999), Caswell &
Kaye (2001), Kaye et al. (2001) and Kaye & Pyke (2003).
A stochastic demographic model for L. bradshawii was
developed by Caswell & Kaye (2001), based on data from an
experimental burning study. Individuals were classified into six
stages based on size and reproductive status: yearlings, small
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
Stochastic LTRE analysis 329
THE STOCHASTIC FIRE ENVIRONMENT
The model for environmental dynamics is based on a two-state
Markov chain for fires (each year is either with fire or without
fire). This generates a four-state Markov chain for the environmental states (0, 1, 2 and 3 or more years post-fire). Let f be the
long-term frequency of fire, and q the temporal autocorrelation coefficient of the fire process (the norm of q determines
the rate of decay of correlation as time increases, the sign of q
determines whether the correlation is of one sign, or oscillates).
In the two-state fire model, the probability of fire in year t + 1
if there was no fire in year t is q ¼ f(1 ) q). The probability of
a fire if there was a fire in year t is p¼q+q (see Caswell 2001,
Section 14.1). The resulting transition matrix for the four environmental states is
0
1
p
q
q
q
B1 & p
0
0
0 C
C:
P¼B
eqn 37
@ 0
1&q
0
0 A
0
0
1&q 1&q
Ifq<0, f must satisfy
Table 1. The population growth rate k calculated from the
environment-specific matrices A[hi] for Lomatium bradshawii (from
Caswell & Kaye 2001)
Years
post-fire
Fisher
Butte k
Rose
Prairie k
0
1
2
‡3
1.020
0.984
0.662
0.869
1.155
1.118
0.483
0.906
&q
1
,f,
1&q
1&q
eqn 38
in order to keep probabilities bounded between 0 and 1
(for details, see Caswell & Kaye 2001). Note that even if
the fire process is i.i.d., so that q ¼ 0, the environmental
process given by eqn (37) is not i.i.d.
LTRE ANALYSIS
The two sites represent the treatments in this study. There is no
information on differences in fire dynamics at the two sites, so
Caswell & Kaye (2001) studied the response of log ks to the
frequency and autocorrelation of fires. Here, I use stochastic
LTRE analysis to explore the differences in log ks resulting
from various hypothetical scenarios of environmental differences. I will use the matrix entries as the vital rates h, there
being no natural lower level parameterization in this model.
Matlab code for the calculations is available in Appendices
S1–S5.
The stochastic growth rate log ks increases with fire frequency for both species. The RP site has a growth advantage,
with log kðRPÞ
> log kðFBÞ
at all fire frequencies (Fig. 2). The
s
s
RP advantage, measured by log ksðRPÞ & log kðRPÞ
increases
s
from c. 0.02 when f ¼ 0 to c. 0.13 when f ¼ 1.
We begin by considering Case 1, and examine the contributions of differences in the vital rates within each environmental
state, as a function of fire frequency. At each fire frequency f, I
calculated the environment-specific sensitivities of log ks,
using eqn 16. I used these sensitivities to calculate the contributions of each of the vital rates, using eqn 22. Finally, I integrated the contributions by summing C(hi) for each
environmental state. Figure 3 plots this sum as a function of
the fire frequency f, with autocorrelation fixed at q¼0.
At f ' 0, the environment is overwhelmingly in state 4
(three or more years post-fire), and the treatment effect on log
0.16
Stochastic growth rate difference
and large vegetative plants, and small, medium and large
reproductive plants. The environment was classified into four
states defined by the time since the most recent fire: the year of
a fire and 1, 2 and 3+years post-fire, and vital rates were estimated in each of these environmental states. The matrices are
given in Caswell & Kaye (2001).
Populations were studied in two sites: Fisher Butte (FB) and
Rose Prairie (RP) in western Oregon. The two sites differed in
quality for L. bradshawii, with RP superior to FB. Population
growth rates were generally higher at RP than at FP (Table 1),
and the stochastic growth rate was higher in RP than in FB at
any fire frequency. The critical fire frequency required to maintain L. bradshawii populations was about 0.8–0.9 at FB, but
only 0.4–0.5 at RP.
The causes of the differences between the sites are not
known. They had similar soils and hydrology but differed in
population characteristics and associated vegetation (Pendergrass et al. 1999). Prior to the initiation of burning treatments
in 1988, plants at Rose Prairie were smaller and produced
fewer flowering structures than at Fisher Butte, although fruit
production per plant was similar at the two sites (Pendergrass
et al. 1999; Kaye et al. 2001).
0.14
Actual
Calculated
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
Fire frequency
0.8
1
Fig. 2. The difference in the stochastic growth rate log ks between
the Rose Prairie (RP) and Fisher Butte (FB) populations of Lomatium bradshawii, as a function of the frequency of fire. Autocorrelation of the fire process q ¼ 0. Growth rates calculated by Monte
Carlo simulation with T ¼ 10 000 iterations.
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
330 H. Caswell
Figure 4 partitions these contributions differently, by
displaying the contributions of each matrix element
summed over all environmental states. At low fire frequencies, there are large positive contributions from an
RP advantage in transitions into the small reproductive
plant stage. At higher fire frequencies, the contributions
are increasingly due to an RP advantage in fertility, especially of medium-sized reproductive plants.
For a more complete example (Case 3), suppose that the two
sites differed in their environmental dynamics. Suppose that
the fire frequencies, autocorrelations and resulting stochastic
growth rates were
0.2
u=1
2
3
4
Contribution
0.15
0.1
0.05
0
−0.05
0
0.2
0.4
0.6
Fire frequency
0.8
1
f
q
log ks
Fig. 3. The contributions of the matrices A[hi] in each environmental
state to difference in the stochastic growth rate log ks between the
Rose Prairie (RP) and Fisher Butte (FB) populations of Lomatium
bradshawii, as a function of the frequency of fire. Autocorrelation of
the fire process q¼0. Growth rates and sensitivities calculated by
Monte Carlo simulation with T¼10 000 iterations.
RP
0.5
)0.5
)0.043
0.7
0.5
0.081
In FB, years with and without fires tend to alternate; in RP,
there is a tendency for runs of years with or without fires.
To decompose this treatment effect, I first constructed the
Markov chain transition matrices from eqn 37, and then calculated the stationary distributions p(RP) and p(FB) as eigenvectors of P. Next, for each site, I generated the i.i.d. transition
matrix Q from eqn 25, and then computed the contributions
C(P) from eqn 31, C(Q) from eqn 32, and C(R) from eqn 33.
Then I calculated the environment-specific sensitivities of log
ks from eqn 16, for both P(RP) and P(FB), and used these to
calculate the contributions C(hi) of the vital rates in each environmental state, using eqn 35. Finally, I summed the C(hi) to
obtain the integrated effect of all vital rate differences in each
environment.
ks is completely due to differences in A[h4]. At the other
extreme, with f ' 1, the environment is almost always in state
1 (the year of a fire) and the treatment effect is due to differences in A[h1]. At intermediate fire frequencies, there are positive contributions from vital rate differences in states 1 and 2,
and negative contributions from differences in state 4.
The decomposition (eqn 21) is approximate because it
neglects nonlinearity in the response of log ks to h. Its accuracy is good in this case, despite the large differences in log ks,
as shown by the comparison of the actual values and those
computed from the sum of the contributions (Fig. 2).
f = 0.01
FB
0.3
0.05
0.1
0
0
−0.05
−0.1
1
2
1
3
4
Row
5
5 6
3 4
2
1
Column
6
2
3
4
5
5 6
3 4
1 2
6
0.5
0.8
0.1
0.2
0
0
−0.1
−0.2
1
2
1
3
4
5
6
5 6
3 4
2
1
2
3
4
5
6
5 6
3 4
2
1
Fig. 4. Matrix plots of the contributions of
the elements of the Ai, integrated over all
environmental states, to the difference in the
stochastic growth rate log ks between the
Rose Prairie (RP) and Fisher Butte (FB)
populations of Lomatium bradshawii.
Results are shown for fire frequencies from f
0.01 to 0.8. Autocorrelation of the fire process q ¼ 0. Growth rates and sensitivities
calculated by Monte Carlo simulation with
T¼10 000 iterations.
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
Stochastic LTRE analysis 331
f
q
log ks
FB
RP
0.9
0.0
0.027
0.1
0.0
)0.113
In this case, log kðFBÞ
> log kðRPÞ
, despite the general
s
s
advantage in vital rates of RP over FB in most environmental
states. Figure 6 presents the contributions C(Q), C(R)
and C(hi), and shows how the contributions of the vital rate
differences are, in this case, overwhelmed by the RP disadvantage due to the stationary distribution of the environment.
The sum of the contributions in Fig. 6 is )0.1326, while the
actual difference in log ks is )0.1395 (an accuracy of 95%,
even with a very large difference in growth rate). This list of
hypothetical examples could be extended, but is sufficient to
show how the theory developed here provides contributions of
environment-specific vital rates integrated over stages (Fig. 3),
of stage-specific rates integrated over environments (Fig. 4),
and of the frequency and autocorrelation patterns of the environment (Figs 5 and 6).
Discussion
The new method presented here provides a general framework
for decomposing treatment effects on the stochastic growth
rate in Markovian environments. It is a direct generalization of
the familiar LTRE approaches for time-invariant and periodic
models. Comparative studies of the stochastic growth rate
require additional data on the stochastic dynamics of the environment, beyond that needed for time-invariant models
(Fig. 1). Many stochastic studies present conditional results;
for example, the study of L. bradshawii provides log ks as a
function of f, q and H but does not estimate the value of log ks
actually exhibited in either of the two sites. To do so would
require long-term data on the stochastic environment, which is
hard to come by. However, such information may possibly be
extracted from historical data (e.g. Smith, Caswell & MettlerCherry 2005; Lawler et al. 2009) or projected from climate
models (Hunter et al. 2007; Jenouvrier et al. 2009a).
0.1
Contribution
0.08
0.06
0.04
0.02
0
−0.02
Q
R
A_1
A_2
A_3
A_4
Fig. 5. The contributions of the independent and identically distributed component of the environment (Q), the autocorrelated component of the environment (R) and the projection matrix entries in each
environmental state A1,…, A4) to the difference in the stochastic
growth rate log ks between the Rose Prairie (RP) and Fisher Butte
(FB) populations of Lomatium bradshawii. Calculations based on the
(hypothetical) situation when the frequency of fires is 0.5 for FB and
0.7 for RP and the autocorrelation of the fire process is q ¼ )0.5 for
FB and q ¼ 0.5 for RP. Growth rates and sensitivities calculated by
Monte Carlo simulation with T ¼ 10 000 iterations. The sum of these
contributions approximates the difference in log ks between treatments.
Such data are only beginning to be collected, but there is
every reason to expect more of them in the future. For example, Svensson, Pavia & Aberg (2009) analyzed the seaweed
Ascophyllum nodosum, comparing log ks between the coast of
Sweden and the Isle of Man. For the Swedish site, but not the
Isle of Man site, they classified years in terms of ice conditions,
with measured frequencies of occurrence over a 12-year period.
0.1
0.05
0
Contribution
Figure 5 shows these contributions. Most of the growth rate
advantage of the RP site can be attributed to an RP advantage
in A[h1] and A[h2] (the year of a fire and the year immediately
following a fire). The difference in the long-term frequency of
environmental states, and the differences in autocorrelation
patterns, make relatively little contribution.
The accuracy of the approximations involved in the LTRE
analysis is good. The sum of the contributions in Fig. 5 is
0.1192, while the actual difference in log ks is 0.1219 (an accuracy of 98%).
Suppose now that, for some reason, a fire prevention program in the RP site reduced the fire frequency to f ¼ 0.1 (well
below the critical threshold for persistence), but a fire management program increased the fire frequency in the FP site to f
0.9.
−0.05
−0.1
−0.15
−0.2
−0.25
Q
R
A_1
A_2
A_3
A_4
Fig. 6. The contributions of the independent and identically distributed component of the environment (Q), the autocorrelated component of the environment (R) and the matrix entries in each
environmental state A1,…, A4) to the difference in the stochastic
growth rate log ks between the Rose Prairie (RP) and Fisher Butte
(FB) populations of Lomatium bradshawii. Calculations based on the
(hypothetical) situation when the frequency of fires is 0.9 for FB and
0.1 for RP and the autocorrelation of the fire process is q ¼ 0 for both
populations. Growth rates and sensitivities calculated by Monte
Carlo simulation with T ¼ 10 000 iterations. The sum of these contributions approximates the difference in log ks between treatments.
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333
332 H. Caswell
If they did the same for the Isle of Man, an LTRE analysis
using the methods presented here would be immediately possible. An interesting possibility is provided by the studies of
Gervais, Hunter & Anthony (2006) on the burrowing owl
(Athene cunicularia) and Henden et al. (2009) on the Arctic fox
(Vulpes lagopus). Both of these studies reported stochastic
growth rates in an environment defined by stages of the cyclic
dynamics of small mammal prey of the focal species. Since prey
populations exhibit their own kinds of dynamics, this raises the
possibility of linking those dynamics to those of the species
relying on them.
The method presented here analyses Markovian environments in which the environmental states have an interpretation (years since fire, flood or rainfall conditions, etc.). Some
stochastic studies, however, randomly select matrices from a
series collected over time (e.g. the early study of Bierzychudek 1982 based on two yearly matrices, or the recent study
by Jenouvrier et al. 2009b based on 44 years of matrices
on emperor penguins). Although these models are actually
Markov chains (see Caswell 2001, Section 14.5, for some
alternate Markov chain specifications for such studies), if
years are simply a random sample of environmental variation, then it is of little interest to know the contribution of
vital rate differences in, say, 1988 compared with 1989 or
1987. In such models, the mean and variance of the vital
rates may be of more interest. Davison et al. (2010), drawing
on the stochastic elasticity results of Tuljapurkar, Horvitz &
Pascarella (2003), have presented an approach to LTRE
analysis in terms of the contributions of differences in the
mean and the variance of the vital rates. That method nicely
complements the approach presented here.
The analysis of L. bradshawii presented as an example here
reveals an understandable pattern of environment-specific contributions that change with the frequency of the environmental
states. In the scenarios explored here, even relatively large differences in environmental autocorrelation made small contributions to treatment effects on log ks. This is not surprising,
given the small impact of changes in autocorrelation on the
stochastic growth rate in this model (Caswell & Kaye 2001). It
is, however, not guaranteed. Given the proper interaction
between environmental states and the stage structure, autocorrelation can have dramatic impacts on the growth rate
(Caswell 2001, Example 14.1). How often this happens in
nature will only be revealed by further comparative studies.
Acknowledgements
I thank Evan Cooch, Josh Goldstein, Michael Neubert, and Catherine Nosova
for discussions, and Gordon Fox and Hans de Kroon for helpful comments on
the manuscript. Supported by NSF Grant DEB-0816514, and funds from the
Ocean Life Institute and the WHOI Arctic Research Initiative.
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Received 13 August 2009; accepted 2 December 2009
Handling Editor: Hans de Kroon
Supporting Information
Additional Supporting Information may be found in the online version of this article:
Appendix S1. Matlab code for first Lomatium example (pdf file).
Appendix S2. Matlab code for first Lomatium example (m file).
Appendix S3. Matlab code for second Lomatium example (pdf file).
Appendix S4. Matlab code for second Lomatium example (m file).
Appendix S5. Matlab data file containing Lomatium example
matrices (mat file).
As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials may be
re-organized for online delivery, but are not copy-edited or typeset.
Technical support issues arising from supporting information (other
than missing files) should be addressed to the authors.
! 2010 The Author. Journal compilation ! 2010 British Ecological Society, Journal of Ecology, 98, 324–333