Spatio-temporal variation in Markov chain models of subtidal community succession

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  • 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

    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


    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

    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-

    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

    given by

    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

    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 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

    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

    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

    (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)] 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


    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 1,...,8; l 1,...,9. The maximum likelihood esti-mates a^a

    tl ij of the entries of A

    (tl ) are

    a^atl ij


    i1 nijtl2

    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

    location l.

    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

    a^al ij

    P8t1 nijtlP8


    i1 nijtl3

    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


    P9l1 nijtlP9


    i1 nijtl4

    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

  • a^aij P9


    t1 nijtlP9l1P8


    i1 nijtl5

    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

    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 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).

    Model selection

    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 &

    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 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

    Time Effect

    FT,FSTG2 = 4356.5df = 770

    p < 0.0001

    Location Effect

    FL,FSLG2 = 7248.1df = 880p < 0.0001

    Location Effect

    FL, FSLG2 = 7011.6df = 880p < 0.0001

    Time Effect

    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

    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) 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

    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









    ce (









    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

    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

    h1 limT!1


    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



    ce (































    = 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 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






























    ce (



    of p


    es o



    Time (years)

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

    HY 1 MYX MYC






    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

    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 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

    Stochastic Markov chains of subtidal communities 675

    2002 Blackwell Science Ltd/CNRS


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