Spatio-temporal variation in Markov chain models of subtidal community succession
Post on 06-Jul-2016
R E P O R TSpatio-temporal variation in Markov chain models
of subtidal community succession
M. Forrest Hill,* Jon D. Witman
and Hal Caswell
*Institute of Theoretical
Dynamics, University of
California, One Shields Avenue,
Davis, CA 95616, USA.
Department of Ecology &
Evolutionary Biology, Brown
University, Box G-W Providence,
RI 02912, USA.
Biology Department MS 34,
Woods Hole Oceanographic
Institute, Woods Hole,
MA 02543, USA.
Fax: +530 752 7297
In this paper we ask whether succession in a rocky subtidal community varies in space
and time, and if so how much affect that variation has on predictions of community
dynamics and structure. We describe succession by Markov chain models based on
observed frequencies of species replacements. We use loglinear analysis to detect and
quantify spatio-temporal variation in the transition matrices describing succession. The
analysis shows that space and time, but not their interaction, have highly significant
effects on transition probabilities. To explore the ecological importance of the spatio-
temporal variability detected in this analysis, we compare the equilibria and the transient
dynamics among three Markov chain models: a time-averaged model that includes the
effects of space on succession, a spatially averaged model that include the effects of time,
and a constant matrix that averages over the effects of space and time. All three models
predicted similar equilibrium composition and similar rates of convergence to
equilibrium, as measured by the damping ratio or the subdominant Lyapunov exponent.
The predicted equilibria from all three models were very similar to the observed
community structure. Thus, although spatial and temporal variation is statistically
significant, at least in this system this variation does not prevent homogeneous models
from predicting community structure.
Markov chains, loglinear analysis, environmental stochasticity, community ecology,
Succession, benthic invertebrates.
Ecology Letters (2002) 5: 665675
I N T R O D U C T I O N
Community development (succession, in the broad sense of
the word) is a process of species replacement. The
probabilities of replacement of one species by another
determine the rate and direction of succession. These
probabilities can be used to define a Markov chain, which
provides a dynamic framework from which to calculate the
rates, sequences, and patterns of succession (e.g. Waggoner
& Stephens 1970; Horn 1975; Usher 1979; McAuliffe 1988;
Yeaton & Bond 1991; Tanner et al. 1994; Gibson et al. 1997;
Hill 2000; Wootton 2001a,b). The most detailed of these
analyses, based on the most extensive data, are for marine
benthic communities, including coral reefs (Tanner et al.
1994; 1996), the rocky intertidal (Wootton 2001a,b), and
hard-substrate subtidal communities (Hill 2000 and the
Most Markov chain models of communities have
assumed that the transition probabilities are constant in
time and homogenous in space, which leads to easily
analysed linear time-invariant Markov chains. But trans-
ition probabilities are not really constant in time or
homogenous in space, and while constant transition
matrices may describe some sort of average environment,
several authors have doubted their adequacy as models of
community dynamics (e.g. Usher 1979; Li 1995; Logofet
et al. 1997; Childress et al. 1998). For instance, Usher
(1979) showed that successional rates varied temporally in
a Ghana termite community and argued that time-varying
Markov chains were required to model termite succes-
sional processes. Facelli & Pickett (1990) argued that
homogeneous Markov chains fail to account for historical
effects on transition probabilities and that only higher-
order Markov models, in which transitions depend on the
past as well as the current community state, can
adequately describe community dynamics, although they
confused temporal variation with higher-order Markov
Ecology Letters, (2002) 5: 665675
2002 Blackwell Science Ltd/CNRS
The existence of statistically significant spatial or tem-
poral variation begs the question of how important that
variation is to community dynamics. The null hypothesis in
such tests is that transition probabilities are exactly constant
over space or exactly invariant through time. Given enough
data, arbitrarily small amounts of variation will lead to
rejection of the null hypothesis. That variation, however,
could be so small as to be biologically irrelevant in terms of
the structure and dynamics of the community.
Here, we follow a rigorous statistical analysis of spatio-
temporal variation with an examination of the effects of
such variation on both asymptotic and transient community
dynamics. We do this by comparing models that include the
effects of spatial variability on succession but which average
over time, models that include temporal variability but
which average over space, and models that average over
both space and time. In the only previous comparison of
this type, Tanner et al. (1996) used simulations to compare
first-order, second-order and semi-Markov models of a
coral reef community and found that all models gave similar
results. They did not, however, perform statistical tests to
determine if transition probabilities among the models were
Studying the effects of environmental change on succes-
sion requires statistical methods to detect and quantify
spatio-temporal variation in transition probabilities. Caswell
(1988, 2001), drawing on the work of Bishop et al. (1975),
introduced the use of loglinear models for such tests in the
related context of detecting spatio-temporal variation in
demographic projection matrices. More recently, informa-
tiontheoretic methods for model selection based on the
Akaiki Information Criteria (AIC) have increased the power
of this approach (Caswell 2001). Loglinear models are the
more powerful descendants of the tests developed by
Anderson & Goodman (1957), to test the time-homogeneity
of Markov chains. Although the AndersonGoodman test
has been used on models of ecological communities (e.g.
Usher 1979; Tanner et al. 1994; Gibson et al. 1997; Wootton
2001a), the loglinear approach is more general, and can be
extended to more complex experimental designs (Caswell
To date, loglinear analysis has not been applied in its full
generality to the problem of characterizing spatio-temporal
variation in a Markov chain model. We do so here, focusing
on a subtidal community of sessile invertebrates (sponges,
sea anemones, polychaetes, bryozoans and ascidians) and
crustose coralline algae in the Gulf of Maine. Empirical
studies of subtidal communities have shown that species
diversity is controlled by is a complex network of biotic and
abiotic factors, such as predation, competition, disease,
sedimentation and current flow (Dayton et al. 1970; Sebens
1986; Scheibling & Hennigar 1997; Genovese & Witman
1999; Witman & Dayton 2001). These factors are known to
vary in space and time (Witman 1996), however, there is
little information about how this variation affects species
composition and community dynamics.
M A R K O V C H A I N M O D E L S
Markov chain succession models describe a community in
terms of a set of points (referred to here as patches), each of
which can be occupied by one of a set of species or functional
groups of species. The state of the community at time t is given
by a probability vector x(t), whose ith element xi is the
probability that a point is occupied by species i at time t. The
vector x(t) satisfies 0 xi 1 and Si xi 1. We supposethat the transition probabilities might vary in space and time.
The dynamics of the community in year t at location l are
xt 1 At ;l xt 1where A(t,l ) is a column-stochastic transition matrix descri-
bing successional dynamics for the time interval (t, t + 1) at
spatial location l. ( We use subscripts to denote matrix or
vector elements, and superscripts to distinguish matrices
specific to times, locations, or both.) The matrix entry at ;l ij
gives the probability that a patch is occupied by species i at
time t + 1, given that it was occupied by species j at time t. If
the transition matrix is homogeneous, so that A(t,l) A, andif A is primitive, then x(t) converges from any initial
condition to a structure proportional to the right eigenvec-
tor u corresponding to the dominant eigenvalue k of A.Thus u is a prediction of community structure in a constant
M E T H O D S
The data for our model were collected at yearly intervals
over a nine-year period from 1986 to 1994 on subtidal rock
walls at 3033 m depth on Ammen Rock Pinnacle, 100 km
east of Portsmouth, New Hampshire, on Cashes Ledge in
the Gulf of Maine, USA ( Leichter & Witman 1997). They
consist of a series of colour photos chronicling the spatial
distribution of sessile species on the rock wall substrate
through time. Nine permanently marked 0.25 m2 quadrats
(0.625 0.4 m), positioned horizontally along a 20 m spanof the rock wall habitat, were photographed each year with a
Nikonos camera mounted on a quadrapod frame (as in
Witman 1985). Quadrats were separated by a horizontal
distance of 15 m. We made 31.25 20 cm2 colour printsof the high resolution colour slides, and used these prints to
A total of 14 species, each occupying at least 0.5% of the
study area, were recorded in the quadrats. We grouped these
666 M. F. Hill, J. D. Witman and H. Caswell
2002 Blackwell Science Ltd/CNRS
species into 11 state categories (Table 1). Nine states
correspond to a single species, one corresponds to bare
rock (empty space), and one corresponds to a functional
group consisting of ascidians (Aplidium pallidum and Ascidia
callosa), bryozoans (Parasmittina jefferysi and Idmidronea
atlantica), and a polychaete (Spirorbis spirorbis). This functional
group was identified by an objective analysis showing a
high level of functional similarity among these species
Transition data were obtained by superimposing a
rectangular lattice of 600 points, at 1 cm intervals, onto
the quadrat photographs. This corresponds to approxi-
mately a 2 cm interval on the actual substrate ([(0.25 0.06)] 2.04). This scale is approximately equivalent to thesize of the smallest organisms in the community. We
consider each point as the centre of a patch with an area of
4 cm2. Since individuals of some subtidal species may be
larger than 4 cm2, a single individual can occupy more than
one patch. Many of the larger species, however, are
colonial sponges, which can be reduced in size by
predation, disease, or competitive overgrowth. This means
it is possible for a species that occupies four patches at
time t to occupy only three patches at time t + 1. Thus
each transition represents a change in the species state of a
given patch within a quadrat.
We recorded the species occupying each patch on each
quadrat in each year, for a total of approximately 42 000
points (the species occupying a few points were unidenti-
fiable from the photos; these points were excluded from the
To estimate the transition matrices, we construct a four-
way contingency table N of size 11 11 8 9, in
which patches are classified by their state (S ) at time t,
their fate (F ) at time t + 1, the time (T ) of the
observation, and the location (L) of the quadrat. The
entry nijtl in cell (ijtl ) of the table gives the number of
patches making the transition from state j to fate i at time
t in location l (Caswell 2001).
From N, we construct four sets of transition matrices:
1. A set of 72 space-time specific matrices,
A(tl ), t 1,...,8; l 1,...,9. The maximum likelihood esti-mates a^a
tl ij of the entries of A
(tl ) are
That is, a^atl ij is the number of individuals moving from state
j to state i at time t and location l, divided by the
total number of individuals starting in state j at time t and
2 A set of 9 spatial matrices, A(l ), l 1,...,9, in which theeffects of time are ignored. The maximum likelihood
estimates a^al ij of the entries of A
(l ) are
3 A set of 8 time-varying matrices, A(t ), t 1,...,8, in whichthe effects of space are ignored. The estimates a^a
tij of the
entries of A(t ) are
4 A set consisting of a single matrix, A, in which both
temporal and spatial effects are ignored. The entries a^aij of A
are estimated by
Table 1 Subtidal species identified in thenine quadrats located at 30 m depth on
Ammen Rock pinnacle in the Gulf of Maine.
Model states are identified in the column
State ID. The number of points counted per
state is shown in the right-hand column. The
state BR (x1) corresponds to bare rock. The
state FG (x8) is a group of 4 ecologically
equivalent species (Hill 2000)
xi Model States Species type State ID Number
1 Bare Rock BR 4491
2 Hymedesmia 1 Sponge HY 1 14875
3 Hymedesmia 2 Sponge HY 2 1226
4 Myxilla fimbriata Sponge MYX 4525
5 Mycale lingua Sponge MYC 3001
6 Metridium senile Sea anemone MET 1298
7 Urticina crassicornis Sea anemone URT 992
8 Aplidium pallidum Ascidian FG 3350
Ascidia callosa Ascidian
Parasmittina jeffreysi Bryozoan
Idmidronea atlantica Bryozoan
9 Crisia eburnea Bryozoan CRI 11074
10 Filograna implexa Polychaete FIL 2219
11 Coralline Algae Encrusting algae COR 875
Stochastic Markov chains of subtidal communities 667
2002 Blackwell Science Ltd/CNRS
Markov chain models like these assume that, while
transitions may vary in space or over time, they do not
respond to variation in species composition, at either the
quadrat level or the local, patch-neighbourhood level. Thus
the above methods of model parameterization do not
include the effects of local species interactions on
transition probabilities. To model these effects would
require a nonlinear Markov chain formulation (Hill 2000).
We are currently developing such models to investigate the
importance of local interactions on large-scale pattern
formation in the subtidal community. Here, however, we
are concerned with determining whether transition proba-
bilities vary significantly in space and time (for whatever
reason) and how this variability affects community deve-
S T A T I S T I C A L S I G N I F I C A N C E
O F S P A T I O - T E M P O R A L V A R I A T I O N
The transition probabilities in each of the matrices (2)(5)
characterize the rates and patterns of species replacement at
the place and time where the matrix was measured. Thus
spatio-temporal variability in succession appears as spatio-
temporal variation among transition matrices. Our first task
is to evaluate the statistical significance of this variation. We
do so using loglinear analysis (Bishop et al. 1975; Caswell
In loglinear analysis the logarithm of the number of
transitions, nijtl, from State j to Fate i at Time t in Location
l (i.e. the log of the cell frequencies in N ) is modelled as a
linear function of the effects of F, S, T, L, and their
interactions. We define models by listing the highest order
interactions included in the model (Caswell 2001). Because
the models are hierarchical, the inclusion of an interaction
implies the inclusion of all lower order interactions
involving those variables. For example a model that includes
the interaction STL (i.e. State Location Time) must alsoinclude the effects of S, T, L, ST, SL, and LT on the
transition frequencies in N.
In the present context, the response variable is Fate (i.e.
species transitions) and the explanatory variables are State,
Time and Location. Thus all models will include the F and
STL terms ( Fingleton 1984). The null hypothesis in tests for
differences among Markov chains is one of conditional
independence, i.e. that Fate at time t + 1, conditional on the
State at time t, is independent of time T and location L (i.e.
there are no FT and FL interaction terms). Thus the null
model is designated as FS, STL, which in mathematical
terms can be represented as
log nijtl u uFi uSj uT t uLl uFSij uST jt uSLjl uFi uTLtl uSTLjtl 6
where u represents the effect of all transitions on the cell
frequencies of N, uS( j ) represents the effect of the jth initial
state on cell frequencies, uFS( ij ) represents the effect of the
interaction of the jth initial state and the ith fate on cell
frequencies, etc. The parameters of the model are estimated
by maximum likelihood methods. See Caswell (2001) for a
general discussion of this methodology or Silva et al. (1991)
for a demographic example.
We fit the following series of loglinear models using the
Statistica software package (StatSoft, Inc. 2000) (model:
FS, STL: null model
FSL, STL: transition probabilities vary in space
FST, STL: transition probabilities vary in time
FSL, FST, STL: transition probabilities vary in space and
FSTL: space and time interact in determining probabilities
The effects of time and location on transition probabil-
ities are tested by comparing these models as shown in
Fig. 1. Beginning with the null hypothesis FS, STL at the top
of the figure, we include the location effect by adding the
term FSL (and hence FL, because the models are
hierarchical). The effect of these terms is evaluated by
comparing the likelihoods of the models FS, STL and FSL,
STL. If the improvement in likelihood (DG2) is sufficientlylarge (it has a v2 distribution with degrees of freedom equalto the difference (Ddf) between degrees of freedom of thetwo models) we reject the hypothesis that the FSL
interaction is zero. Similarly, we introduce time effects by
adding the terms FT and FST and evaluate their effects on
the likelihood by comparing the models FS, STL and FST,
We also test for the effects of location by adding the FSL
interaction to the model (FST, STL) that already includes the
time effects, and test the time effects by adding the FST
interaction to the model (FSL, STL) that already includes the
location effects. Finally, we test for the time locationinteraction by comparing the models FSL, FST, STL and
FSTL (Caswell 2001).
Adding parameters to a loglinear model will always produce
a better fit to the data. Thus choosing the best model, i.e.
668 M. F. Hill, J. D. Witman and H. Caswell
2002 Blackwell Science Ltd/CNRS
the model that best approximates the mechanisms gener-
ating the transition data, requires a measure that includes
both goodness-of-fit and parsimony. One such measure,
with rich theoretical support, is the Akaike Information
Criteria (AIC; Burnham and Anderson 1998). For loglinear
models AIC can be calculated as
AIC G 2 2df 7where G2 is the goodness of fit likelihood statistic and df is
the degrees of freedom of the model (Christensen 1990).
The loglinear model that minimizes AIC is the best
compromise between parsimony and goodness of fit. Note
that likelihood ratio tests using DG2 compare nested models.In contrast, AIC can also compare non-nested models and
is based on an optimization criteria (Burnham & Anderson
For ease of interpretation, it is helpful to scale AIC values
relative to the best model (DAIC ). In general, DAIC 2implies that the model, while not as good as the best model,
has substantial support and should be investigated further.
Alternatively, if DAIC 10, the model has no support andshould be excluded form further investigation (Burnham &
A graphic representation of the tests for the time and location
effects, and the AIC values, is shown in Fig. 1. The effect of
Location L is highly significant whether it is evaluated with
the effect of time excluded or included in the analysis. The
effect of Time T is also highly significant whether location is
excluded or included in the analysis. The interaction of time
and location, however, is not significant (P 0.9998). Thisimplies that the temporal variation produces parallel effects
on the transition matrices at all the locations.
Comparison of AIC values shows that the best model is
FSL, FST, STL (AIC )6491.5). Thus both location andtime are included in the best model of patch transitions in
FT,FSTG2 = 4356.5df = 770
p < 0.0001
FL,FSLG2 = 7248.1df = 880p < 0.0001
FL, FSLG2 = 7011.6df = 880p < 0.0001
FT, FSTG2 = 4119.5df = 770p < 0.0001
Time x Location Interaction
FTL, FSTLG2 = 5,828.5df = 6160p = 0.9998
FS, STLG2 = 17,196.1
df = 7810AIC = 1576.1
FST, FSL, STLG2 = 5,828.5
df = 6160AIC = -6491.5
FST, STLG2 = 12,839.6
df = 7040AIC = -1240.4
FSTLG2 = 0df = 0
AIC = 0
FSL, STLG2 = 9,948.0
df = 6930AIC = -3912.0
Figure 1 Tests for the effects of time (T ),location (L), and their interaction in a
loglinear analysis of the subtidal transition
data. Each box designates a particular model
and shows its goodness-of-fit G 2 statistic, its
degrees of freedom, and its AIC value. The
top box is the null model (Fate depends only
on State). The lower boxes represent models
that include higher-order interactions
between Fate and T, L, or both. Terms
added to each model, along with the
corresponding changes in G 2 (DG 2) anddegrees of freedom, are shown along the
Stochastic Markov chains of subtidal communities 669
2002 Blackwell Science Ltd/CNRS
rocky subtidal communities. As DAIC for the saturatedmodel (FSLT ) is over 6000, an interaction effect of time location on patch transition probabilities is not supported
and is excluded from further consideration. Note that the
model FSL, STL has a much lower AIC value than FST,
STL, indicating that location has a larger effect on transition
probabilities than does time.
E C O L O G I C A L C O N S E Q U E N C E S
O F S P A T I O - T E M P O R A L V A R I A T I O N
Spatial and temporal variation are both statistically signifi-
cant. This fact alone, however, does not reveal how
biologically important that variation is, nor what its
consequences are for community structure and dynamics.
To examine this, we must compare models that include
spatial and or temporal variation with models that do not.Here, we compare three Markov chain models: a time-
averaged model in which transitions vary in space, a spatially
averaged model in which transitions vary over time, and a
homogeneous model averaged over both time and space.
If transitions varied in space but not in time, succession
at location l would be described by the time-invariant
matrix A(l ) specific to that location. The asymptotic
community structure would be given by the eigenvector
u(l ) corresponding to the dominant eigenvalue of A(l ),
normalized to sum to 1. Community structure would vary
from location to location. A typical community would be
described by the mean vector uu L)1 Sl u(l ), and vari-ability among locations would be described the variation
among the u(l ).
If transitions varied in time but not in space, the
community would be described by a spatially homogeneous
but time-varying model, of which the observed time-specific
matrices A(t ), t 1,...,8, would represent a sample. Toexplore the effects of this variation, we constructed a
Markov chain in a random environment (Cogburn 1986)
based on these matrices. In this model, one of the A(t ) is
chosen independently, with probability 1 8, at each timestep. Figure 2 shows a realization of this process; note that
each species settles into a characteristic pattern of variation.
In our case, the matrices A(t ) are all primitive (Caswell
2001), and a direct calculation shows that any product of
two or more of them is strictly positive. Thus, they form
an ergodic set, and the community structure vector x(t)
converges to a fixed stationary distribution, independent
of the initial condition (Cohen 1976b; Tuljapurkar 1990;
Caswell 2001). This stationary distribution describes both
the mean and the variability in community composition
generated by the temporal variation. We estimated the
stationary distribution by simulation, discarding 200
iterations to eliminate transients and then treating the
following 10 000 iterations as a sample of the stationary
distribution. We used the average over this sample to
characterize the expected community structure in the
Finally, suppose we were to ignore the advice of the
loglinear model, eliminate both spatial and temporal
variation, and use the constant matrix A to describe the
community. The asymptotic community structure would be
given by the eigenvector u corresponding to the dominant
eigenvalue of A, normalized to sum to 1.
Observed and predicted community composition
Although they are statistically different, the models A(l ), A(t ),
and A yield very similar expectations of asymptotic
community structure (Fig. 3). The main difference is that
the homogenous model, A, predicts a slightly higher mean
abundance of the bryozoan Crisia and a slightly lower mean
abundance of the polychaete Filograna than the non-
homogenous models (A(l ) and A(t )). From a practical
viewpoint, however, this difference is biologically trivial.
The asymptotic structure predicted by the three models
agrees well with the observed community composition. To
quantify this, we used data from 4 of the 9 quadrats to
estimate the space-varying, time-varying, and homogeneous
Markov chains, and compared their equilibrium predictions
(as described above) with the observed community
structure in the remaining 5 quadrats. We repeated this
process for all 126 possible ways of selecting 4 out of 9
quadrats. To measure the agreement we calculated the
product-moment correlation coefficient rp between the
predicted equilibria and the observed structures (Table 2).
Even when parameters are estimated from sharply reduced
sample sizes, the equilibrium predictions of all three
models explain about 80% of the variability in abundance
100 125 150 175 200
BRHY 1 MYXMYCCRI
Figure 2 An example of community dynamics in a stochasticenvironment projected by the time-varying Markov chain model.
The trajectories show the fraction of patches occupied through
time by Hymedesmia 1 (HY 1), Crisia (CRI), Myxilla (MYX), Mycale
(MYC), and bare rock (BR).
670 M. F. Hill, J. D. Witman and H. Caswell
2002 Blackwell Science Ltd/CNRS
among the species in the community. When the models are
estimated based on data from all 9 quadrats, and compared
to the observed structure in the same quadrats1 the
proportion of variance explained rises to over 95% for all
three models. Thus, the equilibrium predictions are highly
correlated with observed abundances and there is no
significant difference in the ability of the three models to
predict observed community structure.
Transient dynamics depend on initial conditions, and can be
characterized in many different ways. We examine two here:
the long-term rate of convergence to the equilibrium and
the dynamic response to a major perturbation.
The rate of convergence to equilibrium in time-invariant
models can be measured in several ways (Cohen et al. 1993;
Rosenthal 1995). Here, as did Tanner et al. (1994) and
Wootton (2001a), we use the damping ratio
q k1jk2j 8
where k1 and k2 are the first and second largest eigenvaluesof A. Community structure converges to the equilibrium, in
the long run, exponentially like exp(t log q) (Caswell 2001).The closer the second eigenvalue is in magnitude to the first,
the slower the rate of convergence. The half-life of a
perturbation is given by log 2 log q. Because our modelsare Markov chains, k1 1, and q is completely determinedby |k2|.
The analogous measure of convergence in a time-varying
model is the second-largest Lyapunov exponent (Tuljapurkar
1990, p. 30). The largest Lyapunov exponent is familiar in
demography as the stochastic growth rate (Tuljapurkar 1990;
Caswell 2001); it is given by
Tlog jjAT1 A0x0jj 9
where x(0) is an arbitrary initial vector, A(0),...,A(T)1) is a
sequence of matrices, and |||| is any vector norm. Becauseour model is a Markov chain, h1 0, since x(t) is restrictedto the unit simplex. The second largest exponent is
calculated by a similar numerical approach, but starting
with two initial vectors and following the rate of growth of
the area of the parallelogram that they define. We used the
algorithm described in Ott (1993, p. 136), with T 1.0 106.
The results are shown in Table 3. In all three models,
deviations from equilibrium decay at an asymptotic rate of
from 16% to 23% per year (a half-life of 34 years). The
location-specific matrices yield slightly faster convergence
rates, while the rates of the homogeneous matrix and the
time-varying model are almost identical. Incidentally, these
half-lives are similar to those that can be calculated from the
damping ratios reported by Tanner et al. (1994) for coral
= Space-varying= Time-varying
Figure 3 Comparison of the equilibrium community structure
predicted by the spatially varying matrices A(l ), time-varying
matricies A(t ), and the homogeneous matrix A. For the set of
space-vaying matrices, the predicted mean abundance of species i is
uui L)1 Sl ui(l ), where ui(l ) is the predicted fraction of patchesoccupied by species i in quadrat l. For the set of time-varying
matrices, the predicted mean abundance of species i is
xxi T )1 St xi(t ), where xi(t ) is the simulated abundance ofspecies i at iteration t, and T 10 000 is the total number ofiterations. For the homogeneous matrix, the predicted mean
abundance of species i is give by the ith element of the dominant
eigenvector u. Error bars represent 1 SD. See Table 1 for
Table 2 Mean and standard deviation of Pearsons correlationcoefficient, rp, between equilibrium predictions of the Markov
chain models and observed abundances. Models were parameter-
ized using transition data from 4 quadrats and observed frequen-
cies obtained from abundance data from the 5 remaining quadrats.
Standard deviations for rp were calculated over all 126 possible
combinations of 4 out of 9 quadrats
Model Mean SD
Spatial 0.896 0.039
Temporal 0.905 0.032
Homogeneous 0.905 0.038
1It may seem that this would be comparing predictions with the same dataused to make those predictions, and that the result would be an artifactual
agreement between the two (cf. Facelli & Pickett 1990; Wootton 2001a).
This is not so. The equilibrium distribution depends only on the transition
matrix A. Each column of A (say, column j) is calculated from transitions of
patches beginning in state j (see equations (2)(5)). Column j is independent
of how many patches are in states other than j, and of the transitions made
by those patches. Thus A is independent of the relative abundance of patch
states in the data from which it is estimated, as is its prediction of the
relative abundances at equilibrium.
Stochastic Markov chains of subtidal communities 671
2002 Blackwell Science Ltd/CNRS
reefs (36 years), but longer than those implied by the
damping ratios for the intertidal community studied by
Wootton (2001a) (0.7 years).
The damping ratio and Lyapunov exponent give long-
term, asymptotic rates of convergence. They do not
characterize dynamics immediately following perturbations.
To compare the models short-term responses to a large
perturbation, we simulated each model from an initial
condition of 100% bare rock, corresponding to a major
disturbance that eliminated all species (Tanner et al. 1994;
1996; used the same manipulation in their analysis of coral
reef communities). For the space-varying model, we ran
simulations for each location matrix A^A(l ), l 1,,9, and
averaged the results over all locations. For the time-
varying model, we ran 1000 simulations of the stochastic
environment model (matrices chosen randomly, iid, from
A^A(t ), t 1,,8) and averaged the results over all simula-tions.
Figure 4 shows the transient dynamics of mean species
abundances for the first 30 years. Simulations for all three
models are remarkably similar. For the most part,
transient changes in species abundance predicted by the
spatially varying and homogeneous models fall within 1
standard deviation of those predicted by time-varying
D I S C U S S I O N
Our results add two pieces to the discussion about Markov
chain models of succession. First, they provide the most
comprehensive documentation to date of the statistical
significance of temporal and spatial variation in transition
probabilities. Although temporal variability has long been a
concern about Markov chain models (Usher 1979; Facelli
& Pickett 1990; Childress et al. 1998), ours is the first
study to apply modern tools of loglinear analysis and
model selection to the problem. We found that the effects
of space and time are not only statistically significant, but
that the best model, as identified by AIC, includes both
Table 3 The rates of convergence for the homogeneous model(defined by the matrix A), the spatially varying model (A(l ), l 1,...,9), and the time-varying model (matrices chosen randomly, iid,
from A(t ), t 1,...,8). Because the matrix for each location givesits own convergence rate, the mean and 95% confidence interval
for log q are shown for the spatially varying model
rate 95% CI
A log q 0.165 4.2A(l ) log q 0.232 [0.348, 0.117] 3.0A(t ) h2 0.169 4.1
0 10 20 30 0 10 20 30 0 10 20 30
HY 1 MYX MYC
URT MET FG
CRI FIL COR
Figure 4 Simulated recovery dynamics of
the community starting from 100% bare
rock for the spatially varying, time-varying,
and homogeneous models. Each plot repre-
sents time trajectories for a single species
predicted by the three models. The points at
the end of each plot is the observed
abundance averaged over all quadrats (error
bars are 95% confidence intervals). The
states HY 2 and BR are not shown, however,
the dynamics of these states among the three
models are identical. See Table 1 for
672 M. F. Hill, J. D. Witman and H. Caswell
2002 Blackwell Science Ltd/CNRS
The advantage of the loglinear approach is that it allows
us to test the effects of multiple factors (i.e. Time and
Location) on transition probabilities. In the simplest cases,
loglinear analysis is equivalent to the AndersonGoodman
test (Anderson & Goodman 1957), which is a likelihood
ratio test for the effects of a single factor (e.g. time) on
transition probabilities. The AndersonGoodman test has
been applied to demographic models (Bierzychudek 1982;
Cochran 1986) and to community Markov chain models
(Tanner et al. 1994; Wootton 2001a). It cannot be directly
applied to multifactorial designs, except by ad hoc methods
that are analogous to using all possible pairwise t-tests in
place of an ANOVA in a factorial experiment. While some
authors continue to use the Anderson-Goodman test to
analyse Markov chains, it is not an appropriate method for
characterizing the effects of time and space on transition
probabilities, and it should be retired and replaced by
Any study that, like ours, collects transition data within
quadrats faces the issue of independence of samples, which
is assumed in the significance tests. If samples are not
independent, the G 2 values are inflated. Methods exist to
correct for dependence (Altham 1976; Cohen 1976a; Brier
1980), and we can use these to see if our significance levels
could be an artifact of dependence among adjacent patches.
Suppose that, unbeknownst to us, the patches in the
sampled quadrats were not independent, but instead were
clusters of n dependent patches. Their dependence can be
measured by the intraclass correlation coefficient rI (the
proportion of variance in the response due to variability
among clusters, e.g. Kempthorne 1957, p. 228). If patches
within clusters are independent, rI 0. If patches within acluster respond as identical copies of each other, rI 1.Then the critical value for significance tests should be
increased from v2df ;a to (1 + (n ) 1)rI) v2df ;a (Brier 1980).
Thus, for each of our tests, we can explore the cluster size
that would be required to destroy the statistical significance
of the time and location effects (at the a 0.05 level). Wefind that in the case of the most extreme possible
dependence (rI 1), clusters would have to be from 5 to8 times larger than our patches. There is no reason to
believe that clusters of patches respond in lock-step on this
scale (if there was, we would not have designed our
sampling as we did). We conclude that the significance of
spatial and temporal variation in these data is robust to
possible dependence among patches.
Our second conclusion seems paradoxical in light of our
statistical results. We found that including spatial or
temporal variability in the models has little effect on com-
munity dynamics or equilibria. Models integrated over space,
time, or both make similar predictions about both asymp-
totic and transient dynamics, and the equilibrium predictions
agree closely with observed community composition. The
resolution of this apparent paradox comes from recognizing
that the statistical significance of spatial or temporal
variation does not imply that they are biologically important,
because even trivially small effects can refute the hypothesis
that, say, transition probabilities in several locations are
exactly equal. See Johnson (1999) or Royall (1997) for dis-
cussions of common misinterpretations of significance tests.
In this case, spatial and temporal variation are not
important, at least as far as transient dynamics or equilibria
are concerned. Other aspects of community structure might,
of course, differ among the models.
The close proximity and the spatial arrangement of the
quadrats suggest that they all probably receive similar larval
recruitment signals and experience similar temperature,
salinity, and current regimes. This is supported by the lack
of a significant Location Time interaction. We conjecturethat the location effects are due mainly to species
interactions resulting from small-scale compositional differ-
ences between quadrats. This finding provides some
justification for the analysis of homogeneous models of
succession, certainly in this community (Hill 2000) and
perhaps elsewhere (Tanner et al. 1994; Wootton 2001a,b).
Ecologists have argued that species interactions that
occur at the local scale can generate large scale patterning
in community structure (Levin 1992). Wootton (2001c)
looked at this effect by comparing predictions of a spatially
explicit cellular automaton model with a homogenous
Markov chain of a rocky intertidal mussel bed. He found
the cellular automata (CA), which models local interactions
explicitly, did a better job of predicting the distribution of
gap sizes within mussel beds, but that both models did
equally well in predicting community structure. While it is
not surprising that a spatial model would predict the
observed distribution of open space better than a non-
spatial model (although he never says how well the CA
predicts the observed spatial pattern of species), the fact
that the CA and the homogeneous Markov model gave
similar predictions of species abundance patterns is
consistent with our results.
To explore the effects of local species composition on
transitions in our model will require the development of
nonlinear Markov chain models. Nonlinear Markov chain
models for simple communities with small numbers of
interacting species have been studied theoretically by
Caswell & Cohen (1991a,b, 1993, 1995), Barradas et al.
(1996), and Caswell & Etter (1999). Although nonlinear
models can in general give rise to multiple steady states,
limit cycles, and chaotic dynamics, these Markov chain
models do not exhibit any of those phenomena. However,
they were developed to describe changes in the species
composition of patches, rather than the replacement of
individuals, and thus may behave differently from a
nonlinear version of the models we describe here. We
Stochastic Markov chains of subtidal communities 673
2002 Blackwell Science Ltd/CNRS
will describe those nonlinear versions, methods for
parameter estimation, and the effects of nonlinearity on
community dynamics, transient dynamics, and resilience
and reactivity, in a subsequent paper.
A C K N O W L E D G E M E N T S
The authors would like to thank J. Cohen, G. Flierl,
M. Neubert, and J. Pineda for their advice and comments,
J. Leichter and S. Genovese for their help in collecting
photo quadrat data, and K. Sebens for collaborating on
offshore cruises. This research was supported by NSF
grants DEB-9119420, DEB-9527400, OCE-9811267, OCE-
9302238, and DBI-9602226, EPA grant R82-9089, and
NOAAs National Undersea Research Program, University
of Connecticut Avery Point (NURCUCAP). Woods
Hole Oceanographic Institution contribution 10766.
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Editor, M. Pascual
Manuscript received 19 April 2002
First decision made 21 May 2002
Manuscript accepted 17 June 2002
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