spatio-temporal variation in markov chain models of subtidal community succession

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.

*Correspondence:

Fax: +530 752 7297

E-mail: [email protected].

Abstract

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.

Keywords

Markov chains, loglinear analysis, environmental stochasticity, community ecology,

Succession, benthic invertebrates.

Ecology Letters (2002) 5: 665–675

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

present paper).

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

dependence.

Ecology Letters, (2002) 5: 665–675

�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

significantly different.

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-

tion–theoretic 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 Anderson–Goodman 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

2001).

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 ¼ 1. We suppose

that the transition probabilities might vary in space and time.

The dynamics of the community in year t at location l are

given by

xðt þ 1Þ ¼ Aðt ;l Þ xðtÞ ð1Þwhere 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 aðt ;l Þij

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) ¼ A, and

if 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

environment.

M E T H O D S

Field data

The data for our model were collected at yearly intervals

over a nine-year period from 1986 to 1994 on subtidal rock

walls at 30–33 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 span

of 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 1–5 m. We made 31.25 · 20 cm2 colour prints

of the high resolution colour slides, and used these prints to

identify species.

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


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

(Hill 2000).

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)] ¼ 2.04). This scale is approximately equivalent to the

size 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

analysis).

Parameter estimation

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 ¼ 1,...,8; l ¼ 1,...,9. The maximum likelihood esti-

mates aaðtl Þij of the entries of A(tl ) are

aaðtl Þij ¼ nijtl

P11i¼1 nijtl

ð2Þ

That is, aaðtl Þij 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

location l.

2 A set of 9 spatial matrices, A(l ), l ¼ 1,...,9, in which the

effects of time are ignored. The maximum likelihood

estimates aaðl Þij of the entries of A(l ) are

aaðl Þij ¼

P8t¼1 nijtl

P8t¼1

P11i¼1 nijtl

ð3Þ

3 A set of 8 time-varying matrices, A(t ), t ¼ 1,...,8, in which

the effects of space are ignored. The estimates aaðtÞij of the

entries of A(t ) are

aaðtÞij ¼

P9l¼1 nijtl

P9l¼1

P11i¼1 nijtl

ð4Þ

4 A set consisting of a single matrix, A, in which both

temporal and spatial effects are ignored. The entries aaij of A

are estimated by

Table 1 Subtidal species identified in the

nine 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


aaij ¼P9

l¼1

P8t¼1 nijtl

P9l¼1

P8t¼1

P11i¼1 nijtl

ð5Þ

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-

lopment.

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

1988, 2001).

Loglinear analysis

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 also

include 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 ¼ u þ uFðiÞ þ uSðjÞ þ uT ðtÞ þ uLðl Þ

þ uFSðijÞ þ uST ðjtÞ þ uSLðjl Þ þ uFðiÞ þ uTLðtl Þ

þ uSTLðjtl Þ ð6Þ

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:

interpretation).

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

time

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 sufficiently

large (it has a v2 distribution with degrees of freedom equal

to the difference (Ddf) between degrees of freedom of the

two 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,

STL.

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 · location

interaction by comparing the models FSL, FST, STL and

FSTL (Caswell 2001).

Model selection

Adding parameters to a loglinear model will always produce

a better fit to the data. Thus choosing the best model, i.e.



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 ¼ G 2 � 2ðdf Þ ð7Þwhere 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

1998).

For ease of interpretation, it is helpful to scale AIC values

relative to the best model (DAIC ). In general, DAIC £ 2

implies 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 and

should be excluded form further investigation (Burnham &

Anderson 1998).

Loglinear results

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 ¼ 0.9998). This

implies 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 ¼ )6491.5). Thus both location and

time are included in the best model of patch transitions in

Time Effect

FT,FST∆G2 = 4356.5

∆df = 770p < 0.0001

Location Effect

FL,FSL∆G2 = 7248.1

∆df = 880p < 0.0001

Location Effect

FL, FSL∆G2 = 7011.6

∆df = 880p < 0.0001

Time Effect

FT, FST∆G2 = 4119.5

∆df = 770p < 0.0001

Time x Location Interaction

FTL, FSTL∆G2 = 5,828.5

∆df = 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

Null Model

Saturated Model

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) and

degrees of freedom, are shown along the

arrows.



rocky subtidal communities. As DAIC for the saturated

model (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 ¼ 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 ¼ 1,...,8, would represent a sample. To

explore 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 time

step. 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

time-varying model.

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

0.6

0.4

0.2

0

BRHY 1 MYXMYCCRI

Time

Abu

ndan

ce (

frac

tion

of

occu

pied

pat

ches

)

Figure 2 An example of community dynamics in a stochastic

environment 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).



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

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 ¼ k1

jk2jð8Þ

where k1 and k2 are the first and second largest eigenvalues

of 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 models

are Markov chains, k1 ¼ 1, and q is completely determined

by |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

h1 ¼ limT!1

1

Tlog jjAðT�1Þ Að0Þxð0Þjj ð9Þ

where x(0) is an arbitrary initial vector, A(0),...,A(T)1) is a

sequence of matrices, and ||Æ|| is any vector norm. Because

our model is a Markov chain, h1 ¼ 0, since x(t) is restricted

to 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 ¼ 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 3–4 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

Abu

ndan

ce (

frac

tion

of

occu

pied

pat

ches

)

0.5

0.25

0

BR

HY

1

HY

2

MY

X

MY

C

ME

T

UR

T

FG

CR

I

FIL

CO

R

= Space-varying= Time-varying

= Homogeneous

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 ¼ L)1 Sl ui(l ), where ui

(l ) is the predicted fraction of patches

occupied by species i in quadrat l. For the set of time-varying

matrices, the predicted mean abundance of species i is

�xxi ¼ T )1 St xi(t ), where xi(t ) is the simulated abundance of

species i at iteration t, and T ¼ 10 000 is the total number of

iterations. 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

abbreviations.

Table 2 Mean and standard deviation of Pearson’s correlation

coefficient, 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 data�used 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.



reefs (3–6 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 AA(l ), l ¼ 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

AA(t ), t ¼ 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

model.

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

effects.

Table 3 The rates of convergence for the homogeneous model

(defined by the matrix A), the spatially varying model (A(l ), l ¼1,...,9), and the time-varying model (matrices chosen randomly, iid,

from A(t ), t ¼ 1,...,8). Because the matrix for each location gives

its own convergence rate, the mean and 95% confidence interval

for log q are shown for the spatially varying model

Model

Convergence

rate 95% CI

Half-life

(years)

A log q ¼ 0.165 4.2

A(l ) log q ¼ 0.232 [0.348, 0.117] 3.0

A(t ) h2 ¼ 0.169 4.1

0.4

0.2

0

0.1

0.05

0

0.1

0.05

0

0.04

0.02

0

0.04

0.02

0

0.04

0.02

0

0.2

0.1

0

0.4

0.2

0

0.1

0.05

0

Abu

ndan

ce (

frac

tion

of p

atch

es o

ccup

ied)

Time (years)

0 10 20 30 0 10 20 30 0 10 20 30

HY 1 MYX MYC

URT MET FG

CRI FIL COR

Spatially-varying

Time-varying

Homogeneous

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

abbreviations.



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 Anderson–Goodman

test (Anderson & Goodman 1957), which is a likelihood

ratio test for the effects of a single factor (e.g. time) on

transition probabilities. The Anderson–Goodman 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

loglinear analysis.

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 ¼ 0. If patches within a

cluster respond as identical copies of each other, rI ¼ 1.

Then the critical value for significance tests should be

increased from v2df ;a to (1 + (n ) 1)rI) v2

df ;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 ¼ 0.05 level). We

find that in the case of the most extreme possible

dependence (rI ¼ 1), clusters would have to be from 5 to

8 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 conjecture

that 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



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

NOAA’s National Undersea Research Program, University

of Connecticut – Avery Point (NURC–UCAP). 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



spatio-temporal variation in markov chain models of subtidal community succession

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