THE IMPORTANCE OF THE VARIANCE AROUND THE MEAN EFFECT SIZE OF ECOLOGICAL PROCESSES

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CONCEPTS & SYNTHESISEMPHASIZING NEW IDEAS TO STIMULATE RESEARCH IN ECOLOGY

Ecology, 84(9), 2003, pp. 2335–2346q 2003 by the Ecological Society of America

THE IMPORTANCE OF THE VARIANCE AROUND THE MEAN EFFECTSIZE OF ECOLOGICAL PROCESSES

LISANDRO BENEDETTI-CECCHI1

Dipartimento di Scienze dell’Uomo e dell’Ambiente, University of Pisa, Via A. Volta 6, I-56126, Pisa, Italy

Abstract. Experiments in ecology are usually designed to provide tests of hypotheseson the influence of the mean intensity of causal processes, whereas the variance aroundmean effects has been largely overlooked as a causal force in biological assemblages.Repetition of experiments in space and time provides an estimate of this variability atspecific scales, but does not explain how changes in variance generate structure in assem-blages and the extent to which variance and mean intensity interact. This paper seeks toidentify suitable procedures for empirical analyses on the influence of variance and meanintensity of predictor ecological variables on spatial and temporal patterns in natural pop-ulations. A survey of the ecological literature indicates that temporal variability in studiesof disturbance and in analyses of consumer–resource interactions is generally expressed interms of frequency of events. This is inappropriate, as frequency confounds the variancewith the mean effect size of a process. A possible solution to the problem involves ex-perimental designs in which levels of intensity and those of variability are chosen inde-pendently over explicit spatial or temporal scales and treated as fixed, orthogonal factors.Examples are offered for various scenarios of consumer–resource interactions along withindications for statistical tests of hypotheses. Such novel approaches have important ram-ifications for understanding variability in a wide range of ecological contexts and forpredicting the response of assemblages to increased environmental fluctuations, includingthose expected under modified climate conditions.

Key words: consumer–resource interactions; disturbance; ecological models; ecological pro-cesses, variance vs. frequency; effect size; experimental design; interaction webs; mean intensity cf.variance of ecological processes; patterns, temporal and spatial; random vs. fixed effects in ecologicalexperiments; simulations and trophic interactions.

INTRODUCTION

Concepts of natural variability continue to have astrong impact in ecology, influencing the philosophicaland methodological foundations of the discipline andits social implications (Pickett and White 1985, Strong1986, Pimm 1991, Gaines and Denny 1993, Kareivaand Bergelson 1997, Huston 1999, Landres et al. 1999).Variation is increasingly viewed as a genuine propertyof ecological systems, and the notion that unexplainedvariability conveys at least as much ecological infor-mation as regularities in nature is now widespread (Ch-esson 1986, Horne and Schneider 1995). In attemptsto explain natural variability, ecologists are increas-ingly concerned about the need to understand the caus-es of variation in the magnitude of ecological processesover explicit spatial and temporal scales (Dayton 1971,

Manuscript received 8 July 2002; revised and accepted 3 Jan-uary 2003. Corresponding Editor: A. M. Ellison.

1 E-mail: [email protected]

Levin 1992, Underwood and Petraitis 1993, Menge etal. 1994, Schneider 1994, Berlow et al. 1999). Al-though the term ‘‘variability’’ can be vague (Kareivaand Bergelson 1997), it embraces both variation in spa-tial and temporal patterns in populations and assem-blages (response variables) and heterogeneity in theprocesses that generate and maintain the patterns (pre-dictor variables). Elucidating the linkages betweenthese two facets of natural variability is of fundamentalimportance to advance ecological understanding and toincrease accuracy and precision of ecological predic-tions.

In principle, both predictor and response variablesmay have distributions with unknown parameters suchas the mean and the variance. A clear distinction there-fore exists between dependent and independent vari-ables with respect to these parameters (Fig. 1). Tra-ditionally, estimates of variance are used in ecologicalstudies to describe spatial and temporal patterns in at-tributes of populations (Andrewartha and Birch 1954,

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2336 LISANDRO BENEDETTI-CECCHI Ecology, Vol. 84, No. 9

FIG. 1. Distinction between mean and variance in predictor and response variables and how they are addressed in ecologicalstudies. The thickness of black arrows is proportional to relative importance. Gray arrows and the question mark identifygaps in ecological research that are addressed in the present paper. See Introduction and Appendix A for further details andfor references.

den Boer 1968). It is the variance of response variables,usually in association with changes to the mean, thathas attracted most of the attention of ecologists. In aseminal paper, Taylor (1961) documented the existenceof a positive relationship between the mean and thevariance in measures of density for various natural pop-ulations. Pielou (1969) provided conceptual and ana-lytical procedures to interpret changes to the mean andto the variance in density of populations by distin-guishing between intensity and grain of pattern. Thisdistinction focused attention on spatial dispersion ofindividuals to link pattern to process in clumped pop-ulations (see also Underwood 1996). Other authorshave attempted to identify the ecological mechanismsunderlying the relationship between the mean and thevariance (e.g., Hanski 1987, Perry and Woiwod 1992)and several exhaustive reviews have been published onthis topic (Hurlbert 1990, McArdle et al. 1990,McArdle and Gaston 1992, Gaston and McArdle 1993).

In contrast, comparably fewer studies have focusedattention on the variance of predictor variables as acausal explanation for the observed patterns (Butler1989, Benedetti-Cecchi 2001). This discrepancy re-

flects the common practice in hypothesis testing of de-fining effect sizes with reference to the mean intensityof predictor variables. It also reflects difficulties withthe design and execution of experiments to test hy-potheses of variation of ecological processes and, ul-timately, the lack of coherent ecological theory relatingthe variance of processes to the mean and the varianceof response variables.

Incorporating the variance of ecological processesinto explanatory models requires understanding howchanges in the magnitude of causal forces impact pop-ulations and assemblages over explicit spatial and tem-poral scales. A profitable way to tackle this issue is toreplicate experiments in space and time. This approachis recommended whenever tests of hypotheses aboutthe generality of a process are of concern (e.g., Un-derwood and Petraitis 1993, Beck 1997). Examples areoffered by some studies of competition and consumer–resource interactions conducted in aquatic and terres-trial habitats (Connell 1983, Schoener 1983, Sih et al.1985, Menge et al. 1994, Reader et al. 1994, Benedetti-Cecchi et al. 2000, Thrush et al. 2000). Although rarelyused, procedures are available to estimate variance

September 2003 2337VARIANCE OF ECOLOGICAL PROCESSES

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components of statistical interaction terms that measurethe consistency (or lack thereof) of a process in spaceand/or time (Searle et al. 1992).

This analytical framework is extremely important toadvance ecological understanding of variable phenom-ena. There are, however, unresolved issues about thesensitivity of variance components to spatial and tem-poral changes in the values of the target variable. Alarge interaction term can result from changes in theoverall value of a response variable (e.g., the abun-dance of a population) across experimental sites or pe-riods—changes that occur independently of changes inthe magnitude of the specific predictor variable beinginvestigated. Furthermore, there can be logistical dif-ficulties in estimating variance components over verysmall or very large scales from replicated experiments(Carpenter 1990, Carpenter et al. 1995), and no clearprocedure has yet been developed to relate these quan-tities to predictive theoretical models. Finally, repeti-tion of experiments across sets of sites or periods oftime extracts random sources of variation that requireexplanation on their own. That is, there is no quanti-tative relationship between the variance componentsestimated from experiments and the causes of spatialor temporal variability. Experimental tests of hypoth-eses that treat variability as a fixed effect are necessaryto derive quantitative, predictive relationships betweenthe response of populations (or assemblages) to chang-es in the regime of variation of predictor variables.

An alternative approach for testing hypotheses aboutthe effects of variable processes is that of manipulatingthe frequency of events (e.g., predation or physical dis-turbance) in field experiments (Navarrete 1996, Collins2000, McCabe and Gotelli 2000). The frequency of aprocess over explicit temporal scales is a measure ofvariability (the number of events per unit of time orspace), but it confounds the temporal variance of aprocess with its mean effect size (or intensity) (seeVariance vs. frequency of ecological processes, below).In a previous study (Benedetti-Cecchi 2000a) I haveshown the importance of distinguishing between meanand variance of effects in consumer–resource interac-tions using conceptual, simulative, and correlativeanalyses. Teasing apart these two sources of variation,however, requires new approaches to natural variabilityand innovative experimental designs.

The main purpose of the present paper is to identifythe logical, experimental, and analytical tools neces-sary for establishing cause–effects relationships be-tween changes in variance (in addition to mean values)of predictor variables and the mean and the varianceof response variables, such as abundance of popula-tions, over explicit spatial and temporal scales (this isthe area of research implied by the question mark inFig. 1). This goal is pursued by reviewing current ap-proaches to the understanding of ecological variabilityand by making an existing model that relates variabilitybetween patterns and processes (Benedetti-Cecchi

2000a) more workable and transparent and, finally,identifying appropriate experimental designs to testpredictions of the model. The hope is that of stimu-lating theoretical and empirical research on the influ-ence of variable processes in a wide range of ecologicalcontexts. This should result in increased capabilities ofpredicting the response of populations and assemblagesto increased environmental fluctuations, includingthose expected under modified climate conditions.

VARIANCE VS. FREQUENCY OF

ECOLOGICAL PROCESSES

A review of the recent ecological literature (Appen-dix A) indicates that conceptual and empirical attemptsto separate the influence of intensity from that of var-iability of ecological processes are limited. Amongthose studies that have considered the two sources ofvariation simultaneously, frequency was largely usedto describe the variability (temporal variability in mostcases) of a process. I will show that this approach isinappropriate, as frequency confounds variability withintensity of a process.

Frequency is the number of events per unit of timeor space. Evaluating the effects of changing the fre-quency of a process on some response variable inevi-tably involves contrasts between spatial or temporalextents that differ in the number of times an event hasoccurred. Under this scenario, the two contrasting con-ditions differ not only in terms of the number of timesan event takes place, but also in terms of the overallintensity with which a process operates over the spe-cific spatial or temporal scale considered. This is il-lustrated by the few experimental studies on distur-bance and consumer–resource interactions that haveexamined variation in these processes explicitly.

In an elegant study that simultaneously examined theinfluence of frequency, intensity, and spatial extent ofdisturbance on macroinvertebrates colonizing artificialsubstrata in streams (McCabe and Gotelli 2000), tem-poral variability was imposed by abrading the substra-tum once or twice a week. Two different levels of in-tensity were applied for each frequency in an orthog-onal design: (1) low intensity (by abrading the sub-stratum with a wire brush) and (2) high intensity (byscraping with a paint scraper). This design was notappropriate to separate the effects of temporal vari-ability from that of intensity of disturbance, becausetreatments exposed to different frequencies of distur-bance also differred in terms of the total number ofscrapings (and therefore overall intensity) imposed dur-ing the course of the experiment (lasting 43 days), re-gardless of the procedure used to abrade the substra-tum.

Similar problems arise with experimental studies onconsumer–resource interactions. Early attempts to ex-plore the effects of variable consumer–resource inter-actions in assemblages included studies on the influ-ence of large herbivores in the Serengeti (McNaughton

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1983) and effects of bluegill sunfish on macroinver-tebrates in a North Florida lake (Butler 1989). Thestudy of Butler contrasted constant vs. temporally var-iable regimes of predation by maintaining the meanintensity of predation comparable across treatments.That is, effects due to changes in temporal variance ofpredation were properly examined without the con-founding effect due to differences in intensity acrosslevels of temporal variability. The experiment, how-ever, did not examine variation in intensity at all, sothat it was impossible to partition mean effects andtemporal variability of predation. This problem wasaddressed later by Navarrete (1996) in a study of pre-dation by whelks on intertidal sessile invertebrates ofrocky shores in Oregon. Two orthogonal treatmentswere included in the Navarrete study in order to assessthe separate and combined effects of intensity and tem-poral variability of predation: (1) intensity (with 2, 4,and 8 whelks/experimental unit) and (2) frequency(with three levels: low, medium, and high). Frequencyof predation was manipulated by allowing predatorsinside cages for two months and removed for eitherfour (low frequency) or two (medium frequency)months, or maintained inside cages throughout the ex-periment (high frequency). Each frequency of preda-tion was crossed with each level of intensity and re-peated cyclically for about two years.

The study of Navarrete (1996) had the merit of in-corporating effects of mean intensity (expressed on ayearly basis) and temporal variability of predation inthe same analytical framework. The author himself,however, acknowledged that the different combinationsof intensity 3 frequency overlapped in the regime ofpredation imposed to populations of prey (Navarrete1996: 306). That is, expressing temporal variability ofpredation as the frequency of events confounded tem-poral variation with intensity.

A distinction between frequency and temporal var-iance is instrumental in separating effects due to theoverall intensity of a process with its inherent vari-ability over explicit scales. Thus, the alternative wouldbe to focus on temporal variance rather than frequency(e.g., Butler 1989) and to devise an experimental designwhere this factor is made orthogonal to intensity ofpredation (see Criteria for experiments: Temporal var-iability, below). As indicated by the results of Navar-rete (1996), strong interactive effects between the meaneffect size of predation and its inherent temporal var-iability can have dramatic influences on assemblagesof species (see also Sih and Ziemba 2000).

As I have emphasized elsewhere (Benedetti-Cecchi2000a), understanding these relationships is key to de-velopment of better models of consumer–resource in-teractions and to reconcile contrasting results betweentheoretical and empirical studies of food webs. Below,I examine this concept further and propose alternativeexperimental designs that can solve the problem of con-founding intensity with variability (either temporal or

spatial) of foraging in consumer–resource interactions.Furthermore, I show that the approach can be extendedto the analysis of intensity and variability of predictorvariables for a wide range of ecological processes.

A FRAMEWORK FOR DIFFERENTIATING

BETWEEN INTENSITY AND VARIANCE

OF ECOLOGICAL PROCESSES

The impact of variable ecological processes on spa-tial and temporal patterns in density of populations canbe explained in terms of (Benedetti-Cecchi 2000a): (1)the inherent spatial and temporal variance of the pro-cess considered, (2) the residual variance of the pop-ulation (i.e., the variability due to other processes), (3)the mean intensity of the process (e.g., strong vs. weakbiotic interactions), and (4) the abundance of the targetpopulation (dense vs. sparse populations).

Changes in both intensity and variance of a processcan increase, decrease, or have no detectable effect onthe variance (whether spatial or temporal) in densityof the target population. The specific outcome is largelydependent upon the magnitude of the variance of thecausal process relative to the underlying variability ofthe population and the scaling relationship between themean and the variance of the response variable. Therelative contribution of the scaling relationship, in turn,depends on the magnitude of change imposed by theprocess on the mean abundance of the population.

Consider the extreme case where a population ofprey is abundant and distributed homogeneously in ahabitat in the absence of predation (Fig. 2A). If a con-sumer impinges on this population with a large effectsize (i.e., it removes a large proportion of the prey)and if the spatial variance of the process is also large,the resulting effect is a dramatic increase in spatialvariance in density of the population of prey (Fig. 2B).A similar increase is expected if the effect size of theconsumer is small, but spatial variance of foraging islarge (Fig. 2C). Whether such an effect is larger than,equal to, or smaller than that resulting from a largemean impact of consumers depends on the scaling re-lationship between the mean and the variance in abun-dance of prey. If a large reduction in mean abundanceof prey results in a strong dampening effect on spatialvariance, then a consumer that removes few individualscan have a larger impact on spatial heterogeneity oftheir resource than a more voracious consumer. In con-trast, negligible effects on the spatial distribution ofprey are expected when the variance of the trophicinteraction is negligible, either under a large (Fig. 2D)or a small (Fig. 2E) mean effect size of foraging.

At the other extreme there is the case of a sparsepopulation of prey that is highly aggregated in space(Fig. 3A). It is unlikely that a consumer can increasespatial variability in abundance of prey further undersuch circumstances. It is, however, possible for a con-sumer that greatly reduces the mean abundance of preyto decrease also the spatial variance of the target pop-


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FIG. 2. Possible effects of different regimes of foragingon an abundant resource that is distributed homogeneouslyin space. Panel (A) depicts the baseline scenario. Large var-iance in consumption increases spatial heterogeneity in dis-tribution of the resource regardless of the intensity of theprocess.

FIG. 3. Possible effects of different regimes of foragingon a sparse resource that is distributed heterogeneously inspace (panel A). Intense foraging is expected to reduce spatialpatchiness in abundance of the resource regardless of thevariance of the process (panels B and D). This is driven bythe scaling relationship between the mean and the variancein abundance of the resource. Feeble foraging (panels C andE) is unlikely to affect the spatial distribution of the resourceunder these circumstances.

ulation uniquely as a consequence of the scaling re-lationship between the mean and the variance. This canbe expected under both homogeneous and heteroge-neous regimes of foraging (Fig. 3B and D), but notwhen the mean intensity of foraging is weak (Fig. 3Cand E).

There is a wide range of possible intermediate sce-narios between the two cases illustrated above. Forexample, where prey are aggregated as in Fig. 3A, butsome individuals occur outside clumps, predators canincrease spatial variance in distribution of the resourceif they are more effective in removing isolated indi-viduals. This can possibly occur only if the intensityof predation is small (intense predation would neces-sarily affect also individuals in clumps), and if pre-dation is spatially homogeneous (isolated individualsmust be removed systematically by predators for spatialvariance to increase). In contrast, if consumers foragepreferentially on clumps ignoring isolated individuals,then spatial variance in density of the resource wouldbe reduced. This effect would be larger under a regimeof intense predation that operates homogenously acrossclumps.

Simulation experiments and correlative analyseshave been used to explore the effects of different com-

binations of variance in trophic interaction, effect size,residual variability and mean abundance of populationsof prey on small-scale spatial variance in these popu-lations (Benedetti-Cecchi 2000a). These analyses pro-vide support for the conceptual model summarizedabove (Fig. 2 and 3) and reconcile contrasting resultsbetween theoretical and empirical studies of food webs.That is, whereas several models of consumer–resourcedynamics predict that strong trophic interactions en-hance spatial and temporal variability in abundance ofprey and weak interactions dampen fluctuations (May1973, McCann et al. 1998), results from field experi-ments have also indicated that feeble interactions cangenerate heterogeneity in populations of prey (e.g.,Berlow 1999). This discrepancy does not necessarilyreflect lack of coherence between theoretical and em-pirical approaches to consumer–resources interactions.Consumers foraging with a large mean intensity caneither enchance or reduce spatial heterogeneity in abun-dance of resources in simulation experiments, whereasconsumers that operate with a small mean intensity canincrease spatial heterogeneity when foraging is veryvariable and the resources has low residual variance

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(Benedetti-Cecchi 2000a). Having indentified these re-lationships using conceptual, simulation, and correla-tion analyses, it is now time to devise appropriate ex-perimental designs to test specific hypotheses aboutcasual effects under natural settings.

CRITERIA FOR EXPERIMENTS

Variance–mean relationships in predictor variables

Before proceeding further, recall that the mean andthe variance of ecological variables are often related.Lack of independence may not be limited to dependentvariables, as discussed earlier (see Fig. 1 and referencesin the Introduction, above), but may also occur withrespect to predictor variables. Attempts to tease apartthe effect of changing the mean intensity of a processfrom the effect of altering its variance may thereforeappear naive and inappropriate. I argue that this par-adox is only apparent. Independence, both within andbetween treatments, is a requirement of most commonanalytical procedures and it is consistent with the logicof experimental design. Orthogonal designs, for ex-ample, break the covariance among the independentvariables, removing any possible pattern of correlationamong them (see Petraitis [1998:190] for a discussionfocused on ecological experiments). This does not im-pair the ability of the experiment to examine relation-ships among correlated variables, as provided by testsof interaction terms. Independence is necessary tomake correct inferences from experiments, but does notmean that the predictor variables must be naturally in-dependent. Otherwise it would be difficult to conceiveany useful role of experimental ecology, since manyfactorial experiments include predictor variables thatare correlated in the real world.

Conversely, there are realistic cases in which chang-es in mean intensity of a process can occur indepen-dently of changes in spatial or temporal variance. Forexample, new individuals may recruit to a clumpedpopulation of consumers to form new aggregations inthe habitat. In this case the mean intensity of foragingwould increase whereas spatial variance would not (as-suming that the spatial arrangement of consumers re-flects the spatial patterning of foraging). In contrast, ifthe new individuals recruited in the existing aggrega-tions, then also spatial variation in foraging would alsoincrease (see Fig. 12 in Underwood [1996]).

Given the considerations above, it seems appropriateto illustrate how effects due to changes in variabilityand intensity of ecological processes can be examinedexperimentally. I offer two examples. The first focuseson the effect of altering the temporal variance of apredictor variable on the mean response of a dependentvariable. The second example illustrates how spatialheterogeneity of a process can be manipulated in orderto assess its influence on estimates of spatial varianceof a dependent variable. In principle, each of the twoexperiments above could be adapted to examine effects

on the mean and on the variance (temporal or spatial)of response variables. A brief overview of the analyt-ical procedures that might be used to analyze the pro-posed experiments is provided in Appendix B.

Temporal variability

A possible solution to the problem of confoundingtemporal variability with intensity in studies of pre-dation or disturbance is that of contrasting cases wherethe specific process operates on a regular basis vs. sit-uations where events are distributed heterogeneouslyin time. The basic requirement is that the total numberof events is equal under different regimes of temporalvariability for a specified period. Different levels oftemporal variability can then be made orthogonal todifferent levels of intensity of the process (e.g., dif-ferent density of predators in experimental units) infactorial experiments where intensity and temporal het-erogeneity are two fixed factors. This would enabletests of hypotheses about main effects and interactionsof the two relevant sources of variability on patternsin abundance of the target variable.

For example, suppose there are observations indi-cating that in a given system consumers invade patchesof prey with average densities ranging from 1 to 5individuals per unit of space and temporal variance inabundance in the range of 0 to 10 in different sites overa year. This is a realistic scenario for consumers in-habiting intertidal habitats of rocky shores, includingcarnivores such as whelks and herbivores like limpets(Fairweather 1988a, b, Navarrete 1996, Benedetti-Cec-chi et al. 2000). An experiment can then be proposedto compare regimes of foraging with various degreesof intensity and temporal heterogeneity, using naturalobservations to identify appropriate levels for com-parison (e.g., Petraitis 1998). Temporally independentobservations would then be necessary to estimate tem-poral variance in the system, in order to decide whatsort of variation should be included in the experiment.Because measures of temporal variability are oftenbased on spatial replicates, particular attention shouldbe paid to remove the confounding effect of spatialvariation from estimates of temporal variance (Under-wood 1992, Stewart-Oaten et al. 1995).

A realistic experiment would consist of replicatepatches of habitat for each of the following treatments(Fig. 4): (1) intensity (2 or 4 consumers per experi-mental unit, fixed and orthogonal), (2) temporal vari-ability (zero, low and high, fixed and orthogonal tointensity). Consumers can be maintained at the desireddensities using fences, cages, or any other suitable tech-nique, with appropriate procedural controls for artifacts(e.g., Peterson and Black 1994). Response variableswould be densities of prey and any other species ex-pected to respond to the experimental treatments eitherdirectly or through indirect effects (Strauss 1991,Wootton 1994, Benedetti-Cecchi 2000b). The scope ofthe experiment can then be expanded by replicating at


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FIG. 4. Experimental design proposed to disentangle intensity from temporal variability in consumer–resource interactions.The temporal patterning of addition/removal of consumers in a single replicate is illustrated for each treatment over a periodof 12 months, with intensity expressed as the average number of consumers (predators) present in a given experimental unitfor the period of study. Data are the number of predators each month. Temporal variability is expressed as a variance-likequantity (K 2 ). There are two levels of intensity (2 or 4 consumers per unit, on average) and three levels of temporal variability(with slight differences between densities). Intensity and temporal variability are fixed orthogonal factors in the experiment.

different spatial and temporal scales to test hypothesesabout the generality (or lack thereof) of the processesinvestigated.

Admittedly, there can be problems in identifying re-alistic sequences of density of consumers to obtain thedesired levels of temporal heterogeneity. This requiresrefined observations and a great deal of understandingof the natural history of the system being investigated.Furthermore, there are several possible sequences ofdensity of consumers for any given level of temporalvariability. If there are reasons to believe that the orderby which predators are manipulated in experimentalunits matters, then the experiment may be expanded toincorporate different sequences for each level of lowand large variability.

Spatial variability

I focus on trophic interactions rather than distur-bance, although the concepts discussed here apply toboth processes. The choice is determined by the factthat whereas a number of studies have considered theimportance of spatial variation in consumer–resourceinteractions (e.g., Fairweather 1988a, b, Commito etal. 1995, Sommer 2000, Adler et al. 2001), comparablyfewer studies have investigated spatial variation in in-

tensity of disturbance. The problem of separating themean effect size of consumers from spatial variationin intensity of the trophic interaction has been ap-proached only with simulation and correlation analysesuntil now (Benedetti-Cecchi 2000a). What kind of ex-periments would enable empirical tests of hypothesesabout intensity and spatial variation in trophic inter-actions? A possible solution would be that of compar-ing the variance of the target variable among experi-mental units with consumers to that calculated fromunits that have no consumers at all (Fig. 5). Each con-trast would provide a measure of the effect size ofconsumers, expressed ln( / ), under a specific re-2 2S S1C 2C

gime of foraging (where ‘‘2C’’ indicates that consum-ers are excluded from some areas, and ‘‘1C’’ indicatesthat consumers are maintained at known densities inother areas). Intensity of foraging would be manipu-lated by comparing units with no consumers with thosethat have either small or large average densities of pred-ators/grazers. The influence of spatial patterning of for-aging can be assessed by contrasting units that haveeither similar or different numbers of consumers forany particular average density, to create homogeneousand heterogeneous patterns of foraging, respectively.By choosing appropriate densities of consumers it

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FIG. 5. Experimental design proposed to disentangle intensity from spatial variability of foraging. Intensity of foragingis manipulated by comparing experimental units with no consumers with those that have either small or large average densityof consumers (numbers in boxes under ‘‘Experimental treatments’’). Levels of spatial variability in foraging are created bycomparing units that have either similar or different numbers of consumers for any level of average density. The effect ofconsumers (carnivores and herbivores) is expressed as ln( / ), where is the estimate of the variance in abundance2 2 2S S S1C 2C 1C

of the resource among replicates in the presence of consumers and is the estimate of the variance in the absence of2S2C

consumers. Each contrast must be replicated several times in order to create a factorial experiment where intensity and spatialvariability of foraging are two fixed orthogonal factors.

would be possible to design factorial experimentswhere both intensity and variability in foraging are twofixed, orthogonal factors.

The example considered here (Fig. 5), illustrates asimple case involving two levels of intensity (averagedensities of 3 vs. 0 and 6 vs. 0 consumers per unit ofspace) and two levels of spatial variability in foraging(2 vs. 0 and 8 vs. 0). Variability is calculated as avariance-like quantity from the density of consumersin each pair of experimental units. Obviously, there arevarious ways to create the desired contrasts by usingdifferent densities of consumers and possibly more thantwo densities for each level of intensity and spatialvariability. This results in a drastic increase in the num-ber of experimental units needed to complete the ex-periment. In the particular example presented here,having four replicates for each treatment would resultin a total of 64 experimental units, which is not morethan the number of units used in many published ex-periments of grazing or predation. From these units,16 independent observations would be obtained for theanalysis. There is, however, the need to include pro-cedural controls for artifacts if fences or cages are used,which would require some extra units.

The experimental design presented here has a num-ber of advantages over previous experiments that havebeen used to investigate the consequences of spatialvariation in foraging. First, it solves the problem of

confounding the mean intensity of foraging with spatialvariation. Second, tests of hypotheses about possibleinteractive effects between the two processes arestraightforward, because intensity and spatial variabil-ity are orthogonal factors in the design. Third, it allowsthe calculation of an effect size on spatial variance ofpopulations of resources that is analogous to proposedmeasures of ‘‘interaction strength,’’ whose propertieshave been reviewed recently (Paine 1992, Osenberg etal. 1997, Berlow et al. 1999).

RAMIFICATIONS TO OTHER ECOLOGICAL SCENARIOS

Though my analysis of variability has focused onconsumer–resource interactions and natural distur-bance, the logical, methodological, and analytical pro-cedures discussed here have implications for a widerange of ecological contexts, including environmentalmanagement and conservation. Human activities re-distribute the intensity of many ecological processesin space and time, altering their natural variance at thescale of the landscape (Landres et al. 1999). Manage-ment of domestic grazing systems, for example, ad-vocates the uniform distribution of grazing over rela-tively large areas (Berg et al. 1997, Adler et al. 2001).This can have profound, albeit unidentified, effects onspatial and temporal pattern in the vegetation and maynot be the optimal choice to minimize impacts and topreserve species diversity. Management of natural re-


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serves, in contrast, sometimes contemplates the use ofrotational schemes of protection (e.g., Castilla 1999,Edgar and Barret 1999). This choice is based on con-siderations of the effect of changing the intensity of aprocess, whether detrimental to populations, such asharvesting, or positive, such as protection. There is,however, no consideration of the potential effects thatchanges to the temporal variance of these processeshave at local and regional scales.

Changes to the temporal variance of natural pro-cesses, in addition to their intensity, are expected alsounder modified climate conditions. Models of climatechange predict increased probabilities of occurrence ofextreme events such as storms and changes in the tim-ing of their occurrence (Smith and Buddemeier 1992,Michener et al. 1997). Experimental studies investi-gating possible ecological responses to climate changeoften incorporate treatments applied at constant rates(e.g., manipulation of levels of temperature, nutrients,and/or water), to simulate forecasted climate scenarios(e.g., De Valpine and Harte 2001). These experimentsdo not consider the possible impact due to changes inthe variance of processes, which is expected due to thepredicted increase of occurrence of extreme events, asexplicitly recognized by De Valpine and Harte (2001).An experimental design such as that illustrated in Fig.4, where temperature, nutrients, or precipitation, in-stead of predators, is the manipulated factor, wouldenable the simultaneous test of hypotheses aboutchanges in intensity and variability of processes relatedto climate change.

DISCUSSION

There is increasing evidence that variance aroundthe mean effect of causal processes plays a prominentrole in generating pattern in populations and assem-blages (Chesson and Case 1986, Davis 1986, Butler1989, Menge et al. 1994, Horne and Schneider 1995,Navarrete 1996, Thrush et al. 2000). Experimentalstudies have indicated that variability in predation,grazing, or physical disturbance can have importanteffects on the composition, abundance, and diversityof assemblages, in addition to influencing spatial andtemporal patchiness of populations. The pioneeringwork of Butler (1989), for example, has elucidated howtemporally variable predation can introduce both spa-tial and temporal heterogeneity in abundance of pop-ulations of prey, compared to a constant regime of pre-dation. Later work by Navarrete (1996) has providedfurther evidence of the importance of variable preda-tion, indicating that frequency of foraging can affectthe composition of developing assemblages in additionto maintaining spatial variability in abundance of prey.Sommer (2000) has shown that spatial heterogeneityin grazing can enhance microalgal diversity in labo-ratory experiments.

The frequency of physical disturbance can also haveimportant effects in assemblages. Collins (2000) has

shown that increasing the frequency of fires in tallgrassprairie reduces the temporal stability of the vegetation,with ‘‘stability’’ defined as lack of statistically detect-able directional change in composition and abundance.In contrast, frequent fires enhanced year-to-year fluc-tuations in composition of assemblages associated withthe vegetation, but introduced no directional trend overlarger temporal scales. Unlike tallgrass prairie, fre-quency of disturbance per se had no detectable effecton assemblages of macroinvertebrates in streams(McCabe and Gotelli 2000), although species richnesswas greater in disturbed treatments than in undisturbedcontrols, regardless of whether frequency, extent, ormode of disturbance was manipulated.

Whereas some experimental evidence on the con-sequences of variable ecological processes has accu-mulated in the literature, the implications of hetero-geneity in patterns of distribution, abundance, and di-versity of species are much less appreciated. The ev-idence provided so far suggests that spatial andtemporal patchiness affects some attributes of popu-lations, including persistence and productivity. In ter-restrial habitats, for example, patchiness in the vege-tation can concentrate resources enhancing primaryproduction (Pickett and Cadenasso 1995, Adler et al.2001). Aggregation of individuals may also reduce theprobability of attack from predators while increasingefficiency of feeding and physiological tolerance tophysical stress (Turner 1989, Bertness and Leonard1997, Landres et al. 1999). There are, however, costsassociated with the habit of living in clumps. Aggre-gation may enhance, rather than decrease, the vulner-ability of a population to consumers, foster the deple-tion of resources and the spread of diseases, and, ul-timately, reduce survival (e.g., Floater 2001).

The issues above imply complex dynamics of pro-cess and pattern, indicating that cause–effect relation-ships are neither univocal nor directional. There arefeedbacks whereby spatial and temporal patchiness canaffect both mean intensity and variability of the pro-cesses that act as proximate causes of variation in pop-ulations and assemblages. This is particularly evidentin consumer–resource interactions. For example, spa-tial pattern of the vegetation can have strong influenceson the distribution, behavior, and efficiency of foragingof grazers in both terrestrial and aquatic habitats (Mc-Naughton 1983, Duffy and Hay 1991, Banks 1998).How these feedbacks will affect the interplay betweenvariation in predictor variables and long-term spatialand temporal patterns in assemblages remains largelyunexplored. The empirical studies discussed so far haveprovided a first grasp on the topic, but understandingthe mechanisms underlying the dynamics of variableecological interactions requires more than focusing onthe frequency of events.

The logical, experimental, and analytical proceduresaddressed in this and a previous paper (Benedetti-Cec-chi 2000a) provide a framework to explain heteroge-

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neity in response variables as a function of variationin intensity and variability of predictor variables. Theimportance of the variance around the mean effect sizeof ecological processes, the need to separate the twosources of variation in experiments, and the likely in-teractive nature of intensity and variability have beenrecognized by previous studies (McNaughton 1983,Chesson 1986, Butler 1989, Navarrete 1996). A the-oretical and empirical framework allowing tests of ex-plicit hypotheses is, however, slow to develop, possiblyas a consequence of the inherent complexity of thetopic and the use of ambiguous concepts such as thefrequency of causal events. Frequency, however, maystill provide a focus for experiments when it is im-possible to identify appropriate spatial or temporalscales to examine relationships between intensity andvariability of ecological processes.

In a recent paper Adler et al. (2001) emphasized theneed to consider the magnitude of spatial variability ingrazing relative to preexisting levels of heterogeneityin the vegetation, in order to provide sensible predic-tions of the influence of herbivory on spatial patchinessof plants. Using consumer–resource interactions as areference, I have shown that the ratio of the varianceof a predictor variable to residual variance of the re-sponse variable cannot capture the full range of out-comes of ecological interactions (Benedetti-Cecchi2000a). The intensity of a process and the relationshipbetween the mean and the variance in the measuredquantity must also be considered. For example, a con-sumer that forages heterogeneously in space removingonly a small proportion of the resource can introducemore variation in populations of prey than a consumerthat forages with the same degree of spatial hetero-geneity, but with a larger overall intensity (Benedetti-Cecchi 2000a). This effect is driven by the scalingrelationship between the mean and the variance and itis more likely to occur in sparse than abundant re-sources. In contrast, a consumer that forages with alarge mean effect size over a sparse and intrinsicallyvariable resource is likely to dampen spatial variationin abundance of prey (Benedetti-Cecchi 2000a).

The procedures advocated in the present paper canbe extended to the analysis of a wide range of pro-cesses, from physical disturbance to biological inter-actions, in both terrestrial and aquatic habitats. Where-as repeated experimentation in space and time is im-portant to estimate the magnitude of spatial and tem-poral variability of ecological processes, experimentaltests of hypotheses that treat variation as a fixed effectare essential to understand how variance in ecologicalprocesses generates structure in populations and as-semblages. Having recognized the pervasiveness ofscale-dependent variability of patterns and processes,it is now timely to establish quantitative, causal rela-tionships between these two prominent features of nat-ural systems using appropriate experimental proce-dures.

ACKNOWLEDGMENTS

This study benefited by comments and constructive criti-cism from M. J. Anderson, J. A. Commito, Aaron Ellison, F.Micheli, Tony Underwood, and an anonymous reviewer. Dis-cussions with I. Bertocci, E. Maggi, and S. Vaselli helped toclarify my ideas about some of the concepts expressed in thispaper. Financial support was provided by the University ofPisa.

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APPENDIX A

A summary of concepts of frequency, intensity, and variance of predictor variables in the ecological literature is availablein ESA’s Electronic Data Archive: Ecological Archives E084-055-A1.

APPENDIX B

An overview of procedures for the analysis of the experiments proposed in this paper is available in ESA’s Electronic DataArchive: Ecological Archives E084-055-A2.

Documents

THE IMPORTANCE OF THE VARIANCE AROUND THE MEAN EFFECT SIZE OF ECOLOGICAL PROCESSES