estimating community stability and ecological …estimate the strengths of interactions between...

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Ecological Monographs, 73(2), 2003, pp. 301–330q 2003 by the Ecological Society of America

ESTIMATING COMMUNITY STABILITY AND ECOLOGICALINTERACTIONS FROM TIME-SERIES DATA

A. R. IVES,1,4,5 B. DENNIS,2,4 K. L. COTTINGHAM,3,4 AND S. R. CARPENTER1,4

1Department of Zoology, University of Wisconsin, Madison, Wisconsin 53706 USA2Department of Fish and Wildlife Resources and Division of Statistics, University of Idaho,

Moscow, Idaho 83844-1136 USA3Department of Biology, Dartmouth College, Hanover, New Hampshire 03755 USA

4National Center for Ecological Analysis and Synthesis, University of California,Santa Barbara, California 93101-5504 USA

Abstract. Natural ecological communities are continuously buffeted by a varying en-vironment, often making it difficult to measure the stability of communities using conceptsrequiring the existence of an equilibrium point. Instead of an equilibrium point, the equi-librial state of communities subject to environmental stochasticity is a stationary distri-bution, which is characterized by means, variances, and other statistical moments. Here,we derive three properties of stochastic multispecies communities that measure differentcharacteristics associated with community stability. These properties can be estimated frommultispecies time-series data using first-order multivariate autoregressive (MAR(1)) models.We demonstrate how to estimate the parameters of MAR(1) models and obtain confidenceintervals for both parameters and the measures of stability. We also address the problemof estimation when there is observation (measurement) error. To illustrate these methods,we compare the stability of the planktonic communities in three lakes in which nutrientloading and planktivorous fish abundance were experimentally manipulated. MAR(1) mod-els and the statistical methods we present can be used to identify dynamically importantinteractions between species and to test hypotheses about stability and other dynamicalproperties of naturally varying ecological communities. Thus, they can be used to integratetheoretical and empirical studies of community dynamics.

Key words: community matrix; community stability; multivariate autoregressive process; reac-tivity; resilience; stationary distribution; stochastic population model; time-series analysis; vectorautoregressive process.

INTRODUCTION

The stability of ecological communities depends onat least three components of community structure: di-versity (species richness), species composition, and thepattern of interactions among species. Much of the cur-rent scientific discussion of community stability fo-cuses on diversity, because the relationship betweendiversity and community stability is central to argu-ments for preserving ecological diversity (Ehrlich andDaily 1993, Schulze and Mooney 1993, Tilman andDowning 1994, Schlapfer and Schmid 1999). However,the role of diversity in community stability is intimatelylinked to species composition and the pattern of inter-actions among species (Vitousek 1990, Walker 1992,Lawton and Brown 1993, Vitousek and Hooper 1993,Tilman 1996, Hooper and Vitousek 1997, Tilman et al.1997, Ives et al. 1999b, 2000b). Composition is im-portant because the presence of particular species with-in a community could change how the community re-sponds to particular environmental perturbations (Mc-

Manuscript received 26 June 2001; revised 25 February 2002;accepted 26 February 2002; final version received 12 April 2002.Corresponding Editor: N. J. Gotelli.

5 E-mail: [email protected]

Naughton 1977, 1985, Frank and McNaughton 1991).For example, a zooplankton species that is particularlytolerant to low pH could make the total zooplanktonbiomass in a community more stable against acidifi-cation (Frost et al. 1995). The pattern of interactionsamong species is important because the responses ofindividual species to environmental perturbations de-pend not only on the direct effect of the perturbationon their reproduction and mortality rates, but also onthe indirect effects acting through changes in the abun-dances of other species in the community (May 1974,Holt 1977, Paine 1980, Pimm 1984, Abrams 1987, Car-penter et al. 1994b, Ives 1995b, McCann et al. 1998,Klug et al. 2000). Thus, understanding community sta-bility requires understanding the combined effects ofdiversity, composition, and the patterns of interactionsamong species.

Community stability may be measured in many dif-ferent ways. Most common measures of stability re-quire systems to have an equilibrium point, or at leasta well defined ‘‘normal’’ state from which deviationscan be measured. Resilience (sensu Pimm 1984) ismeasured by the characteristic rate at which the abun-dances of populations within a community return toequilibrium following a disturbance. Resistance is a

302 A. R. IVES ET AL. Ecological MonographsVol. 73, No. 2

measure of the magnitude of displacement of popula-tion abundances from equilibrium when subjected to adisturbance. Both of these concepts of stability can bemodified for cases in which there is no strict equilib-rium point (Steinman et al. 1992, Cottingham andSchindler 2000). Nonetheless, they are still based onthe responses of systems away from their normal states.It is difficult to apply these measures to systems thatare continuously buffeted by environmental fluctua-tions such that the normal state includes frequent dis-turbances.

An alternative approach is to view the dynamics ofa community as a stochastic process and to define sta-bility in terms of properties of a stochastic model ofpopulations within the community. Such a stochasticmodel describes how the means, variances, and otherstatistical moments of population densities changethrough time. The equilibrium of a stochastic model isthe joint stationary distribution of population abun-dances, and analogues of the deterministic concepts ofthe return time, resilience, and resistance can be definedwith reference to this stationary distribution. As wewill describe below, community stability may be mea-sured by several different properties of the stochasticmodel. All of these properties share the advantage ofbeing readily estimated directly from data. This is im-portant, because disentangling the effects of diversity,species composition, and species interactions on com-munity stability requires robust statistical techniquesto analyze data from different ecological communities.

Our goals here are twofold. First, we describe dif-ferent properties of stochastic processes that can beused to define stability. We focus on a particular typeof stochastic process: first-order multivariate (or vec-tor) autoregressive processes, abbreviated MAR(1)(Judge et al. 1985, Reinsel 1997). In MAR(1) pro-cesses, the changes in population abundances from timepoint t 2 1 to the next time t depend only on thepopulation abundances at time t 2 1 and environmentaldisturbances occurring between time t 2 1 and t; pop-ulation abundances and environmental disturbances be-fore time t 2 1 have no direct effect. The first-orderprocess implies that enough information can be ob-tained about a community at a single point in time topredict the immediate changes in species’ abundances.In addition, MAR(1) processes assume that the inter-actions among species, and between species and en-vironmental variables, are linear (at least after variableshave been suitably transformed). MAR(1) models pro-vide relatively simple approximations to nonlinear,non-first-order processes and therefore can be used todescribe the general stochastic properties of complexcommunities (Ives 1995a, b).

Our second goal is to provide the statistical tools toestimate the strengths of interactions between speciesand the stochastic properties of the dynamics of com-munities from time-series data. For a community witha given number of species, the dynamics depend on

how each species is affected by environmental fluc-tuations, measured in terms of changes in the species’population growth rates. The dynamics also depend onhow changes in the abundance of species caused byenvironmental fluctuations in turn cause changes in thepopulation growth rates of other species via species-species interactions. Interactions between species canbe viewed as a filter that amplifies, either weakly orstrongly, the variability in species’ population growthrates caused by environmental fluctuations (Ives et al.2000b). Thus, estimating the stability properties of acommunity necessarily requires estimating the strengthof interactions between species. We approach this sec-ond goal by first describing statistical procedures forparameter estimation and model selection. We then il-lustrate these procedures by applying them to an ex-ample data set taken from a whole-lake experimentdesigned to assess the response of planktonic com-munities to environmental disturbances.

Road map

Developing both the theoretical measures of stabilityand the statistical methods to apply these measures todata is a formidable task, and some readers may wantto skim some sections of this article. Therefore, wewill begin with a road map. The destination is an anal-ysis of seven years of data taken at weekly intervalsover the summer from three experimentally manipu-lated lakes. The data set contains many of the featuresand puzzles found in other ecological data sets. Pre-senting a brief overview of the data here will help tomotivate the theoretical and statistical development inthe first parts of the article.

The data consist of weekly samples of zooplanktonand phytoplankton, which for the analyses were dividedinto two zooplankton groups (Daphnia and non-Daph-nia) and two phytoplankton groups (large and smallphytoplankton). Daphnia are large, effective herbi-vores, and small phytoplankton are particularly vul-nerable to herbivory, so we anticipated strong inter-actions between Daphnia and small phytoplanktongroups. The experimental manipulations involved re-configuring the initial fish composition of one lake sothat the fish community was dominated by planktivo-rous fish that reduced Daphnia abundance, while theother two lakes were dominated by piscivorous fish thatreduced the abundance of planktivorous fish, and hencehad high Daphnia abundance. Then, over the course ofseven years, nutrients were added to two of the lakes,the lake dominated by planktivores and one of the lakesdominated by piscivores.

The objective of the experiment was to ask how dif-ferences in the fish food webs among lakes altered theresponse of the planktonic community to nutrient ad-dition (Carpenter et al. 2001). Here, however, we focuson a different question: How stable are the planktoniccommunities in the different lakes? Not surprisingly,the two lakes subjected to nutrient addition showed

May 2003 303ECOLOGICAL INTERACTIONS AND TIME SERIES

greater variability over the seven years of the experi-ment as they responded to this perturbation. But weare specifically interested in the question: If these directresponses to the nutrient perturbations were factoredout, thereby creating a theoretical level playing fieldfor comparison, which lake would be the most stable?We are not interested in the stability of the lakes in theface of the nutrient perturbation per se, but instead areinterested in the processes that affect the variability ofthe planktonic communities independently of this spe-cific perturbation. When subjected to continuous en-vironmental perturbations from all sources, which lakeis most stable, and how do differences in the interac-tions within and among planktonic groups in the dif-ferent lakes dictate differences in stability?

We address these questions by fitting a MAR(1) mod-el to the data, and extracting different measures of sta-bility from the fitted model. In the article, we begin bydescribing the relevant theoretical properties ofMAR(1) processes (section First-order autoregressivemodels). Although MAR(1) models are linear (at leastin transformed variables), we show how they can bederived as approximations to nonlinear systems. Wethen describe three different types of measures of sta-bility (section Stability properties of MAR(1) models).The first is based on the variability of the stationarydistribution of the stochastic process, whereas the sec-ond is based on the rate at which the stochastic processconverges to the stationary distribution. The third, re-activity, is a measure of how strongly population abun-dances are pulled towards the mean of the stationarydistribution. Much of the mathematical development ofthis section is relegated to the Appendix.

We then present statistical methods for fittingMAR(1) models to data. To begin, we show how toextend the standard MAR(1) model to account for someof the complexities likely to be encountered when ad-dressing ecological data (section Model modifications).We show how measured environmental factors, suchas nutrient addition in the lake data set, can be factoredinto MAR(1) models. We also consider observationerror, the error unavoidable in ecological data that re-sults from our inability to measure the ‘‘true’’ state ofa system and the environmental variables that affect it.Methods for parameter estimation are then presented,including techniques to calculate approximate confi-dence intervals for not only the parameters of theMAR(1) model, but also aggregate parameters, specif-ically, our measures of stability (section Model selec-tion and statistical inference). After presenting theneeded statistical methods, we apply them to the lakedata set (section Limnological example).

Much of this work is mathematically nontrivial, andwe have not shied away from giving formal mathe-matical results that are needed to implement and mod-ify our analyses. To make the concepts more approach-able, however, we end each section with a descriptivesummary of the major results from the section. Also,

while we present formal results, we generally do notgive proofs, but instead give references where similarresults are derived. Finally, the programs used in ouranalyses of the lake data are available in ESA’s Elec-tronic Archive (Supplement).

FIRST-ORDER AUTOREGRESSIVE MODELS

Below, we review some basic concepts of first-orderautoregressive processes. We begin with the univariatecase for a single species, and then develop the multi-variate case. We also relate the autoregressive modelsto their deterministic counterparts; this makes it pos-sible to show the correspondence between concepts ofstability in stochastic models to the more familiar con-cepts applied in the stability analysis of deterministicmodels.

Univariate AR(1) process

A deterministic model frequently used to describedensity-dependent population growth is a discrete-timeGompertz model (Reddingius 1971, Dennis and Taper1994)

n 5 n exp[a 1 (b 2 1)ln n ].t t21 t21 (1)

Here, nt is population abundance, a is the intrinsic rateof increase, and b governs the strength of density de-pendence, with no density dependence correspondingto b 5 1. As b decreases, density dependence increases,and when b , 0 overcompensating density dependenceoccurs in which population trajectories tend to show‘‘boom–bust’’ patterns, with high and low densities al-ternating between successive time points. On a log scale,the model is a simple linear difference equation

x 5 a 1 bxt t21 (2)

where xt 5 ln nt. The Gompertz equation is only oneof many simple models that have been applied to eco-logical systems (e.g., Zeng et al. 1998). We start withthe Gompertz equation, because it is a log-linear modelthat often gives a reasonable first-order approximationto other nonlinear models.

Provided b ± 1, this model has an equilibrium pointx` given by

x 5 a/(1 2 b).` (3)

The condition for the equilibrium to be stable can beseen from the solution of the model given by Eq. 2.Starting from the initial point x0 and substituting for ain terms of x`, it is easy to derive through recursionthat

tx 5 x 1 b (x 2 x ).t ` 0 ` (4)

Thus, provided zbz , 1, xt will converge to x`, withconvergence occurring more quickly the closer zbz isto zero. If 0 , b , 1, convergence will occur mono-tonically, while damped two-point oscillations will oc-cur when 21 , b , 0.


FIG. 1. Examples of two univariate AR(1) processes (Eq.5) with autoregression parameter (A) b 5 0.6 and (B) b 50.95; a 5 0 in both panels. Each panel shows 10 trajectoriesstarting from the point x0 5 28. The distribution of processerrors Et is normal with mean 0 and s2 5 1, which is depictedas the shaded area at the right in both panels. The stationarydistribution is also normal and is shown as the heavy line tothe right of the panels.

A stochastic version of Eq. 2 is the univariate AR(1)process

X 5 a 1 bX 1 E .t t21 t (5)

Here, Xt is log abundance at time t (in upper case todenote a random process), and Et is a normal randomvariable with mean 0 and variance s2, and with E1, E2,. . . all independent. The random variable Et represents‘‘process error’’ or fluctuations caused by unspecifiedstochastic forces. In ecological terms Et represents en-vironmentally driven fluctuations in the per capita pop-ulation growth rate and thus gives the environmentalvariability experienced by the population (see Denniset al. 1991, Dennis and Taper 1994). The univariateAR(1) process has been an important model in the the-ory and statistical testing of density dependence (Den-nis and Taper 1994 give a review). Properties of AR(1)processes can be found in texts that emphasize the sta-tistical theory of time series (Harvey 1993, Box et al.1994).

Fig. 1 gives two examples of univariate autoregres-sive processes for b 5 0.6 and b 5 0.95. Each paneldisplays 10 separate realizations of the same process.We will refer to this figure below to illustrate propertiesof autoregressive processes.

Univariate stationary distributionThe equilibrium for the AR(1) process Xt is the sta-

tionary distribution, which is the probability distribu-

tion approached by the distribution of Xt as t → `provided zbz , 1. Note that zbz , 1 is the same criterionfor stability as for the corresponding deterministicmodel given in Eq. 2. The stationary distribution forXt is a normal distribution with mean m` and variancey` given by (Box et al. 1994)

m 5 a /(1 2 b) (6)`

2 2y 5 s /(1 2 b ). (7)`

The mean of the stationary distribution is the same asthe deterministic point equilibrium (Eq. 3). The vari-ance y` around the mean is directly proportional to theenvironmental variability, s2. It also depends on b, withvalues of b closer to zero corresponding to less pop-ulation variability. Note that this coincides with valuesof b that produce rapid returns to equilibrium in thecorresponding deterministic model (Eq. 4). In Fig. 1,the stationary distributions are given at the right marginof each panel (heavy line) along with the distributionof the process error Et (shaded).

Univariate transition distribution

The transition distribution is the conditional proba-bility distribution for Xt at any time t given that at timezero the initial condition is known to be X0 5 x0. Thetransition distribution is normal, and both the mean andthe variance depend on time t:

tm 5 m 1 b (x 2 m ) (8)t ` 0 `

2 2 2 2 2 t21y 5 s [1 1 b 1 (b ) 1 · · · 1 (b ) ]t

2 t1 2 (b )2 2 t5 s 5 y [1 2 (b ) ]. (9)`2[ ]1 2 b

Here mt 5 E[Xt z X0 5 x0] denotes the expected valueof Xt, and yt 5 E[(Xt 2 mt)2 z X0 5 x0] denotes the var-iance of Xt given the initial condition X0 5 x0. Theformula for mt is identical to the solution trajectory forxt in the deterministic model (Eq. 4). Note that if zbz, 1, then mt converges to m` and yt converges to y` ast becomes large. For small values of t, the transitiondistribution starts as a spike of probability near theinitial size x0 and then broadens to the stationary dis-tribution as t becomes large. Thus, in Fig. 1 all trajec-tories begin at the same point, but then depart fromeach other as the transition distribution approaches thestationary distribution.

Multivariate AR(1) process

If there are p interacting species, a multivariate ver-sion of the univariate AR(1) model (Eq. 5) is

X 5 A 1 BX 1 Et t21 t (10)

where Xt is a p 3 1 vector of (log-transformed) pop-ulation abundances at time t, A is a p 3 1 vector ofconstants, B is a p 3 p matrix whose elements bij givethe effect of the abundance of species j on the per capitapopulation growth rate of species i, and Et is a p 3 1


vector of process errors that has a multivariate normaldistribution with mean vector 0 and covariance matrixS. The vectors Et represent stochastic environmentalfactors and are assumed to be independent throughtime. Nonetheless, the elements in Et may covary witheach other, as measured by the off-diagonal elementsof the covariance matrix S. Thus, if a good time in-terval for species i is also likely to be good for speciesj, then the element sij in S will be positive.

MAR(1) models as approximations tononlinear systems

The MAR(1) model can be viewed as a linear ap-proximation to a nonlinear first-order stochastic pro-cess. Specifically, consider the model

X 5 f(X , R )t t21 t (11)

where Rt is a q 3 1 normal random variable with mean0 representing the effects of q environmental variableson the population growth rate of the species, and f isa nonlinear function. If the stochastic process definedby Eq. 11 has a stationary distribution with mean X`,the right-hand side of Eq. 11 can be approximated usinga first-order Taylor expansion around X 5 X` andR 5 0 to give

] fX ø f (X , 0) 1 (X , 0)[X 2 X ]t ` ` t21 `]X

] f1 (X , 0)R (12)` t]R

where (]f/]X)(X`, 0) and (]f/]R)(X`, 0) denote the par-tial derivatives of f with respect to the vectors Xt21 andRt, respectively, evaluated at Xt 5 X` and Rt 5 0. Thus,(]f/]X)(X`, 0) is a p 3 p matrix with the i–jth elementgiven by dfi /dXj, and (]f/]R)(X`, 0) is a p 3 q matrixwith the i–jth element given by dfi /dRj. This approxi-mation is identical in form to the MAR(1) model givenin Eq. 10, as can be seen by letting B 5 (]f/]X)(X`,0), A 5 f(X`, 0) 2 BX`, and Et 5 (]f/]R)(X`, 0)Rt.Note that this explicitly defines the elements of B interms of the interaction strengths between species; bij

equals dfi /dXj, the change in the log population growthrate of species i with respect to changes in the logpopulation abundance of species j. This approximationis contingent on the stochastic process being stationary.This is not an important restriction to our discussion,however, because all of the concepts of stability thatwe address apply to stationary processes.

In addition to being a linear approximation to non-linear stochastic processes, Eq. 10 gives the stochasticequivalent of the deterministic multispecies Gompertzmodel. More generally, the matrix B contains the sameinformation as the ‘‘community matrix’’ calculated todetermine the stability properties around equilibriumof nonlinear deterministic models (May 1974, Pimm1982). To see this, consider the deterministic versionof the nonlinear stochastic model in Eq. 11

n 5 h(n )t t21 (13)

where nt is a p 3 1 vector of (untransformed) popu-lation densities, and h is a nonlinear function. The com-munity matrix of Eq. 13 is (]h/]n)(n`), the partial de-rivative of h with respect to n evaluated at the deter-ministic equilibrium, n`. The elements of (]h/]n)(n`)have the form dhi /dnj. Letting fi 5 ln hi and xj 5 ln nj,

n n] f ](ln h ) ]h ]hj ji i i i5 5 5 (14)) ) ) )]x ](ln n ) h ]n n ]nj j i j i jx n n n` ` ` `

with the last step using the equality hi(n`) 5 ni at equi-librium. Thus, the matrix B 5 (]f/]X)(X`, 0) differsfrom the community matrix (]h/]n)(n`) in having everyrow i multiplied by ni and every column j multipliedby (1/nj). It is straightforward to show that matrix Band the community matrix have identical eigenvalues.Since the eigenvalues of the community matrix deter-mine the population dynamics around equilibrium, ma-trix B contains the same information as the communitymatrix about population dynamics.

In summary, the MAR(1) model given by Eq. 10 isa linear approximation to more general nonlinear first-order autoregressive processes. The matrix of speciesinteractions B is similar to the community matrix usedto calculate the stability properties of nonlinear deter-ministic models. It is not surprising, therefore, that use-ful stability properties of the MAR(1) model can bedefined in terms of properties of the matrix B, as weshow next.

MAR(1) stationary distribution

The MAR(1) process Xt has a stationary distributionprovided all eigenvalues of matrix B lie within the unitcircle (Harvey 1989, Reinsel 1997). This is identicalto the criterion for stability of the point equilibrium inthe corresponding deterministic model (May 1974).When the process error of the MAR(1) process is nor-mally distributed, the stationary distribution is a mul-tivariate normal distribution with mean vector m` andcovariance matrix V` given by

21m 5 (I 2 B) A (15)`

V 5 BV B9 1 S (16)` `

where I is the p 3 p identity matrix, and B9 denotesthe transpose of B. Eq. 16 is a matrix equation thatcannot be symbolically solved for the covariance ma-trix V`. However, one can use the ‘‘Vec’’ operator onboth sides of Eq. 16 to obtain an explicit equation forthe elements of V`. The operation Vec(A) creates acolumn vector out of any matrix A by stacking columnsof A on top of each other, with the first column on thetop and the last column on the bottom. The often-handyalgebra of the Vec operator is reviewed by Searle(1982), and Judge et al. (1985:949–950) provide somebasic identities. The resulting explicit equation for theelements of V` is


21Vec(V ) 5 (I 2 B J B) Vec(S). (17)`

The symbol J denotes the Kronecker or direct matrixproduct; M J N is formed by scalar multiplying N inturn by each element of M and arranging the resultsinto a large matrix. Thus, in Eq. 17 B J B is the p2

3 p2 matrix given by

b B b B · · · b B11 12 1p

b B b B · · · b B21 22 2pB J B 5 . (18) _ _ _

b B b B · · · b Bp1 p2 pp

Eq. 17 is the multivariate analog to the univariate sta-tionary variance equation (Eq. 7).

MAR(1) transition distribution

The transition distribution of Xt conditional on theinitial value of X 5 x0 is a multivariate normal distri-bution with mean vector mt and covariance matrix Vt

that change with time:

tm 5 m 1 B (x 2 m ) (19)t ` 0 `

2 2 t21 t21V 5 S 1 BSB9 1 (B )S(B )9 1 · · · 1 (B )S(B )9.t

(20)

Eq. 20 can be vectorized to obtain

2Vec(V ) 5 [I 1 B J B 1 (B J B) 1 · · ·t

t211 (B J B) ]Vec(S)

t 215 [I 2 (B J B) ][I 2 B J B] Vec(S). (21)

Eqs. 19 and 21 reduce to the univariate counterpartsEqs. 8 and 9. It is straightforward to show that theeigenvalues of the Kronecker product B J B are equalto the products of all the pairs of eigenvalues of B:lilj, i 5 1, . . . , p, j 5 1, . . . , p (Seber 1984). Thus,if all eigenvalues of B lie within the unit circle, thenall of the eigenvalues of B J B also lie within the unitcircle. Therefore, Bt → 0 and (B J B)t → 0, and con-sequently mt → m` and Vt → V` as t → `.

STABILITY PROPERTIES OF MAR(1) MODELS

The stability of a MAR(1) model can be measuredin numerous ways, each highlighting a different aspectof the dynamics. Here, we constrain the discussion tostationary processes (in which all the eigenvalues of Blie within the unit sphere), thus excluding the possi-bility of species extinction. Our question is thereforehow stable is the system, rather than whether the systemis stable or unstable. All three of the types of measuresof stability that we present in detail have close con-ceptual and mathematical relationships, and they allinvolve the interplay between short-term environmen-tal fluctuations (vector Et in Eq. 10) and interactionsbetween species (matrix B).

Our first measure of stability quantifies how the var-iance of the stationary distribution compares to the

variance of the process error (environmental variabil-ity). The more stable the system, the lower the varianceof the stationary distribution relative to the variance ofthe process error. The second measure of stability de-pends on the rate at which the transition distributionapproaches the stationary distribution. This is com-parable to stability measured by characteristic returnrates in deterministic systems (May 1974); rapid ap-proach to the stationary distribution (i.e., the stochasticequilibrium) corresponds to a more stable system. Ourfinal measure addresses the reactivity of the system,defined as the immediate response of a system follow-ing a perturbation (Neubert and Caswell 1997). A high-ly reactive system may frequently move further awayfrom the mean of the stationary distribution, eventhough on average the system remains stable. The morereactive a system, the less stable it is. The mathematicaldetails of these measures of stability are given in theAppendix, and here we just give heuristic descriptions.

Variance of the stationary distribution

Stability measured in terms of the relationship be-tween the variance of the stationary distribution andenvironmental variability can be illustrated using theball-in-basin diagram frequently used to illustrate sta-bility in deterministic systems. In Fig. 2A, the balls inboth the steep and shallow basins are subjected to con-tinuous perturbations that occur randomly in either di-rection. Following each perturbation, the ball in thesteep basin rolls rapidly towards the bottom of the ba-sin, while the ball in the shallow basin rolls to thebottom slowly. As a result, the ball in the shallow basinspends more time away from the basin bottom, leadingto a stationary distribution with greater variance thanthe ball in the steep basin.

For the multispecies case, the covariance matrix ofthe stationary distribution V` depends on the covari-ance matrix of the process error S and the interactionsbetween species encapsulated in the matrix B. As ameasure of stability, it is natural to use the variance ofthe stationary distribution relative to the variance ofthe process error (environmental variability). By stan-dardizing by the variance of the process error, differ-ences between the variances of the stationary distri-butions of different communities can be attributed tospecies interactions. In a relatively stable system, spe-cies interactions will cause the variance of the station-ary distribution to be only slightly greater than thevariance of the process error. In contrast, in a less stablesystem, species interactions will greatly amplify theenvironmental variance, thereby creating large vari-ance in the stationary distribution relative to the var-iance of the process error.

Fig. 3A and B give an example of two systems con-taining two competitors having different stability prop-erties. Both systems experience the same environmen-tal variance; the distribution of process errors is de-picted by the shaded circle giving the 95% probability


FIG. 2. Illustrations of three measures of stability. (A)The pair of balls-in-basins portray stability measured by thevariance in the stationary distribution of the location of theball. If subjected to the same disturbance regime (white ar-rows), the ball in the shallow basin on the right will spendmore time away from the center of the basin, because theforce to return to the center (solid arrows) is lower. Thiscreates a stationary distribution with a greater variance. (B)The pair of balls-in-basins portrays stability measured by thereturn rate of the mean of the transition distribution to themean of the stationary distribution. In the shallow basin, theball will move more slowly to the center of the basin afterbeing disturbed to the basin rim at location 1. (C) The pairof balls-in-basins portrays stability measured by the reactiv-ity. In the steep basin, each successive location (1, 2, and 3)tends to bring the ball closer to the center of the basin thanis the case for the shallow basin.

bounds of the distribution. In Fig. 3A, intra- and in-terspecific competition are relatively strong, leading tolow variance in the stationary distribution compared tothe system in Fig. 3B. Thus, the system in Fig. 3A isthe more stable.

Although in this example it is easy to see the dif-ferences between systems, comparing the variances ofstationary distributions for general, multidimensionalsystems is less straightforward. We use two methods,whose details are developed in the Appendix. The firstinvolves measuring the variance in the stationary dis-tribution along axes given by the eigenvectors of B(Ives 1995a). These eigenvectors are depicted by theaxes in Fig. 3A and B. The variance in the stationarydistribution along an eigenvector corresponding to areal eigenvalue, denoted `, isy

2 2y 5 c /(1 2 l )` (22)

where c2 is the variance of the process error along theeigenvector, and l is the corresponding real eigenvalue.From this expression (as with b in Eq. 7 for a univariateAR(1) process), the closer the value of l is to zero,the lower the variance of the stationary distributionalong the corresponding eigenvector.

A similar result can be obtained for any complexeigenvalues of B. Complex eigenvalues occur in con-jugate pairs, a 6 bi, and the corresponding pair ofeigenvectors defines a plane in the multidimensionalspace containing the stationary distribution. The sumof variances within this plane is

r c r c 2y 1 y 5 (c 1 c )/(1 2 \a 6 bi\ )` ` (23)

where and are variances of the stationary distri-r cy y` `

bution along the two eigenvectors, c r and cc are var-iances of the process error, and \a 6 bi\ is the mag-nitude of the eigenvalues. Thus, in the plane definedby the eigenvectors corresponding to a pair of complexeigenvalues a 6 bi, the closer the magnitude of a 6bi is to zero, the lower the variance of the stationarydistribution.

The second method for comparing the variance ofprobability distributions uses determinants to measurethe ‘‘volume’’ of covariance matrices. For example, intwo-dimensional space the area of the parallelogramdefined by any two vectors equals the determinant ofthe matrix containing the two vectors as columns. Sincethe covariance matrices V` and S give the variancesof the stationary distribution and process errors, re-spectively, the volume of the difference V` 2 S mea-sures the degree to which species interactions increasethe variance of the stationary distribution relative tothe variance of the process error. From Eq. 16

2det(V 2 S) 5 det(V )det(B) . (24)` `

Therefore, the proportion of the volume of V` attrib-utable to species interactions is det(V` 2 S)/det(V`)5 det(B)2. The determinant of a matrix equals the prod-uct of its eigenvalues, so det(B)2 5 (l1l2 · · · lp)2 fora community with p species. Since the magnitudes ofthe eigenvalues are less than one, this implies thatdet(B)2 will decrease with increasing numbers of spe-cies in a system, even if the eigenvalues have com-parable magnitudes. Therefore, to facilitate compari-sons among systems with different numbers of species,we use det(B)2/p 5 (l1l2 · · · lp)2/p. This is simply thesquare of the geometric mean of the eigenvalues.

These two methods for comparing the variances ofstationary distributions (Eqs. 22 and 23 vs. Eq. 24) arerelated, since Eqs. 22 and 23 contain eigenvalues ofB, and Eq. 24 contains det(B), which is the product ofall eigenvalues. The former method can be comparedto the most common measure of stability for deter-ministic models, the characteristic return rate. Thecharacteristic return rate of a deterministic system de-pends on the dominant eigenvalue (i.e., the eigenvaluewith the largest magnitude) of the community matrix


FIG. 3. Illustrations of measures of stability: (A, B), variance of the stationary distribution relative to the distribution ofenvironmental process error; (C, D), characteristic return rates of the transition distribution to the stationary distribution;and (E, F), reactivity. For each pair of panels, the panel on the top gives the case of greatest stability. Panels A and B givethe stationary distribution of a system made up of two competitors, with (A) B 5 [[0.6, 20.2], [20.2, 0.6]] and (B) B 5[[0.85,20.1], [20.1, 0.85]]; the MAR(1) process with these matrices was iterated 200 times to give the 200 points in thepanels. The stationary distribution is given by the ellipse that incorporates 95% of the probability density of the stationarydistribution. The shaded circle gives the 95% bounds of the process error, which is the same in panels (A) and (B). The linesgive the eigenvectors corresponding to the eigenvalues (A) 0.8 and 0.4, and (B) 0.95 and 0.75. Panels (C) and (D) give thetransition distributions for two univariate AR(1) processes for 50 time steps starting at x0 5 28, with (C) b 5 0.6 and (D)b 5 0.95. The means of the transition distributions are given by the heavy lines, and the light lines give the mean 6 1 and2 SD. Panels (E) and (F) give the case of two-competitor systems following MAR(1) processes with (E) B 5 [[0.3, 20.1],[20.1, 0.3]] and (F) B 5 [[0.6, 20.2], [20.2, 0.6]]. The arrows in each panel give the expected density (tip of the arrow)at time t for 10 points at time t 2 1 (base of the arrow) selected from the stationary distribution.

(see the discussion after Eq. 14). Similarly, from Eqs.22 and 23, the variance of the stationary distributionrelative to the variance of the process error is greatestalong the eigenvector corresponding to the dominanteigenvalue of B, max(lB). Note, however, that a com-plete description of the variance of the stationary dis-tribution depends on all of the eigenvalues of B, notjust the one with the greatest magnitude.

Rate of convergence of the transition distribution

When measuring the stability of deterministic mod-els in terms of characteristic return rates, the equilib-rium towards which the system returns is a point or(more rarely) some form of stable limit cycle. In sto-chastic models, the equilibrium is the stationary dis-tribution. The rate of return to the stochastic equilib-rium can be measured by the rate at which the transitiondistribution converges to the stationary distributionfrom an initial, known value of the system. The more

rapid the convergence, the more stable the system. Thisis depicted for balls-in-basins in Fig. 2B, where theinitial locations of the balls (labeled 1) are far fromthe bottom of the basin (the mean of the stationarydistribution).

Different moments of the transition distribution mayapproach those of the stationary distribution at differ-ent rates. For simplicity, consider the univariate case.The rate at which the mean of the transition distributionapproaches the mean of the stationary distribution isgiven by Eq. 8; if Dx0 is the initial displacement of themean of the transition distribution from the mean ofthe stationary distribution m`, then the displacement attime t is Dxt 5 btDx0. Thus, the characteristic rate atwhich Dx approaches 0 increases as zbz gets smaller. Incomparison, the variance of the transition distribution,yt, approaches the variance of the stationary distribu-tion, y`, more rapidly, with the characteristic rate ofreturn scaling with b2 (Eq. 9).


The rate of convergence of the transition distributionfor a MAR(1) process is similar to the univariate pro-cess. The rate of return of the mean vector of Xt isgoverned by B (Eq. 19), and as in the deterministiccase, the dominant eigenvalue of B, max(lB), limits thereturn time of the mean to m`. The rate of return of thecovariance matrix, by contrast, involves the Kroneckerproduct B J B (Eq. 21). The eigenvalues of the matrixB J B are the products of all the pairs of eigenvaluesof B: lilj (i 5 1, . . . , p; j 5 1, . . . , p). When thestationary distribution exists, the eigenvalue pair-prod-ucts all have magnitudes smaller than max(lB). Thus,the elements of the covariance matrix of the transitiondistribution approach their equilibrium values rapidlycompared to the elements in the mean vector. As in thecase of the mean, the rate of return of the covariancematrix of the transition distribution to equilibrium islimited by the magnitude of the dominant eigenvalueof B J B, \max(lBJB)\, which equals \max(lB)\2.

The rate of convergence of the transition distributionto the stationary distribution is illustrated for a uni-variate system (Eq. 5) with b 5 0.6 (Fig. 3C) and b 50.95 (Fig. 3D) graphed as the mean 6 1 SD and 6 2SD of the transition distribution; these give the sameunivariate processes as Fig. 1. In the univariate case,b is the single eigenvalue of the systems. In additionto showing that the mean of the transition distributionconverges to that of the stationary distribution morerapidly when the magnitude of b is smaller, Fig. 3Dalso shows that the variance of the transition distri-bution approaches that of the stationary distributionmore rapidly than the mean.

The rate of convergence of the transition distributionto the stationary distribution must be interpreted dif-ferently from the rate of convergence of the populationsize to equilibrium in deterministic models. In deter-ministic models, rapid return rates to equilibrium in asense make a system more predictable. If there is someunknown perturbation at time t 5 0, the population sizewill be close to equilibrium shortly thereafter if the rateof return to equilibrium is rapid. In stochastic models,the rate of convergence of the transition distribution tothe stationary distribution is measured from someknown observation X0 5 x0. At time t 5 0, we havefull knowledge of the state of the system, because thevariance of the transition distribution is zero. Throughtime, the variance of the transition distribution increas-es monotonically to that of the stationary distribution(Eqs. 9 and 19). Therefore, if the state of the systemis known at a given point in time, information aboutthe state of the system (i.e., the variance of Xt condi-tional on X0 5 x0) is lost more rapidly if the varianceof the stationary distribution returns to steady statemore rapidly. This illustrates that, although there is acorrespondence between the convergence rates to equi-librium in deterministic systems and the convergencerates of the mean of the transition distribution to themean of the stationary distribution in stochastic sys-

tems, convergence rates of higher moments of sto-chastic processes do not have a counterpart in deter-ministic systems.

Reactivity

Neubert and Caswell (1997) warn that the commonlyused concepts of stability involve long-term properties.For example, the rate of return of a deterministic systemto equilibrium given by the dominant eigenvalue onthe community matrix, max(lB), formally gives the as-ymptotic rate of return after short-term dynamics havedissipated following a perturbation. For managementand other applications, properties of short-term systembehavior might be of interest. Neubert and Caswellproposed various measures of transient system re-sponse to perturbations. One in particular is reactivity:a highly reactive system tends to move farther awayfrom a stable equilibrium immediately after a pertur-bation, even though the system will eventually returnto the equilibrium point. They proposed the concept ofreactivity for continuous-time deterministic models,and here we develop the concept for stochastic modelsin discrete time.

Reactivity is illustrated in Fig. 2C. For a highly sta-ble (unreactive) system, disturbances (depicted by theopen arrows) may push a ball away from the bottomof the basin, yet on average the ball is pulled stronglytowards the bottom of the basin between each succes-sive time step. In contrast, in the shallow basin thesame disturbances on a ball starting the same distancefrom the bottom of the basin may leave the ball fartherfrom the bottom, because the ball is not brought strong-ly back towards the bottom between successive dis-turbances. This represents a more reactive system thanthat of the steep basin.

Fig. 3E and F give a numerical depiction of reactivityin the stochastic systems containing two competitorslike in Fig. 3A and B. For each of 10 points, the arrowsshow the expected value of the system at the next timepoint. The system in Fig. 3E is unreactive, with theexpected values of the system moving much closer tothe mean of the stationary distribution between suc-cessive time steps. Conversely, in the more reactivesystem in Fig. 3F, the system moves only weakly to-ward the mean.

We use two methods to quantify the reactivity of aMAR(1) process. Both involve taking the mean of theexpected change in distance from the mean of the sta-tionary distribution between successive time steps,where this average is taken over the stationary distri-bution. Letting Xt21 denote the vector of populationabundances at time t 2 1, the squared Euclidean dis-tance between Xt21 and the mean of the stationary dis-tribution, m`, is \Xt21 2 m`\2 5 (Xt21 2 m`)9(Xt21 2m`). For a given value of Xt21, the expected value ofXt is E [Xt z Xt21] 5 E [A 1 BXt21 1 «t21] 5 A 1« «t21 t21

BXt21. In Fig. 3E and F, if the base of a given arrowis Xt21, the tip of the arrow is the expectation given by


A 1 BXt21. The squared Euclidean distance of thisexpectation from the mean of the stationary distributionis \A 1 BXt21 2 m`\2 5 \B(Xt21 2 m`)\2; here we haveused the relationship that A 5 (I 2 B)m` (Eq. 15).Therefore, the change in the squared Euclidean distanceto m` between Xt21 and the expected value of Xt is\B(Xt21 2 m`)\2 2 \Xt21 2 m`\2.

As the first method for quantifying the reactivity ofa MAR(1) process, we use the expectation of this dif-ference when Xt21 is distributed according to the sta-tionary distribution, scaled by the expectation of \Xt21

2 m`\2. In the Appendix, we show that

2 2E [\B(X 2 m )\ ] 2 E [\X 2 m \ ]X t21 ` X t21 `` `

2E [\X 2 m \ ]X t21 ``

tr[S]5 2 (25)

tr[V ]`

where tr[S] and tr[V`] are the traces of S and V`. Thetrace of a covariance matrix is simply the sum of var-iances of the constituent random variables, so tr[V`] isthe sum of variances of individual species abundances.If the variance of the stationary distribution measuredby tr[V`] is large relative to environmental variancemeasured by tr[S], then the reactivity is relatively high.In other words, high reactivity corresponds to the casein which species interactions greatly amplify the en-vironmental variance to produce a stationary distri-bution with high variances in the abundance of indi-vidual species.

In order to calculate the reactivity as quantified byEq. 25, it is necessary to calculate the covariance ma-trix of the stationary distribution (Eq. 17). Thus, unlikethe previous measures of stability, this requires infor-mation not only about B but also about the covariancematrix of the process error, S, which is involved indetermining the stationary distribution. This restrictioncan be removed using a second method to quantifyreactivity. This involves calculating the ‘‘worst-case’’scenario for reactivity, i.e., the maximum value of thequantity on the left-hand side of Eq. 25. In the Ap-pendix, we show

2 2E [\B(X 2 m )\ ] 2 E [\X 2 m \ ]X t21 ` X t21 `` `

2E [\X 2 m \ ]X t21 ``

# max(l ) 2 1 (26)B9B

where max(lB9B) is the dominant eigenvalue of the ma-trix B9B, the transpose of B multiplied by B. Thus,using max(lB9B) 2 1 gives a method to characterize the‘‘worst-case’’ reactivity of a system that depends onlyon the matrix B.

Summary of stability properties

We have defined three types of measures of stabilityof a MAR(1) process. All three depend on the speciesinteractions given in the matrix B. The first type askshow interactions among species determine the variance

of the stationary distribution relative to the environ-mental variance. Interactions between species act as afilter to amplify the environmental variability given bythe process error Et. When species interactions greatlyamplify the environmental variability, the variance ofthe stationary distribution is much greater than the var-iance of the process error, and the system can be saidto be less stable. We give two ways to quantify thedegree to which species interactions amplify environ-mental variability. The amplification can be measuredby the eigenvalues of B, which quantify the amplifi-cation in the directions given by the corresponding ei-genvectors (Eqs. 22 and 23). The amplification can alsobe summarized by the quantity det(B)2/p, which givesa measure of the overall volume of the stationary dis-tribution relative to the volume of the distribution ofprocess error (Eq. 24).

The second type of measure of stability addressesthe rate at which the transition distribution convergesto the stationary distribution from a known point. Therate of convergence of the mean of the transition dis-tribution is limited by the dominant eigenvalue of B,max(lB) (Eq. 19). Therefore, max(lB) serves the samerole as the dominant eigenvalue of the community ma-trix in the stability analysis of deterministic models.The rate of convergence of the variance of the transitiondistribution depends on the eigenvalues of B J B, themaximum of which equals \max(lB)\2 (Eq. 21). Sincefor a stationary process \max(lB)\ . \max(lB)\2, therate of convergence of the variance is more rapid thanthe rate of convergence of the mean.

The third measure of stability addresses the short-term dynamics occurring from one time point to thenext. Reactivity is a measure of the displacement ofthe expected value of Xt from the mean of the stationarydistribution relative to the displacement of Xt21 at thepreceding time point. Reactivity can be quantified asthe expectation over the stationary distribution of thischange in displacement (Eq. 25). Alternatively, reac-tivity can be quantified for the ‘‘worse-case’’ scenario,which gives an expression for reactivity dependingonly on the matrix of species interactions, B (Eq. 26).The greater the variance of the individual species abun-dances relative to the variance of the process error, thehigher the reactivity of the system, and hence the lowerthe stability. Therefore, just like the first measure ofstability, greater variability of the stationary distribu-tion relative to the environmental variability indicateslower stability.

MODEL MODIFICATIONS

The MAR(1) processes discussed above can be mod-ified to encompass a wide range of ecological situa-tions. In addition to the state variables (variates) of thesystem (log population abundances), we may also haveinformation about covariates, such as temperature, nu-trient inputs, or other environmental factors that affectthe dynamics of the system. These covariates can be


included in the MAR(1) model, thereby making it pos-sible to account for long-term environmental trends andto address questions concerning the response of thesystem to known external disturbances. It is also pos-sible to constrain the MAR(1) based on prior biologicalinformation. For example, elements of the matrix B(giving species interactions) may be set to zero if it isknown that species do not interact, or the sign of anelement could be set to positive or negative if the typeof interaction between species is known. As a furthermodification, the MAR(1) model can be constructed toincorporate observation error. Observation error refersto any variability between the true values of variatesand covariates in the MAR(1) model and the valuesthat are observed. The most obvious source of obser-vation error is measurement error, but more generallyobservation error occurs whenever the true variates andcovariates that interact in the MAR(1) process are notknown precisely. For example, even if air temperaturewere known without error, the temperature experiencedby terrestrial animals would vary with wind speed,background substrate, topography, etc., making airtemperature an imprecise estimate of the animals’ truethermal regime. Observation error is ubiquitous in eco-logical systems and adds some complexities to fittingMAR(1) models to data.

Covariates and constraints

The MAR(1) model in Eq. 10 can be modified toinclude covariates:

X 5 A 1 BX 1 CU 1 Et t21 t t (27)

where Ut is a q 3 1 vector containing the values of qcovariates at time t, and C is a p 3 q matrix whoseelements cij give the strength of effect of covariate jon species i. The covariates Ut can be any factors thataffect the system and may be transformed based uponeither biological information about how the factors af-fect species’ per capita population growth rates or thestatistical properties of the MAR(1) model (see Modelselection and statistical inference). Furthermore, whilecovariates may often be abiotic factors such as tem-perature, biotic factors may also be used. For example,in a study of interactions among phytoplankton andzooplankton, fish predation on the zooplankton mayvary with the (known) number of fish in a lake. Becausethe system of interest consists of the phytoplankton andzooplankton, fish predation can be treated as a covariate(e.g., Ives et al. 1999a); the model includes the effectof fish predation on zooplankton, but does not considerthe reciprocal effect of zooplankton on fish.

The covariates Ut differ from the other sources ofenvironmental variability contained within the processerror, Et, not only by being measured, but also by theirstatistical properties. The process error Et is assumedto be temporally uncorrelated; E1, E2, . . . are all in-dependent. In contrast, the covariates Ut need not betemporally uncorrelated, and in fact generally will not

be. Thus, Ut can incorporate the effects of relativelyslow environmental changes. For example, for data setsin which many samples are taken over the course of aseason, it is reasonable to include seasonality; this canbe done, for example, by including a sinusoidal func-tion of day-of-year, or by including day-of-year and(day-of-year)2 as covariates to model seasonal factorsas a quadratic function (Ives et al. 1999a). Known andmeasured periodic environmental drivers can similarlybe included. In addition, Ut may include categoricalvariables to fit the MAR(1) to data sets collected si-multaneously at different locations. For example, sup-pose the MAR(1) model is being fitted simultaneouslyto the communities in two different lakes, and the re-searcher wants to include the possibility of differencesin mean densities between lakes that are not explainedby differences in measured covariates. The appropriatemodel would include a categorical variable Uj,t equalingzero for one lake and one for the other. The regressioncoefficients Cj calculated for Uj,t give the differencesbetween lakes in the constant terms of the autoregres-sion equations for each variate. Specifically, the con-stant terms for the first lake are A, while the constantterms for the second lake are A 1 Cj.

With the addition of multiple covariates, the numberof parameters in the MAR(1) model can become large,particularly when the number of species is large. There-fore, it is often desirable to reduce the number of pa-rameters in the model based upon biological infor-mation about the system. For example, in a study ofphytoplankton and zooplankton, nutrient input shouldhave no direct effect on zooplankton, and therefore thecoefficients in C for the effects of nutrient input onzooplankton can be set to zero. Similarly, the sign ofelements of B and C may be known a priori. Therefore,when fitting the MAR(1) model these sign constraintsmay be imposed by setting to zero any spurious pa-rameter estimates that arise when fitting a parameter-rich model, or by other techniques for constrained es-timation. There are no set rules about how to structurethe MAR(1) model; ecological knowledge of the par-ticular system under study is essential.

The measures of stability we developed (see sectionStability properties of MAR(1) models) apply only tobasic MAR(1) processes with no covariates (Eq. 10).Nonetheless, there is no ambiguity in applying themto the MAR(1) model with covariates (Eq. 27). Mostof the stability measures depend only on the matrix B,so the estimates of this matrix from Eq. 27 can be useddirectly. One measure of reactivity, Eq. 25, depends onthe covariance matrix of the stationary distribution, andthe stationary distribution in Eq. 27 depends on thecovariates. We can, however, estimate the stationarydistribution that would occur if the covariates were heldconstant by using B and the covariance matrix of theprocess error, S, estimated from the full model withcovariates, Eq. 27, thereby enabling the calculation ofreactivity. When any measure of stability is applied to


a model with covariates, the measures assess the dy-namics of the corresponding MAR(1) model in whichthe effects of the covariates have been factored out.

Observation error

Observation error occurs whenever the true valuesof the variates and covariates of the MAR(1) processare not precisely known. Observation error can be for-mally introduced into the MAR(1) model by separatingthe true values of the variates and covariatesX* U*t t

from those which are observed, Xt and Ut:

X* 5 A 1 BX* 1 CU* 1 E*t t21 t t

U* 5 Q(U* , U* , . . . , j , j , . . .)t t21 t22 t t21

X 5 X* 1 Gt t t

U 5 U* 1 H (28)t t t

where is the true process error, and Gt and Ht areE*tnormal random variables with mean 0 and covariancematrices SG and SH giving the observation error. Thismodel assumes that observation errors are normallydistributed (on a log scale) and are not temporally cor-related. Also, if observation error for different vari-ables taken at the same time are independent, then SG

and SH will be diagonal matrices. The effects of ob-servation error in the covariates on parameter esti-mation depend on the temporal autocorrelation struc-ture of the covariates. Therefore, we assume the co-variates are an autoregressive integrated movingU*taverage (ARIMA) process (Box et al. 1994) governedby the linear function Q in which values of mayU*tdepend on values in previous samples and normal ran-dom variables jt with mean 0 and covariance matricesSj. For generality, we leave the order of the ARIMAprocess unspecified.U*t

The statistical complexity of the MAR(1) model withobservation error can be illustrated using the univariatecase with a single covariate. Writing Eq. 28 in termsof the observed variables Xt and Ut,

X 5 a 1 bX 1 cU 1 (G 2 bG 2 cH 1 E*).t t21 t t t21 t t

(29)

This is no longer an AR(1) process, for two reasons.First, the total error terms Et 5 (Gt 2 bGt21 2 cHt 1

) are not temporally independent, since Et and Et21E*tboth depend on Gt21. Thus, the total error term Et is amoving average process (Box et al. 1994). Second, thetotal error term depends on the parameters b and c. If,for example, the value of b were positive, then the termGt 2 bGt21 would cause the total error to be a movingaverage process with negative autocorrelation. This il-lustration demonstrates that analysis of the MAR(1)model with observation error must explicitly accountfor the covariance structure of the error term that in-cludes both observation and process error.

Observation error can be incorporated into data anal-yses in several ways (Fuller 1987, Carpenter et al.1994a, Hilborn and Mangel 1997). First, if independentestimates of observation error (matrices SG and SH) areavailable, these can be incorporated directly (althoughcare must be taken to estimate SG and SH under thesame transforms as the data). Second, if the varianceof the observation error is known relative to the var-iance of the process error, then this ratio can be in-corporated into the model. Third, the variance of theobservation error can be estimated from the data at thesame time as the parameters and process error are es-timated. This uses information about the correlationstructure of the observed errors (Eq. 28) to estimatethe elements of SG and SH. We recommend against thisthird approach, however, because time-series data fromecological systems are often not extensive enough foraccurate estimates of observation variances. We feel itis better to use even a rough independent estimate ofthe observation variance (in either absolute terms orrelative to the process error) and then perform a sim-ulation sensitivity analysis of the effect of observationerror on the conclusions.

Summary of model modifications

The basic MAR(1) model (Eq. 10) can be modifiedto include covariates Ut for environmental factors otherthan species abundances (Eq. 27). This makes it pos-sible to estimate how the system responds to measuredexternal driving variables. It is also possible to struc-ture the MAR(1) model to disallow interactions (setparameters to zero) or constrain the sign of interactions.Finally, observation error can be included (Eq. 28),although this produces a process that is no longerMAR(1) in the observed variates and therefore mustbe analyzed accordingly. With these modifications,however, the measures of stability that we derived inthe preceding section remain unchanged; stability, asmeasured by the way in which interactions among spe-cies determine the properties of the stochastic process,still depends on the matrix B. Covariates and obser-vation error will certainly affect the estimated modelof the observed process. Nonetheless, the measures ofstability can still be interpreted unambiguously in termsof the stochastic properties of the true (unobserved)system in the absence of variability in the externaldrivers (covariates).

PARAMETER ESTIMATION

Parameters of the MAR(1) model can be estimatedusing conditional least squares (CLS), maximum like-lihood (ML), or a Bayesian framework (Schnute 1994).The first two approaches are described for general eco-logical time-series models by Dennis et al. (1995), andhere we present the specific procedures for the MAR(1)model. In some situations the estimates from CLS andML are identical, and in most situations the estimates


are close. Observation error requires a more compli-cated treatment, which we present in brief.

Conditional least squares (CLS)

The CLS estimates of parameters in the MAR(1)model are those values that minimize the squared dif-ference between the observed population abundancesat time t and those predicted by the MAR(1) model,conditional on the population abundances at time t 21. To make the calculations explicit, it is easiest to statethe MAR(1) model in full matrix form. For a data setwith T 1 1 data points, let X be the T 3 p matrix whosecolumn i contains all of the observed data for speciesi, excluding the last observation at time T; X 5 [X0,X1, . . . , XT21]9. Similarly, let Y be the T 3 p matrixof observed data excluding the first observation; Y 5[X1, X2, . . . , XT]9. Finally, let U 5 [U1, U2, . . . , UT]9be the T 3 q matrix of observed covariates and E 5[E1, E2, . . . , ET]9 be the T 3 p matrix of process errorterms. Then the MAR(1) model of Eq. 27 has the form

Y 5 1A9 1 XB9 1 UC9 1 E (30)

where 1 is the T 3 1 vectors of ones. The CLS estimatesof the parameters minimize, for each species i, thesquared difference between predicted and observedpopulation abundances one time step in advance, (Yi

2 Yi)9(Yi 2 Yi), where Yi 5 (ai 1 1 ), andˆ ˆXB9 UC9i i

Yi, ai, Bi, and Ci are the ith rows of the correspondingmatrices of estimated parameters. Letting Z 5 [1, X,U] be the T 3 (1 1 p 1 q) matrix with 1, X, and Uconcatenated horizontally as columns, and Di 5 [ai, Bi,Ci] be the 1 3 (1 1 p 1 q) vector containing theestimates of ai, Bi, and Ci, the CLS estimates are givenby the formula

21D9 5 (Z9Z) Z9Y .i i (31)

These estimates are exactly the least squares estimatesobtained using standard regression by treating Xt as adependent variable, and Xt21 and Ut as independentvariables. Note, however, that although the CLS esti-mates are the same, inference about the estimates (e.g.,confidence intervals) are not the same as those onewould obtain from standard regression, because thedata are time series (see Model selection and statisticalinference: Parameter confidence intervals).

The CLS estimators are asymptotically unbiased;their expected values equal the true parameter valuesas the sample size T gets infinitely large (Judge et al.1985, Tong 1990). For small sample sizes, however,there can be considerable bias in CLS estimators, andthis bias depends on the initial value X0 of the timeseries and the system itself, with systems having slowconvergence to the stationary distribution typicallyleading to greater bias (Fuller 1996). Unfortunately,there is no easy way to estimate this bias or to deter-mine what are ‘‘small’’ sample sizes. Nonetheless, thepresence of bias and a rough indication of its sign andmagnitude can be obtained from bootstrapping (see

Model selection and statistical inference: Parameterconfidence intervals).

Maximum likelihood

For any set of parameter values, it is possible tocalculate the likelihood of the observed data under aspecific MAR(1) model. The maximum likelihood pa-rameter estimates are those parameter values that pro-duce the greatest likelihood of the observed data (e.g.,Hilborn and Mangel 1997). Under the assumptions thatthe process error terms Et have a multivariate normaldistribution with mean vector 0 and covariance matrixS, and are temporally independent, the ML estimatesare those values which minimize the negative log-like-lihood function given by

pT T 1L 5 ln 2p 1 lnzSz 1 [Y 2 (A9 1 XB9 1 UC9)]

2 2 2

213 S [Y 2 (A9 1 XB9 1 UC9)]9. (32)

Note that, unlike CLS estimation, the elements of Sare treated as parameters and are estimated simulta-neously with the other parameters. If there are no apriori constraints placed on the parameters (e.g., allelements of B are estimated), then by matrix differ-entiation of L it can be shown that the ML estimatesare identical to the CLS estimates. However, when thereare constraints, the ML and CLS estimates differ, al-though in our experience the differences are small. LikeCLS estimators, ML estimators also may be consid-erably biased for small sample sizes.

Obtaining the ML estimates requires minimizing L,which has up to p( p 1 q 1 1) parameters in A, B, andC, and an additional p( p 1 1)/2 parameters in S. TheNelder-Mead simplex algorithm is convenient for min-imization (Press et al. 1992), and many mathematicaland statistical software packages include Nelder-Meadsimplex routines (e.g., Matlab, Mathworks 1996). Al-though the computations are relatively straightforward,for systems with many species the computational timeis much greater than that required to calculate the CLSestimates. Thus, for computationally intensive analy-ses, CLS estimates have advantages over ML estimates,particularly since in practice the estimates seem to bevery close in general.

Observation error

Observation error introduces the two problems thatthe total error is temporally correlated and depends onthe parameters of the MAR(1) process (see Model mod-ifications: Observation error). Observation error there-fore requires explicit attention to the covariance struc-ture of the total error between successive samples.

What are the consequences of not taking observationerror into account when in fact it occurs? Consider thesimple case of the univariate AR(1) model without co-variates (Eq. 5):


FIG. 4. Estimates of parameters b and c in an autore-gressive model with observation error (Eq. 35) from (A, C)100 simulated data sets using CLS estimation ignoring ob-servation error, (B, E) a state–space model assuming obser-vation error only in the variate, and (C, F) a state–space modelassuming observation error in both variate and covariate. Thetrue values are b 5 c 5 0.5, and for the simulation the au-toregression coefficient for the covariate was f 5 0.8. Num-bers in each panel give the mean of the estimates of thesimulated data sets.

X* 5 a 1 bX* 1 E* X 5 X* 1 G . (33)t t21 t t t t

For large sample sizes, the expectation of the CLS esti-mate of b computed from the observed variate Xt is

s s bsyx y*x* x*x*ˆE[b] 5 5 5 (34)s s 1 s s 1 sxx x*x* gg x*x* gg

where syx is the covariance between the observed ran-dom variables Xt and Xt21, sy*x* is the covariance be-tween the and , and sxx, sx*x*, and sgg are theX* X*t t21

variances of Xt21, , and Gt (cf., Fuller 1987:3). Thus,X*t21

the CLS estimate of b is biased towards zero, as alsooccurs in simple linear regression (Fuller 1987). Sinceb 5 1 gives the case of no density dependence (i.e.,the population dynamics exhibit random-walk behav-ior), Eq. 34 implies that in the univariate case obser-vation error falsely increases estimates of the strengthof density dependence (Shenk et al. 1998). For themultivariate case with covariates, the situation is morecomplex, and observation error could either increaseor decrease the naive CLS estimate of density depen-dence.

We will illustrate an observation-error model for aunivariate AR(1) process with observation error in boththe variate and a single covariate. The effects of ob-servation error in covariates on the estimates of co-efficients for both variates and covariates depend onthe temporal autocorrelation of the covariates. There-fore, it is necessary to analyze the covariates to deter-mine their temporal structure; this can be done withstandard model identification procedures (e.g., Box etal. 1994:181–223). For our example, we assume weknow that the covariate follows an AR(1) process. AnAR(1) model with observation error in which a singlecovariate follows an AR(1) process is given by

X* 5 a 1 bX* 1 cU* 1 «t t21 t21 t

U* 5 d 1 fU* 1 jt t21 t

X 5 X* 1 gt t t

U 5 U* 1 h (35)t t t

where and are unobserved, and Xt and Ut areX* U*t t

observed values of variate and covariate, respectively;«t and jt are process errors for the variate and covariate;gt and ht are observation errors for variate and covar-iate; and a, b, c, d, and f are regression coefficients. Inaddition to these coefficients, the parameters of themodel include the variances of «t, jt, gt, and ht, denoted

, , , and , respectively. Note that Eq. 35 gives2 2 2 2s s s s« j g h

a bivariate AR(1) process in which the covariate isU*ttreated as a variate.

Eq. 35 is in state-space form, and therefore for agiven data set the likelihood function can be computedusing a set of recursion equations known as the Kalmanfilter (Harvey 1989). The Kalman filter is a numericalprocedure in which successive estimates of the unob-served values in a time series are calculated se-X*t

quentially by first predicting from andX* X* U*t t21 t21

using the autoregressive model (Eq. 35), and then up-dating these estimates using the observed value Xt. Thisprocedure also computes the likelihood function andtherefore can be used in ML parameter estimation withnumerical minimization. Because the methods arestraightforward and presented excellently by Harvey(1989), we present only the results of a sample analysisof Eq. 35.

We generated 100 data sets from Eq. 35 for the caseof a strongly autocorrelated covariate ( f 5 0.8). Wethen fit each data set using (1) CLS which ignoresobservation error in both Xt and Ut, (2) a state–spacemodel in which observation error in Ut was ignored(i.e., setting 5 0 and treating Eq. 35 as a univariate2sh

AR(1) with covariates), and (3) the state-space modelwith observation error in both Xt and Ut given by Eq.35. To fit the data, we assumed that the observationerror variances, and , were known, but then fit all2 2s sg h

other model parameters. Also, for the state–space mod-els we assumed that the means and variances of thefirst observations of each data set equal the estimatedmeans and variances of the stationary distribution.

Fig. 4 plots the 100 estimates of parameters b and cusing each of the three methods. Ignoring observationerror in both Xt and Ut (Fig. 4A, D) leads to underes-timates of both parameters, while ignoring observationerror in Ut but not Xt (Fig. 4B, E) overestimates b andunderestimates c. These biases depend on the autocor-


relation of the covariate Ut ; when this autocorrelationis zero ( f 5 0), the estimate of b is unbiased, althoughthe estimate of c remains biased downwards (analysesnot shown). Accounting for observation error in bothvariate and covariate, not surprisingly, gave less biasedestimates (Fig. 4C, F), although the estimators areslightly more variable. Nonetheless, the ML estimatesobtained from Eq. 35 are only asymptotically unbiased,and when applied to small samples (with ‘‘small’’ de-pending on the structure of the underlying process) theML estimates can be noticeably biased (Fuller 1987:164–173).

Summary of parameter estimation

Conditional least squares (CLS) estimation providesa computationally simple and fast way to fit a MAR(1)model to data (Eq. 31). The parameter estimates arethose that would be obtained if standard regressionwere applied in which Xt were treated as dependentvariables, and Xt21 and Ut were treated as independentvariables. Maximum likelihood (ML) estimates (Eq.32) are identical to CLS estimates for unstructuredMAR(1) models (i.e., no constraints are placed on theparameters). Even when there are constraints, in ourexperience there is little difference between the CLSand ML estimates.

When there is observation error, estimates obtainedfrom either CLS or ML will be biased. Therefore, ac-counting for observation error is important when de-termining the stability of a system, because bias inparameter estimates will also bias estimates of stability.For example, for simple univariate AR(1) processes,observation error biases the estimate of b towards zero(Eq. 34), thereby making the process appear more sta-ble than it really is by any of the three measures ofstability we propose. Estimation with observation errorcan be performed by modeling the process in state-space form and applying a Kalman filter to computethe likelihood function (e.g., Eq. 35). When there isobservation error in the covariates, parameter estimatesdepend on the temporal autocorrelation structure of thecovariates.

MODEL SELECTION AND STATISTICAL INFERENCE

The preceding section has provided tools to estimatethe parameters of a MAR(1) model, but how shouldthe appropriate MAR(1) model be selected in the firstplace? Model selection has two components: specifyinga particular MAR(1) and then performing diagnosticsto determine whether the MAR(1) model adequatelydescribes the data. Once a satisfactory model is iden-tified, it can be used for statistical inference to obtain,for example, the confidence intervals for the modelparameters. It is also possible to obtain confidence in-tervals for other estimable quantities, in particular themeasures of stability we derived in the section Stabilityproperties of MAR(1) models.

Model selection

The general form of the model, such as what variatesand covariates to use, is dictated by the data in hand,the questions being addressed, and prior biologicalknowledge of the system. We can provide only theobvious rules of thumb that a useful statistical modelis more likely if data sets are long, observation erroris low, and the numbers of variates and covariates aresmall. Here, we deal with model selection once thesebroad decisions have been made. In particular, given aset of variates and covariates, what parameters of theMAR(1) model can be set to zero on statistical grounds,because their non-zero estimates do not improve the fitbetween the model and data?

Two criteria are commonly used in model selection,Akaike’s Information Criterion (AIC) and the BayesianInformation Criterion (BIC):

AIC 5 2L /T 1 2Q /Tmin

BIC 5 2L /T 1 Q log(T )/T (36)min

where Lmin is the value of the negative log-likelihoodfunction calculated at the ML estimates, and Q is thetotal number of estimated parameters. The criteria areapplied by selecting that model with the lowest AICor BIC score. Both AIC and BIC incorporate a ‘‘penaltyfactor’’ for the number of parameters in a model(2Q/T and Q log(T)/T, respectively), thereby favoringa model that includes only those parameters which pro-vide a minimum amount of additional informationabout the system (Box et al. 1994, Dennis et al. 1998).For BIC, the penalty factor is larger, so the BIC selectsmodels with fewer parameters than the AIC.

The number of possible MAR(1) models from whichto select can be large. There are various strategies forsearching for the AIC or BIC selected model; we de-scribe one strategy when we analyze our limnologicalexample (see Analysis of limnological data: model se-lection). Any strategy will necessarily involve esti-mating parameters for many models, and obtaining MLestimates is prohibitively intensive computationally.Therefore, in searching for a lowest-AIC or lowest-BIC model, we suggest using the CLS parameter es-timates rather than the ML estimates in calculating Lmin.The similarity between the CLS and ML estimatesmakes this a pragmatic approach. Computing ML es-timates with observation error (e.g., Eq. 35) is muchmore intensive than ML estimation without observationerror, so we restrict attention to model selection underthe assumption of no observation error. Once a modelis selected, the consequences of observation error forthe parameter estimates can be determined.

Diagnostics

Once a MAR(1) model is selected, it should be ex-amined to determine whether it gives a reasonable de-scription of the data set. The diagnostics are based onthe assumptions underlying the MAR(1), focusing on


the residuals between the observed log densities at t,Xt, and those predicted from the MAR(1) model andthe log densities Xt21 and covariates Ut. Useful diag-nostics are (Box et al. 1994, Fuller 1996):

1) Whether the residuals for each species are tem-porally autocorrelated, which can be determined usingautocorrelation functions (ACF) and partial autocor-relation functions (PACF).

2) Whether the mean and/or variance of the residualsare correlated with any of the variates or covariates ofthe model, which is most easily determined by graphingthe residuals or partial residuals (Neter et al. 1989:392)against the variates and covariates.

3) Whether the interactions among species or be-tween species and covariates are nonlinear, which canbe determined again by graphing the residuals againstthe variates and covariates and looking for curved re-lationships.

4) Whether the residuals are normally distributed,which can be determined from normal probability plots.In fact, the residuals need not be normally distributedto have a well-defined MAR(1) process, and CLS es-timation does not assume that the error terms Et arenormal (Judge et al. 1985). Nonetheless, ML estimationand consequently AIC and BIC model selection do as-sume normality.

5) Whether the model explains much of the varianceof the data. In ordinary (non-time-series) regression, itis usual to report the proportion of the total varianceof the dependent variables explained by the model. Intime-series analysis, values of Xt21 are used to predictXt. Therefore, it is most informative to ask what pro-portion of the change in log density from time t 2 1to time t is explained by the MAR(1) model (e.g., Iveset al. 1999a). This can be measured with the conditionalR2, which is the typical R2 applied to the change in logdensities. Often, a fitted model will explain more ofthe total variance in log density than the variance inthe change in log density between consecutive samples,so the total R2 will generally be greater than the con-ditional R2.

These diagnostics are simply guides to determinewhether the model fit is adequate and to suggest pos-sible measures to take to produce a better model, suchas transforming covariates. If the diagnostics show thatthe model fails, there are numerous tactics that can betried to correct the model (Box et al. 1994).

Parameter confidence intervals

Once a well-fitting MAR(1) model has been found,the confidence intervals for its parameters can be cal-culated, thereby giving a statistical assessment of thevalues of coefficients contained in the model. Calcu-lating confidence intervals must account for the co-variance structure of time-series data. For example,even though the CLS parameter estimates are the sameas those obtained from standard regression by treatingXt as a dependent variable and Xt21 as an independent

variable, the variance of the CLS estimator for a timeseries is not the same as the variance of the corre-sponding standard regression estimator, and thereforethe confidence intervals for the parameter estimateswill be different (Dennis and Taper 1994).

To give a simple example of the consequences ofthe covariance structure of time-series data, considerthe problem of calculating the mean of the stationarydistribution, m` 5 a/(1 2 b), of a univariate AR(1)process given by Eq. 5. The sample mean ` 5 X 5m(1/T ) Xt is an unbiased estimator of m`, while itsTSt51

variance is

T1V[m ] 5 V XO` t[ ]T t51

15 V[(a 1 bX 1 E )0 12T

1 (a 1 b(a 1 bX 1 E ) 1 E ) 1 · · ·]0 1 2

T T211 1 2 b 1 2 b5 V E 1 E 1 · · · 1 E1 2 T2 [ ]T 1 2 b 1 2 b

2 T11 2(T11)s 1 1 2 b 1 2 b5 T 2 2 1

2 2 21 2T (1 2 b) 1 2 b 1 2 b

2s 1ø . (37)

2T (1 2 b)

The factor s2/T is the variance of the estimator of themean that would apply if the data were not temporallycorrelated, while 1/(1 2 b)2 arises from the temporalautocorrelation of Xt. Obviously, when b is close to 1,the variance of the estimator of the mean becomeslarge. This is not surprising, because it will be difficultto estimate the mean of an AR(1) process that is closeto a random walk, or in ecological terms, a populationthat experiences weak density dependence.

There are two approaches to obtain confidence in-tervals for parameters of MAR(1) models. The firstuses bootstrapping (Efron and Tibshirani 1993, Denniset al. 2001). This approach begins by obtaining param-eter estimates; ML estimates can be used, although thecomputational speed of CLS estimation makes CLSmore practical. Parameter estimation produces a T 3p matrix of residuals e, with each of the p columnscorresponding to one of the p species in the system.Under the assumption of the MAR(1) model, the ele-ments in each column should be uncorrelated and iden-tically distributed. Therefore, they can be randomlysampled (with replacement) to create a new set of re-siduals, erand. To maintain the covariance structure oferrors for different species, the residuals e should besampled as rows which contain the p residuals fromthe same time point (one for each species). Thus, erand

is a T 3 p matrix whose rows are randomly sampledrows of the matrix e. A bootstrapped data set is createdby selecting an initial point, x0, calculating the pre-dicted value for x1 using the MAR(1) model with


Xt21 5 x0 and Ut 5 u1, adding the first element of erand,and continuing in this manner for the remaining T 21 points. The initial value x0 can be chosen as eitherthe first observation in the real data set, or the predictedmean of the stationary distribution conditional on theobserved values of the covariates u0; for long data sets,this choice makes little difference, while for short datasets we prefer the former approach if the real data sethas relatively unstable dynamics (e.g., max(lB) . 0.5)and x0 is therefore likely to be far from the mean. Theparameters are then estimated for the bootstrapped dataset. Repeating this procedure with a new set of residualserand for, say, 2000 bootstrapped data sets will produce2000 bootstrapped estimates for each parameter. The1 2 a confidence intervals are the intervals given bya/2 and (1 2 a/2) quantiles of the bootstrapped esti-mates. Joint confidence intervals can similarly be cal-culated from the joint distribution of bootstrapped pa-rameter values. Finally, the bootstrapped data sets canbe used to indicate whether the parameter estimatorsare biased (see Parameter estimation). Bias is indicatedif the average parameter values from the bootstrappeddata sets differ sizably and consistently from the es-timates obtained from the data set from which the boot-strapped data sets are constructed. Methods for com-pensating for bias using bootstrapping are reviewed inEfron and Tibshirani (1993). Any approach, however,would have to be explored and justified for MAR(1)models using numerical simulation studies.

The rationale behind the bootstrapping approach isthat, under the assumption that the fitted MAR(1) mod-el is correct, the residuals form a consistent estimateof the process error distribution (Efron and Tibshirani1993). Therefore, the confidence intervals represent theprobability bounds for obtaining parameter estimatesunder the fitted MAR(1) process. Often one wants toknow if a parameter differs from zero. In practice, ifthe confidence limits do not include zero, the parameterdiffers from zero. Departure from zero can be testeddirectly by structurally setting the parameter to zero,refitting the model, and using the refitted, reduced mod-el to create bootstrapped data sets. The full and reducedmodel are fitted to each bootstrapped data set, and thelikelihood ratio test statistic is calculated. The resultingbootstrapped test statistic values form a consistent es-timator of the distribution of the test statistic under thenull hypothesis that the parameter in question is zero.The likelihood ratio statistic for the actual data is com-pared to the desired percentile of the estimated nullhypothesis distribution to conduct the hypothesis test(Dennis and Taper 1994). Note that this will test wheth-er a parameter differs significantly from zero under theassumption that the values of all of the other parametersare those estimated from the MAR(1) model with theparameter in question set to zero. In other words, thistest is conditional on the estimated values of other pa-rameters.

A second approach to obtaining confidence intervals,called profile likelihood, takes advantage of the as-ymptotic chi-square distribution of a likelihood ratiotest statistic. One constructs a likelihood ratio test ofthe null hypothesis that a parameter is equal to a knownconstant versus the alternative hypothesis that the pa-rameter is not equal to that constant. The hypothesistest is inverted into a confidence interval; the set of allvalues of the constant for which the null hypothesiswould not be rejected at level a is an approximate 100(12 a)% confidence interval for the parameter. The tech-nique requires that the null hypothesis model has astationary distribution; otherwise, the likelihood ratiostatistic does not have an approximate chi-square dis-tribution. Dennis et al. (1995) demonstrate the profilelikelihood technique for multivariate time-series pop-ulation models. The technique can be adapted for CLSestimation using various chi-square statistics that ariseunder this type of estimation (Knight 2000).

Both of these approaches for obtaining confidenceintervals assume that there is no observation error.When there is observation error in the covariates, para-metric bootstrapping can be used. Parametric boot-strapping involves producing the bootstrap data sets bysimulating the fitted model, rather than by resamplingresiduals. The data sets are created under the obser-vation error MAR(1) model (Eq. 28), and the model isthen refitted to each of the bootstrapped data sets. Forlarge data sets with several variates, however, this iscomputationally prohibitive. Asymptotic confidenceintervals for the observation error MAR(1) model canalso be obtained from the log-likelihood function (cf.,Judge et al. 1985), although this suffers from the re-quirement of large sample sizes.

Finally, we note that these procedures all assume thatthe model is known to be correct. In fact, the correctmodel is not known, and therefore the parameter con-fidence intervals are conditional on the model selected.As suggested by Zeng et al. (1998), it is conceivableto bootstrap over the collection of all possible modelsby creating bootstrapped data sets from the model in-cluding all admissible parameters, selecting the best-fitting model using AIC or BIC, and then estimatingcoefficients. However, the numerical intensity of thisprocedure makes it impractical with present-day com-puting power.

Confidence intervals for other quantities

Bootstrapping can be used not only to obtain con-fidence intervals for the parameter estimates, but alsofor any function of the parameters. Natural candidatesof interest are the measures of stability in the sectionStability properties of MAR(1) models. The procedureis the same as that outlined for bootstrapping parameterestimates; construct bootstrapped data sets using thebest-fitting model, estimate the quantity in questionfrom the bootstrapped data sets, and determine the con-


fidence intervals from the resulting distribution of boot-strap estimates.

Summary of model selection

Once the general form of a MAR(1) model is chosen(i.e., what variates and covariates to include), the best-fitting MAR(1) model can be selected based on a num-ber of criteria; we suggest Akaike’s Information Cri-terion (AIC) and the Bayesian Information Criterion(BIC). Model selection involves deciding which pa-rameters to estimate and which to set to zero. Once thebest-fitted model is selected, diagnostics should be per-formed to determine whether the fitted MAR(1) modeladequately describes the data. Confidence intervals forthe parameter estimates can be obtained either by boot-strapping or by profile likelihood. Finally, bootstrap-ping can also be used to obtain confidence intervalsfor any function of the parameters. This makes it pos-sible to compare statistically the estimates of stabilityobtained from different ecological systems.

LIMNOLOGICAL EXAMPLE

To illustrate our measures of stability and the sta-tistical methods for fitting a MAR(1) model, we use adata set of phytoplankton and zooplankton in threelakes. After describing the objectives of the experimentand the data set, we proceed by analyzing the data:selecting a MAR(1) model, obtaining parameter esti-mates, and estimating the stability of the different lakecommunities. We use this example to ask two questionsabout our approach. Do MAR(1) models yield param-eter estimates that are sensible ecologically? And dothe measures of stability inform us about differencesin dynamics among lakes?

Objectives of the experiment

The roles of nutrients and predation in ecosystemdynamics are of great interest in both aquatic and ter-restrial ecology (Pace et al. 1999, Persson 1999, Polis1999). To examine the joint effects of nutrients andpredation in lakes, Carpenter et al. (2001) added nu-trients to two lakes with contrasting food webs, whilea third unmanipulated reference lake was monitored.They found that nutrient enrichment caused a largeincrease in biomass of phytoplankton in Peter Lake, asystem dominated by planktivorous fishes that selec-tively removed large-bodied Daphnia. In contrast, WestLong Lake was dominated by piscivorous bass thatreduced planktivorous fish and thereby increasedDaphnia abundance. Heavy grazing by large Daphniasuppressed the response of phytoplankton to the nu-trient enrichment in West Long Lake.

Description of the experiment and data set

The experiment is described in detail by Carpenteret al. (2001) and will be summarized briefly here. Thefood web of the reference lake, Paul Lake, was dom-inated by piscivorous largemouth bass throughout the

experiment. West Long Lake’s food web was domi-nated by piscivores (largemouth and smallmouth bass)and benthivorous yellow perch. Peter Lake was ma-nipulated by removing bass in 1991 to reconfigure thefood web to dominance by planktivorous minnow spe-cies. Beginning in 1993, Peter and West Long Lakeswere enriched with nitrogen and phosphorus at a ratiochosen to be near the pre-manipulation ratio and main-tain limitation by phosphorus. In this paper, we presentdata from two pre-enrichment years (1991 and 1992)and four years in which nutrients were added (1993–1996). We will refer to Paul, West Long, and PeterLakes as the reference, the low planktivory, and thehigh planktivory lakes, respectively.

All three lakes were monitored weekly during theperiod of summer stratification (approximately midMay to early September each year). The samples startedin midsummer in 1991, leading to nine samples in thatyear, while subsequent years yielded 17, 17, 17, 16,and 15 samples. The vast majority of summer sampleswere taken at 7-d intervals; of the total of 254 intervalsbetween samples during the summer months, 242(95%) were 7 d, whereas 3, 8, and 1 were 5, 6, and 8d, respectively. Although the samples were not all takenat the same interval, because the vast majority were 7d apart, we did not account for variation in time be-tween samples in the analyses. Nutrient additions weremeasured directly. Here, we used additions of the mostlimiting nutrient (phosphorus) as a covariate. Naturalinputs of phosphorus are small relative to the experi-mental inputs (Carpenter et al. 2001). We used potentialplanktivory on zooplankton as a second covariate. Po-tential planktivory was calculated as the biomass offishes that could potentially consume zooplankton,based on species and body size. Only two fish sampleswere taken per year, at the beginning and end of sum-mer stratification. We log-linearly interpolated betweenthese points to obtain estimates of potential planktivoryat the times of plankton samples.

We represented the planktonic food web by fourfunctional groups. Phytoplankton biomass (as mg chlo-rophyll/m2, integrated over the photic zone) was par-titioned into small-particle (passing a 35-mm screen)and large-particle (retained by 35-mm screen) fractions.The small phytoplankton are readily consumed by allzooplankton grazers, while the large phytoplankton aregenerally susceptible only to Daphnia. Zooplanktonbiomass (as mg dry mass/m2, integrated over the entirewater column) was partitioned into Daphnia and non-Daphnia biomass, reflecting the important role ofDaphnia as a grazer in these lakes. Methods for fieldsampling and laboratory analyzes are reported by Car-penter et al. (2001).

MAR(1) model selection

We fit the data from all lakes to the MAR(1) modelof the form


X 5 A 1 DA 1 DA 1 (B 1 DB 1 DB )Xt H L H L t21

1 CU 1 E . (38)t t

Matrix B contains coefficients for the interactions be-tween variates (small phytoplankton, large phytoplank-ton, Daphnia, and non-Daphnia), matrix C gives theeffects of environmental covariates (nutrient input andplanktivorous fish biomass) on each species-group, andA is a vector of constants. Matrices denoted DBH andDBL represent differences between the B matrix for thereference lake and the B matrices for the high and lowplanktivory lakes, respectively; matrices DAH and DAL

are defined similarly. To estimate the elements of ma-trices DB, we used the T 3 4 indicator matrices FH

and FL whose elements for each sample equal one forthe high and low planktivory lakes, respectively, andzero for the other two lakes. Thus, the model includedvariables FH and FL to obtain DAH and DAL, andFH·Xt21 and FL·Xt21 to obtain DBH and DBL, where ·denotes the element-by-element product of the matrices(i.e., if P 5 M·N, then the i–jth element of P is mijnij).

Because data were only collected during the summermonths, leading to a long unsampled interval betweenyears, in the CLS and ML analyses we did not includetransitions from the last sample in the fall and the firstsample in the spring. The ways in which we treatedbetween-year intervals for bootstrapping parameterconfidence intervals and fitting observation-error mod-els are discussed below (see Parameter estimation anddiagnostics).

Analyzing the data simultaneously for all lakes has twoadvantages over fitting each lake separately. First, dif-ferences in the B matrices among lakes are captured ex-plicitly in the matrices DBH and DBL. Therefore, modelfitting and statistical inference to identify differencesamong lakes involve determining whether the elementsof DBH and DBL are non-zero. Second, the parameters ofthe model are estimated using data from all three lakes,thereby providing greater precision in estimation.

Rather than consider all possible models, we con-strained the model on biological grounds. Specifically,we assumed that the planktivory manipulations had nodirect effect on phytoplankton groups, and the nutrientaddition had no direct effect on zooplankton groups.Further, we assumed that the Daphnia and non-Daph-nia zooplankton groups did not have a direct effect oneach other, but only interacted through their sharedconsumption of phytoplankton. Finally, we restrictedsome signs of interactions, excluding negative effectsof phytoplankton on zooplankton and positive effectsof phytoplankton on each other. To implement thesesign restrictions, if the signs of interactions obtainedin a fitted violated the restrictions, we forced the cor-responding coefficient to be zero.

We selected the best model structure based on theAIC criterion in which parameters were estimated usingCLS. To find the model with the lowest AIC, we ran-

domly selected a model by including or excluding co-efficients with equal probability. For each random ini-tial model, we either added a coefficient that was ini-tially excluded or subtracted a coefficient that was ini-tially included. If the AIC of the resulting modeldecreased, we proceeded with the resulting model.Each coefficient in turn was tested, and then the pro-cedure was looped so that coefficients were iterativelytested in sequence. This procedure continued until noimprovements were made in the fit of the model. Werepeated this procedure starting with 500 randomlyconstructed initial models. This procedure identified abest-fitting model and an additional four models withAIC values within 3 of the best-fitting model. Althoughthe close values of AIC do not strongly support se-lecting the best-fitting model over the near alternatives,all models were similar, differing in only the presence/absence of single coefficients. Furthermore, the stabil-ity measures differed little among models; values ofdet(B)2/p differed among all five models by 1.0%, 1.0%,and 3.0% for reference, high, and low planktivorylakes, respectively.

Parameter estimation and diagnostics

Fig. 5 shows the data from all three lakes, and Table1 gives the CLS parameter estimates with 95% confi-dence intervals for the AIC best-fitting model. To ob-tain confidence intervals, we created bootstrapped datasets by starting variates at their observed values foreach year of the data set, thereby making no assump-tions about how densities at the end of one year affectdensities at the beginning of the following year. Normalprobability plots of the residuals (Fig. 6) demonstratenear normality, with the exception of a few points atthe upper and lower tails of the distribution. The overalllevels of planktivory (fish biomass) were very differentamong lakes, and using log-transformed planktivoryreduced this difference. Therefore, we also performedthe analyses using untransformed fish biomass, but theresults were quantitatively very close to those obtainedusing log-transformed fish biomass, so we only reportthe results using log-transformed fish biomass. ThePACFs of the residuals showed little evidence of au-tocorrelation (Fig. 7), although there was a high pos-itive lag-4 partial autocorrelation of residuals of largephytoplankton; we have no biological explanation forthis and believe it is not biologically significant.

The residuals obtained from CLS estimation werenearly normal. Therefore, it is not surprising that theML estimates of model parameters are close to thoseobtained by CLS (Table 2). We calculated ML estimatesonly for the best-fitting model obtained using CLS dueto the computational intensity of finding the best-fittingmodel using ML estimation. Thus, we assumed thatML and CLS estimates are similar enough to give thesame best-fitting model.

We also estimated parameters with the observation-error model of Eq. 28 (Table 2), using independent


FIG. 5. Data from the limnological example. The top four panels for each lake give the log biomasses of the variates:large phytoplankton, small phytoplankton, Daphnia, and non-Daphnia. The total number of sample points for each lake is91. Data are plotted as consecutive samples, with the lines broken between years. The lower two panels for each lake givethe rate of nutrient (phosphorus) addition in the experimentally manipulated lakes and the log-biomass of planktivorous fish.The planktivorous fish biomasses were estimated only at the beginning and end of the summer stratification, so we interpolatedthe data for the analyses.

estimates of the variances of observation errors in var-iates and covariates, and , respectively. For the2 2s sg h

two phytoplankton and two zooplankton variates, weset to 0.04 and 0.16, respectively. These correspond2sg

to standard deviations of 0.2 and 0.4, which are roughlytwice the standard deviations estimated by Carpenteret al. (1994a) for phytoplankton and zooplankton; forthe present analysis we doubled the previous estimatesto magnify the potential effects of observation error.Thus, on a log scale the observed values of the phy-toplankton and zooplankton densities have a roughly95% chance of lying within 60.4 and 60.8 units. Theplanktivory covariate was given an observation vari-ance of 0.36, corresponding to a standard deviation of0.6; this corresponds roughly to the typical observationerror for planktivorous fish obtained using samplingmethods similar to those used for our data (He 1990).Incorporating observation error in a covariate requiresspecifying its autocorrelation structure and treating itas a variate of a state-space model (e.g., Eq. 35). Be-

cause planktivory was interpolated from two pointseach summer, it cannot easily be incorporated into astate–space model. Nonetheless, we treated interpolat-ed planktivory data as an AR(1) process. The first-orderautocorrelation of planktivory, estimated simulta-neously with the other model parameters, was high(0.97); although this is an artifact of the interpolation,the true autocorrelation in planktivory will also be high,since abundances of planktivorous fish will changeslowly over a summer relative to changes in zooplank-ton and phytoplankton abundances. Because nutrientloading was experimentally manipulated, we assumedit had no observation error.

For fitting the observation–error model (Eq. 28), wefactored out changes in variates between the end ofsummer in one year and the beginning of summer inthe next year by restarting the Kalman filter procedureeach year. The initial values of the variates each yearwere set equal to the observed values in the initialsamples, and the initial estimates of the variance of the


TABLE 1. Parameter estimates and bootstrapped 95% confidence intervals for the AIC best-fitting autoregressive model,Eq. 27, fit to the limnological data using CLS estimation.

A) Fit of the model

Variate Total R2 Conditional R2

Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.310.740.910.44

0.270.340.170.25

B) Parameter estimates for interactions between variates

ParameterLarge

phytoplanktonSmall

phytoplankton Daphnia Non-Daphnia

BR (reference lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.50 [0.36, 0.58]0.61 [0.47, 0.68]

0.10 [0.014, 0.20]

20.019 [20.032, 20.004]0.76 [0.64, 0.82]0

00.56 [0.43, 0.63]

BH (high planktivory lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.50 [0.36, 0.58]20.077 [20.19, 0.04] 0.61 [0.47, 0.68]

0.33 [0.15, 0.56]0.10 [0.014, 0.20]

20.019 [20.032, 20.004]0.95 [0.91, 0.97]0

00.56 [0.43, 0.63]

BL (low planktivory lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.50 [0.36, 0.58] 20.36 [20.68, 20.05]0.072 [20.13, 0.21]

0.10 [0.01, 0.20]

20.019 [20.032, 20.004]0.76 [0.64, 0.82]0

20.10 [20.27, 20.04]00.56 [0.43, 0.63]

C) Parameter estimates for the effects of covariates on variates, C

Parameter Nutrient addition Planktivory


0.19 [0.08, 0.33]0.31 [0.25, 0.40]00

00

20.14 [20.26, 20.03]20.047 [20.12, 0.02]

Note: Units are: zooplankton, log(mg dry mass/m2); phytoplankton, log(mg chlorophyll/m2), nutrient addition, mg P·m22·d21;planktivory, log(g).

FIG. 6. Normal probability plots of the residuals for thefour variates from the AIC best-fitting model using CLS pa-rameter estimates.

process error each year were set to the global estimatedvalue (Harvey 1989).

The coefficient estimates in the observation-errormodel are similar to the CLS estimates. The largestdifferences occurred for the intraspecific effect of smallphytoplankton on itself in the low planktivory lake(0.072 for CLS; 0.25 with observation error) and forthe effect of Daphnia on small phytoplankton (20.02for CLS; 20.17 with observation error). Other param-eter estimates increased or decreased, demonstratingthat observation error can have different effects on dif-ferent coefficients in the MAR(1) model.

The parameter estimates from the MAR(1) modelsmake ecological sense. Daphnia had a negative effecton small phytoplankton, as expected from this effectivegrazer (Vanni 1986, Carpenter and Kitchell 1993). Thenon-Daphnia zooplankton had a measurable negativeeffect on small phytoplankton only in the low plank-tivory lake. Small phytoplankton had a positive effecton the non-Daphnia group in all lakes, and had a strongpositive effect on Daphnia in the high planktivory lake,which experienced a strong response of small phyto-plankton to nutrient addition. The lack of a detectableeffect of small phytoplankton on Daphnia in the ref-


FIG. 7. Partial autocorrelation coefficients of residuals(PACFs) for the four variates in the AIC best-fitting auto-regressive model with CLS parameter estimates (Table 1).Lines give the approximate 95% confidence limits calculatedas 62 SE, where SE 5 T21/2 and T is the length of the set ofresiduals (Box et al. 1994:68). The length of the set of re-siduals changed from 237 (lag 1) to 165 (lag 5), becausepartial autocorrelation between points from different yearswere not included.

TABLE 2. Parameter estimates for the AIC best-fitting model fit to the limnological data using ML estimation, Eqs. 27 and38, and the observation-error model, Eq. 28.

A) Parameter estimates for interactions between variates

Variate

ML

Largephyto-

plankton

Smallphyto-

plankton DaphniaNon-

Daphnia

Observation error

Largephyto-

plankton

Smallphyto-

plankton DaphniaNon-

Daphnia

BR (reference lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.500.60

0.10

20.020.770

00.55

0.480.70

0.10

20.170.740

00.60

BH (high plankitovory lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.5020.066 0.60

0.340.10

20.020.950

00.55

0.4820.076 0.68

0.360.10

20.170.970

00.60

BL (low planktivory lake)Large phytoplanktonSmall phytoplanktonDaphniaNon-Daphnia

0.50 20.390.076

0.10

20.020.770

20.1000.55

0.48 20.390.25

0.10

20.170.740

20.1100.60

B) Parameter estimates for the effects of covariates on variates C

Variate

ML

Nutrient addition Planktivory

Observation error

Nutrient addition Planktivory


0.200.3200

00

20.1320.048

0.250.2500

00

20.1420.045

erence and low planktivory lakes is likely due to thelow variance of small phytoplankton in these lakes.Detectable competitive interactions between large andsmall phytoplankton only occurred in the high and lowplanktivory lakes. Nutrient input had greater impact onsmall phytoplankton than large phytoplankton. Surfacearea/volume considerations suggest that small phyto-plankton exploit nutrients more effectively than largephytoplankton (Chisholm 1992). Planktivory hadgreater impact on Daphnia than on non-Daphnia. Thisis consistent with the preference for Daphnia shownby size-selective planktivores (Brooks and Dodson1965, Hall et al. 1976).

Comparison of stability among lakes

We used several measures to quantify the stabilityof the three lakes (Table 3). To measure stability interms of the variance of the stationary distribution rel-ative to the variance of the process (environmental)error, we computed the eigenvalues of B (Eqs. 22 and23) and det(B)2/p (Eq. 24) from the CLS parameter es-timates for each lake. The return rate of the transitiondistribution to the stationary distribution is measuredby the dominant eigenvalue of B, max(lB), which givesthe asymptotic rate of return of the mean (Eq. 19), andthe dominant eigenvalue of B J B, max(lBJB), whichgives the asymptotic rate of return of the variance (Eq.21). Finally, to measure reactivity, we used both


TABLE 3. Stability measures for the AIC best-fitting autoregressive model, Eq. 10, withparameters estimated using CLS.

Parameter Reference High planktivory Low planktivory

Eigenvalues

det(B)2/p

max(lBJB)2tr(S)/tr(V`)max(lB9B) 2 1

0.74 [0.64, 0.82]0.59 [0.47, 0.68]0.54 [0.43, 0.63]0.48 [0.36, 0.58]0.33 [0.27, 0.39]0.55 [0.42, 0.67]

20.58 [20.67, 20.47]20.45 [20.58, 20.25]

0.93 [0.88, 0.96]0.60 [0.49, 0.71]0.54 [0.43, 0.63]0.48 [0.36, 0.58]0.38 [0.31, 0.44]0.86 [0.78, 0.93]

20.24 [20.32, 20.14]0.07 [20.08, 0.30]

0.74 [0.64, 0.82]0.51 [0.37, 0.62]0.48† [0.36, 0.58]0.07† [20.12, 0.25]0.12 [0.03, 0.22]0.55 [0.41, 0.67]

20.60 [20.70, 20.49]20.44 [20.57, 20.25]

† In 43 of 2000 bootstrapped data sets, two eigenvalues for the low planktivory lake werecomplex. These are ignored in calculating the 95% confidence bounds (shown in brackets).

TABLE 4. Stability measured by max(lB) and det(B)2/P forthe AIC best-fitting model with parameters estimated withCLS (Eq. 10), ML (Eq. 38), and observation error (Eq. 28).

Parameter andestimation method Reference

Highplanktivory

Lowplanktivory

max(lB)CLSMLObservation error

0.740.760.74

0.930.930.95

0.740.760.74

det(B)2/p

CLSMLObservation error

0.330.360.38

0.380.400.44

0.120.140.24

2tr[S]/tr[V`] (Eq. 25) and max(lB9B) 2 1 (Eq. 26). Tocalculate 2tr[S]/tr[V`], it is necessary to compute thecovariance matrices for the process error and stationarydistribution for each lake. For the covariance matrix ofprocess errors, S, we used the residuals from the datafor each lake when all three lakes were fitted simul-taneously. We then calculated the stationary distribu-tion from Eq. 17 using the lake-specific matrices B.All of these measures of stability factor out the vari-ability caused by covariates and therefore assess thedynamics around the conditional stationary distributionfor fixed values of the covariates.

Comparison of the eigenvalues of B for the threelakes reveals two patterns. First, the maximum eigen-value in the high planktivory lake is much larger thanthe other two lakes. This is due to the large coefficientof Daphnia on itself, b33, in BH (Table 1), indicatingthat the dynamics of Daphnia in the high planktivorylake are highly autocorrelated and show near-random-walk behavior. Second, the minimum eigenvalue forthe low planktivory lake is much smaller than the othertwo lakes. This is due to the small value of the coef-ficient of small phytoplankton on itself, b22, in BL (Table1), indicating that the dynamics of small phytoplanktonare highly intraspecifically regulated. These two pat-terns explain differences among the results of the re-maining measures of stability.

When measured by det(B)2/p, the low planktivorylake is the most stable of the lakes; the lack of overlapof the 95% confidence intervals of det(B)2/p for the low

planktivory lake and the other two lakes providesstrong statistical support for this difference. In contrast,the other measures of stability identify the high plank-tivory lake as the least stable, but do not distinguishbetween the reference and low planktivory lakes. Thehigh planktivory lake has slower rates of returns of themean and variance of the transition distribution to thestationary distribution (max(lB) and max(lBJB)) and hasgreater reactivity (max(lB9B) 2 1 and 2tr[S]/tr[V`]).This contrast between det(B)2/p and the other measuresof stability can be explained by differences in the ei-genvalues of the B matrices. The dominant eigenvaluecorresponds to the ‘‘slowest’’ dimension of the mul-tidimensional dynamics; specifically, the approach ofthe transition distribution to the stationary distributionis slowest in the direction given by the eigenvectorcorresponding to the dominant eigenvalue. Because thedominant eigenvalue for the high planktivory lake isgreater than the other two lakes, measures of stabilitythat are influenced strongly by the dominant eigenvalueof B distinguish this lake from the other two. Not onlyare the rates of return of the mean and variance of thetransition distribution to the stationary distribution dic-tated by the dominant eigenvalue, reactivity is alsostrongly influenced by the ‘‘slowest’’ dimension of thesystem. In contrast, det(B)2/p measures the size of thestationary distribution relative to the size of the dis-tribution of process error. Because det(B)2/p depends onall of the eigenvalues and hence all of the dimensionsof the system, it is sensitive to the low value of thesmallest eigenvalue found for the low planktivory lake.

To compare estimates of stability obtained from pa-rameter estimates by CLS, ML, and our observation–error model, Table 4 gives estimates of max(lB) anddet(B)2/p for all three parameter estimation procedures.All three sets of estimates are similar, with the greatestdifference being the increase in det(B)2/p for the lowplanktivory lake with the inclusion of observation er-ror, reflecting the increase in the smallest eigenvaluewith observation error (Table 2).

Finally, we determined the sensitivity of the stabilityestimates to observation error by varying the magni-tude of the predetermined observation error relative tothe process error and calculating stability according to


FIG. 8. The sensitivity of two measures of stability of thereference, high planktivory, and low planktivory lakes to ob-servation error. (A) gives the dominant eigenvalue of B,max(lB), and (B) gives det(B)2/p. For the variates and covar-iate (planktivory), the ‘‘relative observation error’’ is the ratioof the observation error variance to the variance of the un-observed process error, g (Eq. 28).

max(lB) (Eq. 22) and det(B)2/p (Eq. 24). We set thevariance of observation errors Gt and Ht (Eq. 28) equalto g times the variances of the process errors Et and jt.Thus, g is related to reliability ratio K (Fuller 1987:5)by g 5 (1 2 K)/K. As g increases from zero, relativelymore of the total error in the observed variates andcovariates is attributed to observation error. We re-stricted the analysis to the model structure of the best-fitting AIC model (Table 1).

Increasing observation error has little effect on max(lB)(Fig. 8A). With increasing values of observation errorvariance, the values of det(B)2/p increase for all lakes (Fig.8B), thus implying that accounting for observation errorleads to decreases in the estimates of stability of the sys-tems. Note, however, that values of det(B)2/p in all lakesincrease in parallel, so observation error has not affectedour conclusions about the relative stabilities of the threelakes as measured by det(B)2/p.

Why are the stability properties of thelakes different?

By all measures of stability, the low planktivory lakeis more stable than the high planktivory lake, whereasthe reference lake has similar stability to the high plank-tivory lake when measured by det(B)2/p but similar sta-bility to the low planktivory lake when measured by theother stability properties. Since the reference lake wasunmanipulated, our measures of stability did not simply

identify the lakes that experienced the greatest experi-mental disturbance. Furthermore, the standard deviationsof log biomasses of small and large phytoplankton, Daph-nia, and non-Daphnia were 0.66, 0.36, 2.0, and 0.52 inthe reference lake; 1.1, 0.66, 1.3, and 0.88 in the lowplanktivory lake; and 1.3, 1.2, 3.8, and 0.92 in the highplanktivory lake. With the exception of Daphnia, the var-iability of plankton in the reference lake was lower thanthe other two lakes, and Daphnia variability was onlymoderately higher than in the low planktivory lake (2.0vs. 1.3). Therefore, our measures of stability did not sim-ply identify the lakes with higher variability. This is be-cause all of the measures of stability are determined bythe endogenous processes summarized by the B matrixin the MAR(1) model. The measures of stability assesshow systems respond to environmental fluctuations, nothow variable the systems are; more stable systems arethose that respond less to the same severity of environ-mental fluctuations.

Visualizing the differences in stability among lakesis complicated by the two sources of environmentalvariability that differed among lakes: variability in thecovariates Ut and process error Et. To portray the datain a way that equalizes both types of environmental var-iability, we generated time series from the equation

ˆx 5 Bx 1 et t21 t (39)

where B is the estimated B matrix (different for dif-ferent lakes), and et are CLS residuals for each lakedivided by their within-lake standard deviation; thisdivision causes the variance of the process error to bethe same for each lake. For the initial point of eachyear, we used the difference between the observed ini-tial point and the mean of the stationary distributionthat would occur if the covariates maintained theiryearly mean values for each year. In order to give aunivariate portrayal of the data, we then projected thetime series xt onto the axis given by the eigenvectorcorresponding to the dominant eigenvalue, max(lB), ofthe B matrix for each lake. Along this axis, the timeseries are described by a univariate AR(1) process withthe autoregression coefficient given by max(lB). Theresulting plots in Fig. 9 are comparable to those ofsimulated AR(1) processes in Fig. 1.

Fig. 9 shows that abundances in the high planktivorylake wander slowly up and down, while abundances inthe other two lakes are drawn more tightly to the meanof the stationary distribution (from Eq. 39, this meanis zero). For all three lakes, the eigenvector corre-sponding to max(lB) is nearly parallel to the axis oflog Daphnia biomass; this is the reason the dominanteigenvalues for all three lakes are close to the coeffi-cient for Daphnia on itself given by b33 in the B ma-trices. Thus, Daphnia is largely responsible for differ-ences in stability among lakes as measured by max(lB),a result similar to that obtained by Cottingham andSchindler (2000) using deterministic measures of sta-bility. The importance of Daphnia is also seen in the


FIG. 9. Univariate depictions of the dynamics of the ref-erence, high planktivory, and low planktivory lakes. The sixlines in each panel give the dynamics in each of six years(Eq. 39). The vertical axis gives the projection of the logbiomasses onto the eigenvector corresponding to the domi-nant eigenvalues of the estimates of B for each lake. Theprojections are standardized such that variances of the resid-uals along the eigenvectors are the same for each lake. Dif-ferences in dynamics caused by differences in the variancesof the covariates in each lake are removed as described inthe text (Eq. 39).

raw data in Fig. 5; despite experiencing quantitativelysimilar perturbations due to nutrient addition as thehigh planktivory lake, the abundances of Daphnia re-mained relatively constant in the low planktivory lake.

Biologically, the lack of self-regulation of Daphniain the high planktivory lake was probably caused byhigh planktivory lowering Daphnia densities relativeto the other lakes. With low Daphnia densities andcorresponding high densities of small phytoplankton,the Daphnia population had the potential to increase,causing Daphnia densities to fluctuate more broadly inresponse to environmental variability.

A further difference in stability among lakes revealedby det(B)2/p is the greater stability of the low plankti-vory lake caused by strong self-regulation of small phy-toplankton (b22 in BL). This occurred even though themean biomass of small phytoplankton was lower in thelow planktivory than in the high planktivory lake dur-ing the experimental manipulations. The stability ofthe low planktivory lake could be a consequence of thehigh Daphnia abundance. Because Daphnia have ahigh phosphorus requirement, phosphorus recycling inlakes dominated by Daphnia may be slow, leading tonutrient limitation in the small phytoplankton (Elser etal. 1996). Nutrient limitation may contribute to the self-limitation term for small phytoplankton, b22, estimatedby the model.

Summary of limnological example

We fit a MAR(1) model to data from three lakes thatwere manipulated as part of an experiment to determinethe response of phytoplankton and zooplankton to nu-trient addition and planktivory. The MAR(1) identifiedinteractions that were anticipated to be important onbiological grounds. According to all of the measuresof stability we used, the low planktivory lake was morestable than the high planktivory lake, whereas the ref-erence lake was indistinguishable from either the lowor high planktivory lakes depending on the measure ofstability used. Analyses incorporating observation er-ror did not change the general conclusions. The dif-ferences in stability among lakes were largely deter-mined by the self-regulation of Daphnia and small phy-toplankton, given by the corresponding diagonal termsof the B matrices.

DISCUSSION

MAR(1) models

We used MAR(1) models to characterize the station-ary distribution and stability properties of multispeciestime-series data. Is a MAR(1) process a biologicallyplausible model of an ecological community? Probablynot, at least not in detail. But the more important ques-tion is: Can a MAR(1) model describe ecological inter-actions within communities and the stochastic charac-teristics of the dynamics they produce? For many com-munities, we believe the answer will be yes.

Many models of multispecies interactions are de-signed to explore different mechanisms that underliepopulation dynamics. Since many of these mechanismsinvolve nonlinearities, they demand nonlinear models(May and Oster 1976, Hastings et al. 1993, Carpenteret al. 1994a, Costantino et al. 1997, Ives et al. 2000a).In contrast to nonlinear mechanistic models, MAR(1)models give simple log-linear depictions of species in-teractions. This is not to say that MAR(1) models arenot mechanistic. They include interactions among spe-cies and interactions between species and environmen-tal factors. But they summarize these interactions as


linear dependencies of log population growth rates onlog densities of species and (possibly transformed) en-vironmental covariates. Thus, while MAR(1) can beviewed as purely empirical descriptions of data, theycan also be used to explore the mechanisms by whichspecies interactions affect multispecies dynamics.

An important advantage of using MAR(1) models isthat their dynamical properties can be computed in astraightforward way. If a MAR(1) model is used as anapproximation of a nonlinear process, then the stationarydistribution computed from the MAR(1) model is likewisean approximation of the stationary distribution of the truenonlinear process. Interpreting a MAR(1) model as a log-linear approximation to a nonlinear process is appropriatewhen analyzing complex multispecies time series withmultiple environmental covariates. Even though the un-derlying biological processes may be nonlinear, identi-fying these nonlinearities will be difficult unless they arevery strong (Ives et al. 1999a). As a log-linear approxi-mation, parameters in the MAR(1) model give the averagestrengths of species–species and species–environment in-teractions. When making comparisons among similartypes of communities, as in our example of planktoniclake communities, potential biases caused by the log-linearity of MAR(1) models will likely be similar amongcommunities, thereby reducing the chances of nonlinear-ities confounding the comparison of stability among com-munities.

Measures of stability

The three types of measures of stability that we de-fined for stochastic systems are similar in spirit to sta-bility concepts applied in deterministic settings. Themost frequently used deterministic measure of stabilityis the characteristic return rate, as measured by thedominant eigenvalue of the community matrix (May1974). The interaction matrix B in a MAR(1) modelserves the same role as the community matrix, with thedominant eigenvalue of B determining the character-istic return rate of the mean of the transition distri-bution to the mean of the stationary distribution. Thesquare of the dominant eigenvalue measures the char-acteristic return rate of the variance of the transitiondistribution, so the variance of the transition distri-bution converges more rapidly to that of the stationarydistribution than does the mean.

The suite of eigenvalues of B also measures the de-gree to which the environmental process error alongthe corresponding eigenvectors is amplified by speciesinteractions, with greater amplification occurring foreigenvalues with larger magnitude. This amplificationcan be understood by referring to the deterministic re-sult that the return rates to equilibrium along the ei-genvectors are set by the corresponding eigenvalues.If eigenvalues are large, deterministic return rates areslow, and in the stochastic system the perturbationscaused by environmental variability are not broughtquickly back to the mean of the stationary distribution.

The same idea underlies the concept of reactivity,which is based on the expected change in the distancefrom the mean of the stationary distribution from onetime point to the next. If, on average, the expectedchange does not bring the population much closer tothe mean, then the system will be less stable. Highreactivity (when deviations do not tend strongly to-wards the mean) coincides with slow average returnrates and eigenvalues of B with large magnitudes.

An important feature to all three types of stability mea-surements is that they separate endogenous and exoge-nous components of system variability. The endogenouscomponent of variability is captured in the B matrix ofspecies interactions, while exogenous sources are givenas either measured environmental factors (covariates) orunmeasured environmental fluctuations (process error).What constitutes endogenous vs. exogenous componentsof variability is explicitly defined by the structure of theMAR(1) model. For example, if we had decided to focusonly on phytoplankton in our analyses, we could havetreated grazing by zooplankton as an exogenous factor,rather than an endogenous factor. In this case, Daphniaand non-Daphnia biomasses would be used as covariatesrather than variates. Treating zooplankton as exogenouscovariates would not change the CLS estimates for thecoefficients of interactions between phytoplankton, al-though ML estimates might change, and the best-fittingmodel (as determined by the AIC or some other criterion)might include different interactions from the best-fittingmodel obtained when treating zooplankton as endogenousvariates.

The measures of stability are most useful to make com-parisons among systems. In our example of planktoniccommunities, our analyses identified the low planktivorylake as having higher stability than the high planktivorylake, with the reference lake similar to one or the otherlake depending on the measure of stability applied. Themost important result of the analyses is that they posethe question: Why are the dynamics of the three lakesdifferent? The analyses give some general insight into theanswer to this question; they finger Daphnia and smallphytoplankton as the key groups in explaining the dif-ferences in stability among lakes. Nonetheless, we viewthis as only the starting point to understanding the stabilityof these planktonic communities.

Other stability properties

We have used components of the stationary distri-bution of population abundances to obtain measures ofsystem stability, but other concepts of stability can bederived from other system characteristics. A commonquestion asked about ecosystems is how much a systemwill change in response to a long-term environmentaltrend, such as climate change or lake eutrophication(Schindler 1974, Schindler et al. 1985, Frost et al.1995). This question can be placed in the MAR(1)framework: How will mean species abundances changewhen there is a change in the mean value of one or


more covariates (Ives et al. 1999a)? All of the measuresof stability we have developed apply to the time-in-variant stationary distribution; even though we haveincluded covariates, we have used these to factor outthe time-dependency on long-term environmentaltrends so we can obtain information about the corre-sponding time-invariant stationary distribution deter-mined by the B matrix. Nonetheless, the problem ofhow the mean of the stationary distribution wouldchange in response to a change in the mean of thedistribution of a covariate is related to the measures ofstability we have derived.

As a simple example, consider the univariate AR(1)process given by

X 5 a 1 bX 1 cU 1 E .t t21 t t (40)

If the covariate Ut is a stationary stochastic processwith long-term mean f`, then the mean of the station-ary distribution of Xt is

a 1 cf`m 5 . (41)` 1 2 b

If a change Df in f` occurs, the corresponding changein m` is

cDfDm 5 . (42)

1 2 b

Thus, smaller values of b imply smaller changes in themean of the stationary distribution relative to thechange in the mean of the covariate Ut. This result issimilar to that for the variance of the stationary dis-tribution; Eq. 5 shows that the variance in the stationarydistribution relative to the environmental varianceequals 1/(1 2 b2), which decreases with b as long asb is positive. In the multivariate case, the change inthe mean of the stationary distribution relative tochanges in the mean of covariates can be calculatedfrom the B matrix (Ives 1995b). Therefore, all of thestatistical techniques we have described can be em-ployed to predict how ecosystems will change in re-sponse to long-term environmental trends.

Another type of question that can be asked is whetherthe composition of a community remains stable in theface of environmental fluctuations, or whether specieswill be lost. In our analyses, we have assumed thatextinction does not occur. Our analyses might givesome information about which species in a communityare at risk of extinction if, for example, species havinglow mean and high variance in abundance are morelikely to drop to extinction. Nonetheless, this questionis not addressed directly. Extinction implies that theMAR(1) process is not stationary and therefore requiresinvestigating the properties of nonstationary processes(e.g., Chesson and Case 1986, Chesson 1994).

Conclusions

The chief advantage of MAR(1) models is that theycan easily link data and theory, thereby making many

theoretical ideas statistically testable. Thus, they allowthe exploration of many theoretical ideas with data. Al-though we have focused on stability, other properties ofspecies interactions could be tested using a MAR(1)framework. For example, it would be possible to testwhich species in a community are most sensitive to aparticular environmental factor, or which species haverelatively large impacts on other species in the commu-nity. It is also possible to use MAR(1) models to predictthe response of communities to novel environmental dis-turbances if the effects of these disturbances on the pop-ulation growth rates of different species are known (Iveset al. 1999a). Finally, although we have applied MAR(1)to community variables (biomass of different functionalgroups), the same techniques can be applied to ecosystemvariables, such as primary productivity and nutrient con-tent in different pools in an ecosystem.

Given the growing concerns about the effects of en-vironmental disturbances on communities over largespatial and temporal expanses, MAR(1) models cangive us insights into changes that we have already ob-served, and changes that we should anticipate.

ACKNOWLEDGMENTS

We thank T. M. Frost, K. Gross, A. M. Kilpatrick, J. L. Klug,and J. Ripa for numerous conversations and insights about sta-bility. Comments by N. J. Gotelli, K. Gross, J. Klug, J. Ripa,N. S. Urquhart, and two anonymous reviewers greatly improvedthe manuscript. N. S. Urquhart suggested Eq. A9, and he kindlyallowed us to include it in the Appendix. This work was startedat the ‘‘Community Dynamics’’ working group at the NationalCenter for Ecological Analysis and Synthesis, Santa Barbara,California, USA, and was supported by additional grants fromthe National Science Foundation to A. R. Ives, B. Dennis, andS. R. Carpenter. The whole-lake manipulation experiments weresupported by the National Science Foundation and the A. W.Mellon Foundation.

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APPENDIX

DERIVATION OF STABILITY PROPERTIES

Variance of the stationary distribution

The first of the two methods we use to measure the varianceof the stationary distribution relative to the variance of thedistribution of the process error (Eqs. 22 and 23) involvestransforming matrix B according to axes given by its eigen-vectors (Ives 1995a). Specifically, B can be written in realcanonical form (Cronin 1980) as B 5 SJS21, where J is ablock-diagonal matrix whose elements consist of either a realeigenvalue of B (the scalar l), or a 2 3 2 matrix of the form[[a, b], [2b, a]] corresponding to a pair of complex conjugateeigenvalues a 6 bi of B. Matrix S consists of the eigenvectorsassociated with the eigenvalues of J; the real eigenvectorscorresponding to real eigenvalues are entered as columns ofS, while complex conjugate pairs of eigenvectors correspond-ing to complex pairs of eigenvalues are entered as two col-umns, one containing the real and the other the imaginaryparts of the eigenvectors. Matrix S can be used to transformthe autoregressive process (Eq. 10) into a new coordinatesystem given by the eigenvectors of B. Let Xt 5 SXt and Et

5 SEt be, respectively, vectors containing linear combina-tions of the population densities and process errors in the newcoordinate system, and let A 5 SA. Then from Eq. 10˜ ˜ ˜ ˜X 5 SX 5 SA 1 SBX 1 SE 5 A 1 JX 1 E . (A.1)t t t21 t t21 t

Thus, the transformation produces a description of the au-toregressive process in which species interactions are encap-sulated in the block-diagonal matrix J. The covariance matrixof the environmental variables in the new coordinate systemis C 5 SSS9.

Since J is a block-diagonal matrix, along the eigenvectorscorresponding to a real eigenvalue l, the dynamics are givenby an equation of the form

˜ ˜ ˜X 5 a 1 lX 1 Et t21 t (A.2)

where Xt and Et are the population densities and process errorsalong the same eigenvector. Thus, from Eq. 7 the variance inthe stationary distribution along this eigenvector is

2 2y 5 c /(1 2 l )` (A.3)

where c2 is the corresponding diagonal element of the co-variance matrix C. This is Eq. 22 in the text.

A similar result can be obtained for complex eigenvalues.In the plane defined by the columns of S consisting of realand imaginary components of a complex set of eigenvectorscorresponding to an eigenvalue pair a 6 bi, the dynamicsare given by

r r c r˜ ˜ ˜ ˜X 5 a 1 aX 1 bX 1 Et t21 t21 t

c r c c˜ ˜ ˜ ˜X 5 a 2 bX 1 aX 1 E (A.4)t t21 t21 t

where superscripts r and c denote values along the axes givenby the real and imaginary components of the complex ei-genvector pair, respectively. Ives (1995a) shows that the sumof variances along these axes is

r c r c 2y 1 y 5 (c 1 c )/(1 2 \a 6 bi\ )` ` (A.5)

where \a 6 bi\ is the magnitude of the eigenvalues a 6 bi.This leads to Eq. 23 in the text.

Reactivity

Reactivity is a measure of how strongly short-term (overone time step) changes tend to bring population abundancestowards the mean of the stationary distribution. Consider firstthe deterministic case. Let Dxt 5 xt 2 x` be a vector of


departures of species abundances from their equilibrium val-ues and consider the linear deterministic model

Dx 5 BDx .t t21 (A.6)

For a given perturbation xt21 from equilibrium, the squaredEuclidean distance between xt21 and x` is \Dxt21\

2 5. How much farther from or nearer to equilibriumDx9 Dxt21 t21

will xt be? Scaling the difference between \Dxt21\2 and \Dxt\

2

by the initial magnitude of the perturbation, we find that

2 2\Dx \ 2 \Dx \ Dx9 B9BDxt t21 t21 t215 2 1. (A.7)2\Dx \ Dx9 Dxt21 t21 t21

The term BDxt21/ Dxt21 is known as the RaleighDx9 B9 Dx9t21 t21

quotient and can be used to calculate the ‘‘worst-case’’ sce-nario for the reactivity of the system (i.e., the maximum dis-tance xt will be from x`). If B has full rank (i.e., no eigenvaluesare zero), then the symmetric matrix B9B is positive definite(i.e., Dx (B9B)Dxt21 . 0 for all nonzero vectors Dxt21). Also,9t21

B9B has all real and positive eigenvalues. The worst-casescenario is when the perturbation Dxt21 happens to be in thesame direction as the eigenvector of B9B associated with thelargest eigenvalue, max(lB9B). In this case, the Raleigh quo-tient attains its maximum value of max(lB9B), and the farthestdistance from equilibrium that xt can advance beyond xt21 ismax(lB9B) 2 1 (Eq. A.7 evaluated at its maximum value).Thus, max(lB9B) is a measure of reactivity, with values ofmax(lB9B) . 1 giving worst-case scenarios in which the sys-tem initially moves farther from equilibrium following a per-turbation. On the other hand, when max(lB9B) , 1 the systemalways moves immediately back towards equilibrium.

This analysis of the deterministic case is restricted to theworst-case scenario. For stochastic systems it is possible toask how far, on average, is the system likely to change fromone time step to the next with respect to the mean of thestationary distribution. Let Xt denote the random variable forthe population size at time t, and let DXt 5 Xt 2 m` be thedifference between Xt and the mean of the stationary distri-bution. The expectation of DXt given DXt21 is E [DXt z DXt21]«t21

5 BDXt21. Let \E [DXt z DXt21]\ denote the squared Eu-«t21

clidean distance of the expectation of DXt given DXt21 fromthe mean of the stationary distribution, m`. Thus, if the Eu-clidean distance of a given value of DXt21 from m` is \DXt21\

2,then the expectation for the change in distance from m` be-tween times t 2 1 and t is \E [DXt z DXt21]\2 2 \DXt21\

2.«t21

Because DXt21 itself is a random variable, reactivity dependson the expectation taken over the distribution of DXt21, andto obtain a measure comparable to reactivity defined for de-terministic systems (Eq. A.7), we define reactivity for sto-chastic systems as

2 2E [\E [DX z DX ]\ ] 2 E [\DX \ ]X « t t21 X t21t21 t21 t21 . (A.8)2E [\DX \ ]X t21t21

The smaller this quantity, the greater the tendency for thepopulation to move towards the mean of the stationary dis-tribution between successive time steps. Below we derive anexpression for this quantity.

Because B9B is a symmetric matrix, it can be decomposedsuch that B9B 5 KqK9 where q is a diagonal matrix con-taining the eigenvalues of B9B, and K is a matrix with col-umns containing the corresponding eigenvectors. The matrixB9B is positive definite and symmetric, and therefore its ei-

genvalues are real and its eigenvectors are orthogonal, so KK95 I. Let Zt21 5 K9Xt21 be the population size measured withrespect to axes given by the eigenvectors of B9B, and let nt21

5 K9mt21 be the mean of Zt21. Then2E [\E [DX z DX ]\ ]X « t t21t21 t21

5 E [DX9 B9BDX ]X t21 t21t21

5 E [{(X 2 m ) 1 (m 2 m )}9X t21 t21 t21 `t21

3 B9B{(X 2 m ) 1 (m 2 m )}]t21 t21 t21 `

5 E [(X 2 m )9B9B(X 2 m )]X t21 t21 t21 t21t21

1 (m 2 m )9B9B(m 2 m )t21 ` t21 `

5 E [(Z 2 n )q (Z 2 n )]Z t21 t21 t21 t21t21

1 (m 2 m )9B9B(m 2 m )t21 ` t21 `

p

5 w g 1 (m 2 m )9B9B(m 2 m ) (A.9)O i ii t21 ` t21 `i51

where wi are the eigenvalues of B9B and gii are the diagonalelements of the covariance matrix of Zt21.

Eq. A.9 gives the general case for an arbitrary distributionof Xt21. To obtain the reactivity of the system averaged overa long time period, we can let the distribution of Xt21 be thestationary distribution, Xt21 5 X`. In this case, the secondterm in Eq. A.9 becomes zero, and gii are the diagonal ele-ments of the covariance matrix of stationary distributiontransformed according to axes given by the eigenvectors ofB9B. Thus, reactivity is given as

p

w gO i ii2 2E [\E [DX z DX ]\ ] 2 E [\DX \ ]X « t t21 X t21 i51` t21 ` 5 2 1.p2E [\DX \ ]X t21` gO iii51

(A.10)

The reactivity of the process at the stationary distributiondepends on the average of the eigenvalues of B9B weightedby the variances of the stationary distribution gii. The worst-case scenario is when all of the variance in the process erroroccurs along the eigenvectors corresponding to the dominanteigenvalue of B9B, max(lB9B). In this case

2 2E [\E [DX z DX ]\ ] 2 E [\DX \ ]X « t t21 X t21` t21 ` 5 max(l ) 2 1.B9B2E [\DX \ ]X t21`

(A.11)

This leads to Eq. 26 in the text.An alternative expression for reactivity in stochastic sys-

tems can be obtained for the case in which Xt21 5 X`.Because the expression is in quadratic form,DX9B9BDX` `

E ] 5 tr[B9BV`] (e.g., Judge et al. 1985:[DX9B9BDXDX ` ``

16), the trace of the matrix product B9BV`. Furthermore,since V` is symmetric, tr[B9BV`] 5 tr[BV`B9]. Finally, fromEq. 16 tr[BV`B9] 5 tr[V`] 2 tr[S]. Thus, since

[\ [DXt z DXt21]\2] 5 ,E E E [DX9B9BDX ]X « X ` `` t21 `

2 2E [\E [DX z DX ]\ ] 2 E [\DX \ ] tr[S]X « t t21 X t21` t21 ` 5 2 .2E [\DX \ ] tr[V ]X t21 ``

(A.12)

This is Eq. 25 in the text.

SUPPLEMENT

Programs and data for many of the analyses of the limnological data set are available in ESA’s Electronic Data Archive:Ecological Archives M073-003-S1. Programs are written in Matlab (MathWorks 1996), and data are given as text files formattedto be accessed by the programs.

estimating community stability and ecological …estimate the strengths of interactions between...

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