Goudarad y Loreau. 20008. Interacctions Biodiversity and Ecosystem Functioning

8/6/2019 Goudarad y Loreau. 20008. Interacctions Biodiversity and Ecosystem Functioning

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Nontrophic Interactions, Biodiversity, and Ecosystem Functioning: An Interaction Web ModelAuthor(s): Alexandra Goudard and Michel Loreau

Source: The American Naturalist, Vol. 171, No. 1 (Jan., 2008), pp. 91-106Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/30114879 .

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VOL. 171, NO. 1 THE AMERICAN NATURALIST JANUARY 2008 (

Nontrophic Interactions,Biodiversity,and EcosystemFunctioning:An InteractionWeb Model

AlexandraGoudard12',*ndMichelLoreau3lt1. Biogeochimiet Ecologiedes MilieuxContinentaux,UniteMixtede Recherche 618,EcoleNormaleSuperieure,6 rued'Ulm,F-75230Pariscedex05, France;2. UniversitePierreet MarieCurie,4 placeJussieu,75252Pariscedex05, France;3. Department f Biology,McGillUniversity, 205 avenueDocteurPenfield,Montreal,QuebecH3A 1B1,CanadaSubmitted ecember9, 2006;Accepted ugust20, 2007;ElectronicallyublishedNovember 2,2007Onlineenhancements:ppendixes,igure.

ABSTRACT: Researchntotherelationshipetweeniodiversityndecosystemunctioninghas mainly ocusedon the effectsof speciesdiversity n ecosystempropertiesn plantcommunitiesand,morerecently,n foodwebs.Although here s growing ecognition f thesignificancef nontrophicnteractionsn ecology, hese nteractionsarestillpoorly tudied heoretically,nd their mpacton biodiversityand ecosystem unctioning s largelyunknown.Existingmodelsofmutualism sually onsideronlyonetypeof species nteraction nddo not satisfymassbalanceconstraints.Here,we presenta modelof an interactionweb that includesboth trophicand nontrophicinteractions nd that respects he principleof mass conservation.Nontrophicnteractions rerepresentedn the form of interactionmodifications.We use this modelto studythe relationship etweenbiodiversityndecosystem ropertieshatemergesrom heassemblyof entire nteractionwebs. Weshow thatecosystempropertiesuchas biomassandproductiondependnot onlyon speciesdiversity utalso on species nteractions,n particular n the connectance ndmagnitudeof nontrophic nteractions, nd that the nature,preva-lence,andstrength f species nteractionsn turndependon speciesdiversity.Nontrophicnteractions lter he shapeof therelationshipbetweenbiodiversitynd biomassand canprofoundlynfluence co-systemprocesses.Keywords:iodiversity, cosystem unctioning,nontrophic nterac-tions, interactionmodifications,massbalance,model.

Therelationshipetween iodiversityndecosystemunc-tioninghas emerged s a centralssuein ecology n thelast decade.Human activities ontribute o speciesex-tinction,andbiodiversityosscancause oss of ecologicalservices Pimmet al. 1995;Vitousek t al. 1997;Salaetal. 2000;Loreauet al. 2001, 2002;Kinziget al. 2002;Hooperet al.2005).Therefore, betterunderstandingfthe effectsof biodiversityn ecosystem ropertiess crit-icallyneeded.The relationship etweenbiodiversity nd ecosystemprocesses asmostlybeenstudied xperimentallyndthe-oreticallyn plantcommunities. heoreticalmodelsusu-allypredicthatprimary roductivityncreases ithplantspecies ichness ut saturatesthigh diversityTilman tal. 1997;Loreau 998,2000).Controlledxperimentson-ducted n differentocalitiesHector t al. 1999;Spehn tal. 2005)or overseveral ears Tilman t al. 2001)oftenexhibit hepredicted attern. hepositive ffects fspeciesdiversity n ecosystem unctioning ave been explainedbytwo mainmechanismsTilman 999;Loreau ndHec-tor2001):a complementarityffect,whichemergesromfacilitation r nichedifferentiation,nd a selection ffectarisingfrom the dominanceof specieswith particulartraits.Thesemodels,which ocuson asingle rophicevel,are based on nichetheoryand plant competition or alimitingnutrient.Foodweb modelswith several rophiclevels,however, redicthatplantbiomassdoesnotalwaysincreasewithplantdiversity nd thatchangesn diversitycan lead to complexchanges n ecosystem unctioning(Th6bault ndLoreau 003;Ives et al. 2005).Recent x-periments Jonssonand Malmqvist 000;DowningandLeibold 002;Duffy2002;Paine2002;Duffyet al. 2003,2005)haveshowed hattrophic nteractionsan indeedprofoundlyffect herelationshipetween iodiversityndecosystemunctioning.Thus, herelationshipetween iodiversityndbiomassor productivity as been mostlystudied n plantcom-munitiesorin food webs.Theonlyformof direct peciesinteraction onsideredn these studies s the trophic n-

* E-mail: [email protected]; -mail:[email protected].

Am. Nat.2008.Vol. 171,pp.91-106. c 2007by TheUniversity fChicago.0003-0147/2008/17101-42291$15.00. ll rightsreserved.DOI:10.1086/523945


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92 TheAmericanNaturalist

teractionexploitationompetitionor a shared esourceis an indirect ffectof the consumer-resourcerophic n-teraction). omeexperiments,owever,uggesthatnon-trophicnteractions,uch as facilitation,mayplayan im-portant ole n ecosystemunctioningMulder t al.2001;Cardinale t al. 2002;Rixenand Mulder2005).Forin-stance,Rixenand Mulder 2005)showed hat waterre-tention ncreaseswithspeciesdiversityhrough ncreasingfacilitation nd leads to increased roductivityn arctictundramosscommunities.Experimentsuggest hatdif-ferentkindsof species nteractions o not actin isolationfromeach other n naturebut co-occurwithin he samecommunityCallawayndWalker 997).Evidenceortheimportance f indirectnteractionss alsoaccumulating.Habitatmodifications one type of indirect nteractionthathasbeenstudied xperimentallyBertnesstal. 1999;Mulder t al.2001;Cardinalet al.2002;Rixen ndMulder2005).For nstance,Bertness t al. (1999)have hown hatalgalcanopyreducesphysical tress uchas temperatureor waterevaporationnd thus haspositive ffectson or-ganismrecruitment,rowth,andsurvivaln rocky nter-tidalcommunities, ut habitatmodification y algal an-opycanalsohavenegative ffectsbyincreasingonsumerpressure.Although here is growingrecognition f the signifi-canceof nontrophicnteractionsn communities ndeco-systems, heseinteractions re still poorlystudied heo-retically, nd we still know littleaboutgeneralpatternsand mechanisms. herefore, n important urrent hal-lenge s to understand ownontrophicnteractionsffecttherelationshipetweenbiodiversityndecosystemunc-tioning.Moregenerally,here s anurgentneed o includenontrophicnteractionsn ecological heory Borer t al.2002;Brunoet al. 2003).Therearesome modelsof mu-tualism Goh 1979;Heithaus t al. 1980;Addicott1981;Ringel t al. 1996;Holland t al.2002),butthesemodelsare specific,as they consideronly one kind of speciesinteractions.We lackgeneralmodelsof interactionwebsthat includealltypesof direct pecies nteractionsinter-ferencecompetition,mutualism, xploitation, ommen-salism, mensalism)swellas their ndirect ffects. implemodelsof mutualism asedon Lotka-Volterraquationsalso have the unrealisticpropertyof leadingto explosivesystemsbecausethey do not respectthe physicalprincipleof mass conservation(Ringelet al. 1996). Mass balance scrucial for understanding he functionalprocessesof nat-ural ecosystems.Thus, it is necessaryto construct inter-action web models that satisfymass balance constraints.Arditi et al. (2005) recentlymade a first step in that di-rectionby addinginteractionmodificationsto a food webmodel; they showed an increasing proportion of super-efficient systems as the magnitude of interaction modifi-cations was increased.Here, we expand this approachto

study he structuralnd functional ropertiesf interac-tionwebsandhence herelationshipsetween iodiversityandecosystem ropertieshatemergen complex cosys-tems.Weneedtheories ndmodels oprovide eneralizationson the role of nontrophicnteractionsn ecosystemunc-tioning.It is therefore ecessaryo construct he mostgeneralpossiblemodelof an ecosystem-a model of aninteraction eb hat ncludes oth rophic ndnontrophicinteractions nd thatrespectshe principle f masscon-servation.Here,wepresent theoreticalmodel hatmeetsthis need.Despitetsgenerality,urmodel s too complexto be analyticallyractable. herefore, e study t usingnumericalimulationshatmimica communityssemblyprocess.Thisallowsus to investigateherelationshipe-tweenbiodiversityndecosystemunctioning ndermorerealistic onstraintshanwouldan analyticalquilibriumstudyof specialcases, n agreementwith Loreau t al.'s(2001)recommendationo study herelationshipetweenbiodiversity nd ecosystem unctioningwith a dynamicapproach.Usingthisassemblymodel,we study he rela-tionshipbetweenbiodiversity nd ecosystem roperties,suchas thebiomass ndproductivityfthevariousrophiclevels, n an interactionweb in comparison ith a foodweb.Thiscomparisonllowsus to examine heeffectsofnontrophic nteractionson the biodiversity-ecosystemfunctioningelationship.hus,our nteraction ebmodelprovides usefulbasis orreaching reater eneralitye-gardinghe impactof speciesdiversity ndspecies nter-actionson the functionalproperties f complexecosys-tems.

ModelandMethodsTheInteractionWebModel

Themodel s anextension f a modeldeveloped yTh&-baultandLoreau2003)for a nutrient-limitedcosystemwiththreetrophicevelscontaining n arbitraryumberof plants,herbivores,ndcarnivores.lantsakeupa lim-itingnutrientn theirrhizosphere,huscreating pecies-specific resourcedepletion zones and allowing plant co-existence under some conditions (Loreau 1996, 1998).These species-specificresource depletion zones may beviewed as physical soil volumes, but they may also beviewed in a more abstractway as different niche spacesavailable o differentspecies.Here,we add nontrophic interactionsto this food webto construct an interactionweb model that satisfiesmassbalance constraints.Nontrophic interactionsareincludedin the form of interactionmodifications:each speciescanmodify the trophic interaction between any two species.


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Nontrophic nteractions,Biodiversity, nd EcosystemFunctioning 93Themodel s described y the followingdynamic qua-tions andfigure1:

dC sdC

qcaHiCHHiC-uCci,t

dH -= a PH.dt j=i- acCHiHHi UHHj= 1

= apPiLiPPiLiLiFPt

dL L idt k VR PiLPiLi,dRsdR - XRR- R-- L.idt VR

+ (1 Xpi)UPi 3 (1 - XHi)UHH,=l i=1+ (1- Xc)uc,Ci=l1+ (1 -

qc)aCHiLiHJjCi+ 3(1 - qH)aHAiPiHPjH,., (1)i=1 i j=1

wherex mxyzlog(1+ Xz)= H 1+ X)"xz. (2)ez= 1 z= 1

Here, S is the number of species per trophic level, n =3S is the total number of species, and Pi,Hi, and C;arethe nutrient stocks of plant,herbivore,and carnivore pe-cies i, respectively.We assumethe stoichiometriccomposition of eachspe-ciesto be constant;hence, its nutrient stock is proportionalto its biomass.Parameteraxys a per capita potentialcon-sumption rate, that is, the intensity of the trophic inter-action between predatorspeciesx and prey species y inthe absenceof interaction modification(axy> 0). Param-eter aPiLis the nutrientuptakerate of plant speciesi. Eachherbivoreor carnivorespeciesx may be more or less spe-

Carnivoresefficiency

Ci

(I q)Herbivores

PlantsNutrientinput-

a%7n

Hy

Cx

Hz

P.

\deathux non-recyclednutrient,xrecycling 1-Ax)

iffusionLi 7 IL RNutrientn species-specificresourceepletionone Soilnutrientpool

loss AR

Figure1: Interaction ebmodel.Solidarrowsepresentutrientlows.Forclarity f thefigure,lowsof nonassimilatedutrient eturnedo thesoilnutrient oolduring onsumptionycarnivoresndherbivoresrerepresentednlyon the lefttrophic hain,and flowsof nutrient itherrecycledr lost from heecosystemollowing eatharerepresentednlyon theright rophic hain.Dottedinesrepresentnteraction odifica-tions.Onlyfiveexamples f interactionmodificationsrerepresentedhereforthe sakeof clarity. or nstance, erbivorepeciesH,modifiesthe rophicnteractionetween erbivorepeciesH,andcarnivorepeciesC,,withamagnitudef interaction odificationfm,,.Themodificationof the nutrient low betweenplant speciesPjand its species-specificresource epletion oneL,correspondso intraspecificompetitionrfacilitation.

cialist ifone ofitspotential onsumptionatesaxysmuchhigherthan the others)or generalistif all its potentialconsumption atesax,areof similarmagnitude),withapreferenceor certainpreyspecies.Theconstants , andqc are the conversion fficiencies f herbivores ndcar-nivores, espectivelyseeapp.A in the online editionofthe American aturalistorparameteralues).Nontrophicnteractionsre ntroducednto themodelby addingnontrophicmodifications f trophic nterac-tions:eachspeciesz is allowed o modify hetrophic n-teraction etween peciesx andy with an effect hat de-pends on both its biomassXz and a magnitudeofinteractionmodification f mxY,fig. 1). ParameterIxy sthenontrophicoefficient:t is thetotalnontrophicffectof allspeciesof thecommunityn thetrophicnteractionbetween pecies andy.Thus, pecies consumes peciesy with a realized onsumption ateaxyAxy.he functionthatdescribes ontrophicffects eq.[2])waschosen uchthat it satisfies everal onditions.First, t is a strictlyn-creasing unctionof both the magnitudeof interactionmodificationmxY, nd biomassXz. Second, if eithermxyr= 0 or X, = 0, then (1 + Xz)mxyz = 1, and speciesz does not affect he trophic nteraction etween pecies


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94 TheAmericanNaturalistx andy. Thus, in the absence of interactionmodifications,tx, = 1, and the realizedconsumptionrateaxYxYs equalto its potential value ax,. Third, the function for non-trophic effects is strictlypositive, so that the sign of therealized consumption rate axyxy does not change. Themagnitudeof interactionmodificationmxyzan be positiveor negativewithout changingthe sign of axyxY.Whateverthe nontrophic effects of other species, the nutrient flowbetween species x and y is never reversed.Fourth,whilethe magnitude of interactionmodification mxyzomes asthe exponent of a power function to keep the functionpositive, we chose a linear dependence on biomass X,:preliminaryresults showed that a power dependenceonbiomass is stronglydestabilizing.A positive myz leads tomultiplicationof the potential consumption ratea,y by afactor (1 + Xz)mxyzl,hereas a negative mxyzeads to di-vision of

axyby this factor.

In the presenceof interactionmodifications,each spe-cies can affect any other species x by modifying one orseveral trophic interactions that involve species x andhence increasingor decreasing he populationgrowthratedXx/dtof species x. In the absence of interaction modi-fications, the only direct species interactionis predation,and our model web reduces to a food web. When inter-action modifications are added, each species can have apositive (facilitation),negative (inhibition), or null effecton the population growth rate of any other species, andthus all types of species interactions are possible (com-petition, mutualism, exploitation, commensalism,amen-salism), including intraspecific (negative or positive) ef-fects (mrxz 0 for species z). Our model web is then afull interaction web that includes both trophic and non-trophic interactions.Our interaction web model respects the principle ofmass conservation:a nontrophic interaction,such as mu-tualism or competition, does not affect the total quantityof matter in the ecosystem as a whole. Interactionmod-ificationschangethe material low betweena resourceanda consumerby multiplyingit by some factor,but there ismass conservation overall.

The model also includes nutrient cycling.The constantux is the loss or death rate of species x, and x is thenonrecycled (lost) proportion of nutrient coming fromspecies x. The variableR is the nutrient mass in the soilnutrient pool with volume V,; Li is the nutrient mass inthe set of species-specificresourcedepletion zones, withtotal volume X, of plantsfrom species i. Nutrient is trans-ported between species-specificresourcedepletion zonesand the soil nutrient pool at a diffusion rate 7 per unittime. In our simulations,y was quite high (y = 10) toallow rapidsoil homogenizationand strongindirectplantcompetition for the limiting nutrient. Parameter is the

nutrientinput in the soil nutrientpool perunit time, andXRis the rate of nutrient loss from the soil nutrientpool.

CommunityAssemblyWe constrainedour interactionweb model as littleaspos-siblein orderto explore ts generalproperties.Accordingly,we randomly assignedthe various biologicalparameters(potential consumption rates, intensities of interactionmodifications,deathrates,nonrecycledproportionsof nu-trient) to a regionalpool of species from a uniform dis-tributionwithin appropriatentervalsandlet the localeco-system assemble spontaneously.The establishmentof aspeciesdependson both its intrinsictraits(parameter al-ues) and its interactions with the other species alreadypresent in the ecosystem.The model was simulatednumericallyusingC+ + pro-gramming, and numerical integration of the dynamicequationswas performedwith a Runge-Kuttamethod oforder 4 and a time step of 0.01 during 1,000 iterations,that is, 100,000 time steps (100,000 numerical integra-tions). The local ecosystemresults from an assemblypro-cess that involves species' successive introductions andeliminations. Species were introduced with a biomassequal to 0.01 that was subtracted from the soil nutrientpool, and they were considered extinct if their biomasswas smaller than 0.005, in which case this biomass wasreturned to the soil nutrientpool. Specieswerepickedatrandom from a regionalspeciespool with speciesrichnessn = 3S (S species at each trophic level) and introducedregularly o the community.Each successive ntroductionoccurred aftera constantperiod irrespectiveof whether anew equilibriumwas reached (thus, if the introductionperiod was 100 time steps, there were 1,000introductionevents during a simulation).Localspeciesrichness s the total number of speciesinthe locallyassembledecosystem(to be distinguished romregionalspeciesrichness).The totalvolume of the soil waskept constant irrespective of local species richness:Voi = VR+ .1 V, where Vi= 0 if species i was notpresent in the community. Thus, when a plant speciesbecameextinct,the volume of its species-specific esourcedepletion zones was set to 0 and added to the volume ofthe soil nutrient pool. When a plant species was intro-duced,the volume of its species-specific esourcedepletionzones was created and subtracted rom the volume of thesoil nutrientpool. Thisvolume allocationrulerespects heconservation of total soil volume while at the same timeallowingdifferentplant speciesto occupycomplementaryresourcedepletionzones in the soil. Complementarity e-tweenplant specieshas both theoreticaland empirical us-tifications(Loreau 1998;Loreauand Hector 2001). To beconsistent, however,this allocationrule requiresthat the


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Nontrophic nteractions,Biodiversity, nd Ecosystem unctioning 95total soil volume be greater han the sum of the volumesof the depletionzones of all possible plant species,whichwas the case in our simulations (app. A). Mass conser-vation was also satisfiedupon extinction of a plant speciesby addingthe nutrient stocks in its residualbiomass andin its resourcedepletionzone to the soil nutrientpool and,conversely,upon introduction of a plant species by sub-tractingthese nutrient stocks from the soil nutrientpool(app. B in the online edition of the AmericanNaturalist).

Communityand Ecosystem ropertiesWe examinedthe relationshipbetween the structureandfunctioning of interaction webs and the impact of non-trophic interactionsby analyzingthe effects of regionalspecies richness, nontrophic connectance, and maximalnontrophicmagnitude(independentparameters f the re-gionalspeciespool) on variouspropertiesof the localeco-systems,that is, local speciesrichness,species richnessateachtrophic evel,proportionsof the varioustypesof spe-cies effectsand species interactions, nteraction web con-nectance,total biomass (totalbiomass of all speciesin thelocal ecosystem),biomass of each trophic level, and pro-duction of each trophic level. When we varied regionalspecies richness,we kept equalnumbersof species (S) atall trophic levels in the regional species pool.Food web connectance of the regional species pool wasdefinedas the number of realized rophic interactionsdi-vided by the number of possible trophic interactionsandwas kept constant in all simulations. Its value was closeto 1 becauseconsumers were assumed to be more or lessgeneralist, with potential consumption rates randomlydrawn from a uniform distribution between 0 and 0.01(and hence with a very smallprobabilityof being exactly0). Note that our definition differsfrom the conventionaldefinition of food web connectancebecausefeedinglinkswithin trophic levels and between plants and carnivoreswere not allowedin our model.Interaction modifications were assigned in two steps:(1) there was a certain probabilitythat mxz # 0; (2) ifmxYz 0,

the magnitude of the interaction modificationwas chosen in a uniform distribution.We call the prob-ability that mxz 0 the nontrophic connectance of theregionalspecies pool, which is theprobability hata speciesmodifies the trophic interactionbetween any two species:

(3)nontrophic connectance =

number of realized interaction modificationsnumber of possible interaction modifications"

The number of possible interaction modifications in the

regional species pool is equal to the number of speciesmutiplied by the number of possible trophic interactionsbetweenspeciesat adjacent rophiclevels in the pool, thatis, 3S(S2 + S2 + S).We also variedthe rangeof values of the magnitudeofinteraction modification. Parameter

mxyzwas randomlytaken between a maximum value called "maximal non-

trophicmagnitude"and a symmetricalminimum equaltominus the maximalnontrophic magnitude.The maximalnontrophic magnitudewas then allowed to take on dif-ferentvalues.Thus,we explored he impactsof nontrophicinteractions by manipulatingboth the nontrophic con-nectance and the maximal nontrophic magnitude of theregionalspeciespool.We analyzedthe community and ecosystem propertiesin the local ecosystemsthat resulted from the assemblyprocess. We measured the proportions of species effects(facilitation, inhibition, or no effect) and species inter-actions (mutualism, competition, exploitation,commen-salism, amensalism,or neutral interaction)based on thesign of the net species effects. The net species effect Ei(including trophic and nontrophiceffects)of species g onspecies i was measuredby the partial derivativeof thegrowthratedXIdt of species i with respectto the biomassXgof speciesg:

a(dXI/dt)E, (4)a a7IfEig 0,the effectofspecies onspecies is a facilitation.If Eg< 0, it is an inhibition. f Eg> 0 and Egi 0, theinteractionbetweenspeciesi and g is a mutualism. fEi,< 0 and Eg< 0, species andg arein competition.fEg> 0 andEgi 0, thespeciesnteractionsanexploitation(includingnontrophicormsof exploitation).f Eig= 0andE, > 0, it is a commensalism.f Ei = 0 andEg< 0,it is an amensalism. ensity-mediatedndirectnteractions(Abrams 995), uchasexploitativeutrient ompetition,do not enterinto the calculation f Eig.The species n-teractions hus definedarephenomenologicalet inter-actions,ustastheyaredefinedraditionallynecology.Aphenomenologicalather han a mechanistic efinitionwasnecessaryo account or thewidevarietyof trophicandnontrophicffectsn asimpleunified ramework.hisallowedus to investigatehe effectsof regional peciesrichness,nontrophicconnectance,and maximalnon-trophicmagnitude n theprevalencendstrength f spe-cieseffectsandspecies nteractions.Interactionweb connectance f the localcommunitywas measuredby the proportion f nonneutral peciesinteractions mongall possible pecies nteractions,hatis, by the proportion f species nteractionsn which at


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96 The AmericanNaturalist

A(I(.)

Q,0)0

o_J

403020100 0 2 4 6 8 10

Local species richness

Plant species richnessHerbivorespecies richnessCarnivorespecies richness

B

E0

200015001000500

0 0 2 4 6 8 10Time (x104)

Nutrient mass in thesoil nutrientpoolTotal biomassPlant biomass

Herbivore biomassCarnivore biomass

Figure 2: Temporal hanges n local speciesrichness(A) and biomass(B) duringthe ecosystemassemblyprocess.Biomassand localspeciesrichnessare shown for the community as a whole (blackdotted lines), all plants (blacksolid lines), all herbivores(blackdashedlines),and all carnivores(gray solid lines). The gray dashed line representsthe nutrient mass in the soil nutrient pool. Regional species richness = 45, nontrophicconnectance= 0.2, maximalnontrophicmagnitude= 0.2.

least one of the two net specieseffects (Eigor Egi) s non-zero:interaction web connectance =

number of nonneutral species interactionslocal species richness (local species richness 1)/2= 1 - proportion of neutral species interactions.

(5)We call "mean value of facilitation" he mean value of allpositive net species effects, and we call "mean value ofinhibition" the mean value of all negative net specieseffects.We calculated he proportions,mean values, and stan-dard deviations of species effects and the proportionsofspeciesinteractionsboth in the communityas awhole andwithin each trophic level. We then consideredonly inter-specific specieseffects and interactionswithouttaking ntoaccount the effect of a species on itself (Ei).

The production of each trophic level was measuredbyits nutrient inflow (nutrientmass per unit time):s scarnivoreproduction = qcacyHjACHHjCi,=1 j=1s sherbivoreproduction = 1:

qHaHPjAHpjPjHi,(6)i=1 j=1

primary production = aiLi~PiLiLii.=1We measured all these community and ecosystem prop-erties during the course of community assembly.After atransition phase, however, the ecosystem systematicallyreached a quasi-stationary egime(fig. 2), that is, a phasewhere aggregatedvariables such as local species richnessand biomass showed small variations around a constanttemporalmean value. Therefore,we comparedthe com-munity and ecosystempropertiesin the quasi-stationaryregime by calculatingthe temporalmean during the last


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Nontrophic nteractions,Biodiversity, nd EcosystemFunctioning 9710% fthe simulationterations. oreachvalueofregionalspeciesrichness,nontrophicconnectance, r maximalnontrophicmagnitude,we performed ightsimulationswithdifferentandom ompositionsf theregionalpeciespool, as is often done in experiments HooperandVitousek 997;Hectoret al. 1999;Knopset al. 1999;Til-manet al.2001),andwecalculatedhe meanandstandarddeviation f the measured ropertiesor theseeightrep-licates.Wepresentbelowfiguresor a relatively ighre-gionalspeciesrichness(45 species, .e., 15 speciespertrophicevel),butthe results requalitativelyimilarwhat-ever he number f species. nthefiguresbelow hat de-scribe he effectsof regional peciesrichness,both non-trophic onnectancendmaximal ontrophicmagnitudewereset to 0.2.

ResultsLocal pecies ichness ndspecies ichness t each rophiclevel nthequasi-stationaryegimencreased ithregionalspeciesrichnessn both food webs and interactionwebs(fig.3A,3B).Therefore,heresults btained s functionsof regionalpecies ichness r localspecies ichnesswereverysimilar.

Impact fNontrophicnteractionsndSpecies ichness nthePrevalencendStrengthfSpeciesnteractionsDiversityfSpeciesnteractionsn InteractionWebs. Non-trophic onnectance as an important ffectnot onlyonecosystemproperties ut alsoon the natureof speciesinteractions.Whennontrophiconnectancencreased,heproportion f 0 specieseffectsdecreased, hilethe pro-

portionsof facilitation nd inhibition ncreased p to aplateau fig.4A). In a foodweb,the only directspeciesinteractions exploitation etween rophicevels(fig.4B,when nontrophic onnectance 0, and fig. 4F). In aninteraction eb, heproportionf neutralnteractionse-creased o 0 as nontrophic onnectancencreased, ndhence nteraction ebconnectancencreasedo 100%.Theproportions f commensalismnd amensalismirst in-creased nd hendecreasedo 0 asnontrophiconnectanceincreased,whilethe proportions f mutualism,ompeti-tion,andexploitationncreasedsnontrophiconnectanceincreasedfig.4B).Thus,nteractionmodificationsreatednontrophicnteractions etween pecies(suchas mutu-alismor competition).

Impact f InteractionModificationsn the PrevalencendStrengthf Species ffects.As eithernontrophic onnect-ance(fig.4A)ormaximal ontrophicmagnitudefig.4E)increased,he meanvalueoffacilitationncreased nd hemeanvalueof inhibitiondecreased;hat s, the meanab-solutevaluesof species ffectsncreased,ndthe standarddeviation fspecies ffects lso ncreased. smaximal on-trophicmagnitudencreased,heproportion f 0 specieseffects irstfluctuated ndeventuallyncreasedfig.4E).Thus,both nontrophic onnectance nd maximalnon-trophicmagnitudehad an impacton the prevalence,strength, ndvariabilityf species ffects.The increasen the meanvalueof specieseffectswithnontrophic onnectances a consequencef the assump-tion thatnontrophic ffectsact multiplicativelyn con-sumption rates. Since the magnitudeof interactionmodificationmxYzs randomlyaken roma uniformdis-tribution,t has the sameprobabilityf beingpositiveor

U)

0)

.-0._)-J

A FOODWEB504030201000 20 40 60 80 100

Regional pecies richness

INTERACTIONWEB504030201000 20 40 60 80 100

Regional pecies richness

Figure 3: Localspeciesrichnessn the quasi-stationaryegimeas a functionof regionalspeciesrichnessn food webs (A; nontrophicconnectance 0, maximalnontrophicmagnitude 0) and in interactionwebs (B; nontrophic onnectance 0.2, maximalnontrophicmagnitude 0.2).Wepresent hemeanand standard eviation or localspecies ichnessfilled ircles),lant speciesrichnessunfilledircles),herbivorepecies ichnesstriangles),ndcarnivorepecies ichnesssquares).


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Figure 4: Proportionsand strengthof species effects (A, C, E) and proportionsof speciesinteractions(B, D, F) in the communityas a whole inthe quasi-stationary egimeas functions of nontrophicconnectance(A, B;regionalspeciesrichness= 45, maximalnontrophicmagnitude= 0.2)and regional species richness in interaction webs (C, D; nontrophicconnectance= 0.2, maximal nontrophicmagnitude= 0.2). E shows theproportions and strength of species effects as functions of maximal nontrophic magnitude (regional species richness = 45, nontrophicconnectance= 0.2). F shows the proportions of species interactions as functions of regional species richness in food webs (nontrophicconnectance= 0, maximalnontrophicmagnitude= 0). A, C, and E show the mean and standarddeviation for the proportionsof neutral effects(graydiamonds,graylines, x 100), facilitation(filledcircles,x 100) and inhibition (filled triangles,x 100), the mean and standarddeviation for themean value of facilitation(unfilledcircles, x 100), the mean value of inhibition (unfilled triangles,x 100), and the standard deviationof specieseffects(unfilled quares,x 10). B, D, andF show the mean and standarddeviation for the proportionsof mutualism(filledcircles,x 100),competition(filled triangles,x 100), exploitation(filledsquares,x 100), commensalism(unfilledcircles,x 100),amensalism unfilled riangles,x 100),and neutralinteractions(graydiamonds,graylines, x 100). Dashed lines (gray-filled utlineddiamonds,x 100) represent nteractionweb connectance.


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Nontrophic nteractions,Biodiversity, nd EcosystemFunctioning 99

negative;hus, hepotential onsumptionate hat t mod-ifies has the sameprobability f beingmultipliedby afactor r divided ythesame actoreq.[2]).Ontheotherhand, he net specieseffectEig f a speciesg on anotherspecies is proportionalo themagnitude f therealizedconsumptionates,ajltij,hatareaffected yspecies .Thedifference etween he multiplicativeffectof interactionmodificationsn the magnitude f realized onsumptionratesand he additive ffectof realizedonsumptionateson the netspecies ffectE,g xplainswhy,on average,on-trophicnteractionsend o increase etspecies ffects.Onaverage,0%of the x,,willbe in therange0-1 and50%will be in therange1-oc.Therefore, ith ncreasingon-trophic onnectance,he arithmeticmeanof therealizedconsumptionatesa,,x, will ncreasinglyxceeda,,, lead-ing to an increasing verage trength f both inhibitionandfacilitation.Thispropertymightbe viewedas a limitation f themathematicalormulation f our model,which nvolvesbothmultiplicativendadditive ffectswhile there s nodistributionf numbersn whichboth theirproduct ndtheir umareequal o 1.But t mightbe realistic, ecausebiological atesdo havea lowerboundof 0 andno upperbound,so thatnontrophicnteractionsmaybe expectedto have he effectspredicted yourmodel.Ultimately,hisissuewill have to be resolvedusing empiricaldataonnontrophic ffects n natural cosystems, ut thesedataarecurrentlyorely acking.

The InteractionWeb Connectance ncreaseswith SpeciesRichness.As regional peciesrichness and hencealsolocalspecies ichness)ncreased,heproportion f 0 spe-cies effectsdecreased, hile heproportionsf facilitationandinhibition ncreasedfig.4C).Accordingly,hepro-portionof neutral pecies nteractions ecreasedo verylow values(fig. 4D), whereas he proportions f com-mensalism and amensalism first increased and then de-creased,he proportions f mutualism nd competitionincreased, nd the proportionof exploitationwas un-changed.Thus, interactionweb connectancencreasedwithspecies ichnesso nearly100%:he morenumerousspeciesare, hemoretheyinteractwith otherspecies.In food webs, the proportion of neutral species inter-actions increased with regional species richness (fig. 4F)whereasthe proportion of exploitation decreased.Thus,the increasein interaction web connectancewith speciesrichness n interactionwebs (fig.4D) is due to the presenceof nontrophiclinks between species.The increasein interactionweb connectance with spe-cies richness s a generalpropertyof interactionwebs thatcan be explained ntuitivelyas follows: as regionalspeciesrichnessincreases, he number of trophic links of a givenspecies increases(as long as consumers are not strictspe-

cialists),which ncreasesheprobabilityorthisspeciesohaveat eastonetrophicinkmodified yanyother peciesin the web. The fact that interactionweb connectancetendsto 100%,however,s due to the assumptionhatconsumers regeneralistsn our model. Other ood webconfigurations ay ead to smallerupper imits.TheStrengthf Species ffects ecreases ithSpeciesRich-ness. As regionalpecies ichnessncreasedfig.4C),themean valueof facilitation ecreased nd the meanvalueof inhibitionncreased;hatis, the mean absolute aluesof specieseffectsdecreased. hus,specieseffects endedto be denser,hat s, proportionally orenumerous, ndweaker sregionalpecies ichnessncreasedninteractionwebs(fig.4C).Weak veragenteractiontrength robablybuffers he destabilizingffectsof interactionweb con-nectanceandspeciesrichness.Note thataveragenterac-tion strength lso decreased ithregionalpecies ichnessin food webs(themean absolute alueof inhibitionde-creased;esultsnot shown).SpeciesEffects ndSpeciesnteractions ithin he VariousTrophicevels. Patternswithin hevarious rophicevelswereverysimilar o those reportedn figure4 for thecommunity s a whole.Theonlydifferenceoncernedheprevalencef facilitation nd nhibition, ndhenceof in-terspecificompetition ndcooperation,mongplantsasnontrophic onnectancencreasedfig.C1 in the onlineeditionof the AmericanNaturalist).n plants, he pro-portionof inhibition ncreasedmorerapidlyhanthat offacilitation, hereasn thecommunitys a whole fig.4A)or in othertrophic evels(resultsnot shown),the pro-portionsof inhibitionand facilitation aried n the sameway.In the samemanner,heproportion f interspecificcompetitionncreasedmorerapidly han that of mutu-alismin plants,whereas heseproportions aried n thesame way in other trophic levels or in the whole com-munity (fig. 4B).

Impacts fNontrophicnteractionsnBiodiversityand Ecosystem ropertiesNontrophicInteractionsand EcosystemProperties. Non-trophic connectance and maximalnontrophicmagnitudehad an important effect on ecosystem properties.Localspecies richness decreasedas either nontrophic connect-ance (fig. 5A) or maximalnontrophicmagnitude (fig. 5B)increased,except at a low level of nontrophic parameters.Plantspeciesrichness decreasedas eithernontrophiccon-nectance (fig. 5A) or maximalnontrophic magnitude (fig.5B) increased.In contrast,consumerspeciesrichnesswasless affectedby variationsin nontrophic connectance;it


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100 The AmericanNaturalist

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Figure5: Local pecies ichness, iomass, ndproductionn thequasi-stationaryegime s functions f nontrophiconnectanceA,C,E;regionalspecies ichness 45, maximal ontrophicmagnitude 0.2) andmaximal ontrophicmagnitudeB,D, F;regional pecies ichness 45, non-trophiconnectance 0.2).Wepresenthemeanandstandard eviationor otal filled ircles)ocal pecies ichness ndbiomassndplant unfilledcircles),erbivoretriangles),ndcarnivoresquares)ocalspecies ichness, iomass, ndproductionperunittime).Dotted inesrepresenthenutrientmass n the soil (diamonds).

decreased s maximalnontrophicmagnitudencreased,exceptat a low levelof maximalnontrophicmagnitude.Thebiomassandproduction f eachtrophic evel de-creasedharply seithernontrophiconnectancefig.5C,5E)or maximalnontrophicmagnitude fig.5D, 5F) in-creased. nteraction ebs withhighlevelsof eithernon-trophicconnectance r maximalnontrophicmagnitudeweresystems n which the biologicalprocessesof pro-

ductionandrecyclingwerevery ow andinorganiclowsprevailed.uchhigh evelsof nontrophiconnectance ndmagnitude represumablybsent n natural cosystems.TheImpacts f Nontrophicnteractionsn Ecosystemro-cessesAreMediated yChangesn RealizedConsumptionRatesandSpeciesnteractions.The decrease n biomassand production t all trophic evels as nontrophic on-


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Nontrophic nteractions,Biodiversity, nd Ecosystem unctioning 101

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Figure 6: Biomass and production in the quasi-stationaryregime as functions of regional species richness in food webs (A, C; nontrophicconnectance= 0, maximal nontrophic magnitude= 0) and in interaction webs (B, D; nontrophic connectance= 0.2, maximal nontrophicmagnitude= 0.2). Wepresentthe mean and standarddeviation for total biomass (filledcircles),otalplant(unfilledcircles),otal herbivore triangles),and total carnivore(squares)biomass and production (perunit time). Dotted lines represent he nutrient mass in the soil (diamonds).

nectance or maximalnontrophic magnitude ncreasescanbe explainedby the impactsof nontrophicinteractionsonrealizedconsumption rates and species interactions(fig.4). First,mean realizedconsumption rates increase withnontrophicconnectance,as mentioned above,which con-tributes to a decrease in the biomass and production atthe next lowertrophiclevel. Thesedeclinesin the biomassand productionof lowertrophic evels cascadeup the foodweb and lead indirectly to decreased carnivore biomassand production. Increasingnontrophic connectance alsoleads to more intensecompetitionbetween consumersfortheir resourceand a smallerresource-use omplementarity,which contributes o a decreasen herbivoreandcarnivorebiomass and production.Second, the proportionsof inhibition and competitionincrease more than those of facilitationand mutualism inplantswhennontrophicconnectance ncreases.Thus,non-trophic interactions tend to make competition betweenplant species stronger,which may also partly explainthedecreasein primary production and plant biomass andhence indirectly n consumerproductionand biomass.

TheBiodiversity-EcosystemunctioningRelationshipsin Food Websand InteractionWebsEffectsof SpeciesRichnesson EcosystemProcesses n FoodWebsand InteractionWebs. In both food webs and in-teractionwebs, total biomass increasedwith regionalspe-cies richnessand hence with local speciesrichness(fig.6A,6B). Plant and carnivorebiomassesincreased n parallel,which suggestsa bottom-up control of plants on carni-vores. In contrast,herbivorebiomasswas less affectedbyspecies richness, which suggests a top-down control ofcarnivoreson herbivores.The soil nutrient concentrationdecreased as species richness increased,which shows abetter exploitationof the limiting nutrientby plants.Production at all trophic levels increasedwith regionalspecies richness (fig. 6C, 6D). Primary production in-creasedwith species richness because of the better ex-ploitation of the limiting nutrient by plants;as a result,the increasein plant biomass was much higher than thedecrease n the soil nutrientstock,andprimaryproductionincreased(eq. [6]; fig. 6A, 6B). Herbivoreproductionde-pends on plant biomass and herbivorebiomass (eq. [6]),


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102 The AmericanNaturalistand herbivorebiomass was top-down controlled;there-fore, herbivoreproduction showed the same pattern asplant biomass (cf. fig. 6A, 6C and 6B, 6D). In the samemanner,carnivoreproduction depends on herbivore andcarnivorebiomasses(eq. [6]), but herbivorebiomass wastop-down controlled; herefore,carnivoreproductionfol-lowed carnivorebiomass. Production(especially n plantsand herbivores) was generallyhigh compared with theinorganic nutrient input, L This results from the highrecycling efficiencyof ecosystemsin the quasi-stationaryregime,after a long assemblyprocess.This high recyclingefficiencyalso explainswhy the amount of nutrientin thesoil often remainedrelativelyhigh.Differences between food webs and interaction webshighlight he role of nontrophic nteractions n ecosystems.The increase n biomass andproductionwith regional(fig.6) and local (fig. 7) species richness was less rapid ininteraction webs than in food webs. Differencesbetweenfood webs and interaction webs were greaterat higherlevels of regionalspeciesrichness(fig.6A,6B).Thepositiveeffect of nontrophic interactions on the averagerealizedconsumption rate as species richness increases explainswhy nontrophiceffects are more important n species-richecosystems han in species-poorecosystems n our model.Higherrealizedconsumption rates make speciesmore ef-ficient but also more competitive.Biodiversity-EcosystemFunctioning Relationships. Westudied the relationshipbetween total biomass and localspecies richness at the quasi-stationary regime thatemergedfrom variations n one of the parametersof theregional species pool, that is, regional species richness,nontrophic connectance,or maximal nontrophic magni-tude (fig. 7). Whatever the parameterdrivingvariationsin local speciesrichness,there was a positive relationshipbetween total biomassand local speciesrichness.However,the shape of this relationshipdiffered. The relationshipwasroughly inear n both food webs and interactionwebswhen regionalspeciesrichnessvaried,but it was nonlinearandconcave-upwhennontrophicconnectanceor maximalnontrophic magnitude varied in interaction webs. Thisconcave-up relationshipbetween ocalspeciesrichnessandtotal biomass is explained by the greatereffect of non-trophic interactions on biomass than on speciesrichness(fig. 5). Thus, our model predictsa positive relationshipbetween speciesrichness and biomass in naturallyassem-bled ecosystems,but with a strong impact of nontrophicinteractions on the shape of the diversity-biomassrela-tionship. Specifically, ontrophic nteractionsareexpectedto decrease he magnitudeof biomass andproduction,butchanges in their frequency or strength are expected toincrease the dependency of biomass and production onlocal speciesrichness.

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Figure 7: Relationshipsbetween local speciesrichnessand total biomassdrivenby variations of differentpropertiesof the regional species pool:regional species richness in food webs (filled circles; nontrophicconnectance= 0, maximal nontrophic magnitude= 0), regionalspecies richness in interaction webs (unfilled circles; nontrophicconnectance= 0.2, maximalnontrophic magnitude= 0.2), nontrophicconnectance in interaction webs (triangles; egionalspecies richness =45, maximal nontrophic magnitude= 0.2), and maximal nontrophicmagnitudein interactionwebs (squares; egionalspeciesrichness= 45,nontrophicconnectance= 0.2).Discussion

Our theoretical model shows that species diversity andnontrophic interactionsaffectstronglyand nonintuitivelythe communityand ecosystem properties n communitiesthat have assembled hroughrepeatedcolonization events.Our main results canbe summarizedas follows.First, increasing regional species richness leads to in-creasedspecies richness,biomass, production, and inter-action web connectancebut decreasedaverage nteractionstrength in local communities. Thus, species-richinter-action webs areexpectedto be moreproductiveand moreconnected but to have weakerspecies interactionson av-erage. Their lower interaction strengthis probablywhatallowsthem to maintain a high diversityand connectance,in agreementwith previoustheory (May 1973;Kokkoriset al. 1999,2002). Theirhigh diversity n turn allows them

to use the limiting nutrient more efficientlyand hencehave a higher production and biomass, as predictedbyexisting theory (Tilman et al. 1997; Loreau 1998, 2000;Tilman 1999).Second, ncreasing he frequencyandmagnitudeof non-trophic interactionsin the regionalspecies pool leads todecreased ocal speciesrichness,biomass,andproduction.These counterintuitive ffectsresult romthe fact that non-trophic connectance and maximalnontrophicmagnitudecontributeto an increase n the magnitudeof trophiccon-


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Nontrophic nteractions,Biodiversity, nd EcosystemFunctioning 103

sumption fluxes, and hence the strengthof speciesinter-actions,on average. ncreasedaverage nteractionstrengthallows fewerspeciesto coexist and imposes a highermor-talityon lowertrophic evels,whicheventuallydeterioratesthe functioning of the whole ecosystem.Third,as a consequence,interactionwebs that includetrophic and nontrophic interactionsare expectedto havea lowerlocal richness,biomass,and productionthan foodwebs that include only trophic interactions, all else (inparticular,regionalspecies richness)being equal. A pos-itive diversity-biomass elationshipemergesfrom the as-semblyof both food webs and interactionswebs,but non-trophicinteractionsareexpected o affectthe shapeof thisrelationship.Thus, our resultsemphasizethe need to take into ac-count nontrophicinteractions n theoreticalecology.Oursimple, generalmodel allows all types of species interac-tions to be incorporatedby adding interaction modifica-tions to a food web. This mass-balance onstrainedmodelof an interactionweb providesa useful tool for studyingthe impact of diversity on the functional properties ofcomplex ecosystemswith more realism.TheBiodiversity-EcosystemunctioningRelationshipn InteractionWebs

Totalbiomass increaseswith regionalspeciesrichness,andhence also with local speciesrichness,in both food websand interactionwebs. In both typesof web, carnivorebio-mass is bottom-up controlledby plants,herbivorebiomassis top-down controlled by carnivores,and total biomassdepends mostly on plant biomass. These control mecha-nisms arethe same as in classical oodwebs,andourresultsconcerning food webs are in qualitative agreementwithprevious theoretical studies (Thdbaultand Loreau2003;Ives et al. 2005). Thebault and Loreau (2003), however,showed that in a two-level food web with generalisther-bivores, plant biomass and herbivore biomass increasenonlinearlywith regional species richness and can evendecreaseat a high level of speciesrichness,whereas n ourfood web model, biomassdoesnot decreaseat ahigh levelof species richness.This difference s probablyexplainedby the presenceof a third trophic level in our work andby the fact that, contraryto recent theoretical(Th6baultand Loreau 2003; Ives et al. 2005) and experimental(Downing and Leibold2002;Duffy et al. 2003) studiesinwhich species richness is controlled,our work providesadynamic approach to biodiversityand ecosystem func-tioning in which both factors result from an assemblyprocess.Our model predictsthatbiomass and productionaregenerally ower in interactionwebs than in food webs.Nontrophic interactionsarelikelyto generatestrongcon-straintson speciescoexistence. The speciespresentin the

community have higher realized consumption rates onaverage,which makes them more efficient but also morecompetitive. Therefore,the probabilityof observing ex-treme resourceexploitation and negative effects on eco-systemproperties s higher.SpeciesDiversity,Species nteractions,and EcosystemProcesses

Our work shows that speciesinteractionsdepend on spe-cies richness:both the strengthandthe prevalenceof theseinteractions are diversity dependent. In particular,themean absolute values of interspecificfacilitation and in-hibition effects are expected to decrease with increasingspecies richness. These resultsagreewith existing theory(Kokkoriset al. 1999, 2002). The analysisof naturalfoodwebs (Neutel et al. 2002) suggeststhat naturalecosystemsare characterizedby a majorityof weak interactionsanda minority of strong interactions.Our model also predictsthat interaction web connect-ance increaseswith species richnessand that the propor-tions of the various types of species effects and speciesinteractionsdepend on speciesrichness.A higher speciesdiversityincreases the probability or each species to in-teract with any other species. Although the relationshipbetweentrophicconnectanceandspeciesrichnesshasbeenwell studied in food webs (Martinez1992;Montoya andSole 2003), we lack knowledge concerning nontrophicconnectance and interaction web connectancein ecosys-tems.Speciesdiversity s known to affectecosystemfunction-ing through functional complementaritybetween speciesthat use different resources (Tilman et al. 1997; Loreau1998;Loreau and Hector 2001). Our model furthersug-geststhat it can affectecosystemfunctioningin more sub-tle waysby changingthe nature,prevalence,and strengthof species interactions. For instance, it can enhance eco-system processesby increasing he probabilityof facilita-tion (Cardinale et al. 2002). Recent experiments haveshown that a form of facilitation,wherebya species en-hances the access of other species to resourcesthroughbiophysicalmodifications,affectsthe productivityand thediversity-productivity elationship n bryophyte commu-nities (Mulderet al. 2001;Rixenand Mulder2005). Thus,speciesinteractions, n particularnontrophicinteractions,should be given more attention in studying the relation-ship between biodiversityand ecosystem properties.

Incorporating ontrophic nteractionsinto TheoreticalEcologyOur work providesa consistentecosystemmodel that in-corporates nontrophic interactions in the form of inter-


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104 The AmericanNaturalistaction modifications.Although interactionmodificationswere regarded by Wootton (1994) as a class of indirecteffects, they may be viewed as either direct or indirecttrait-mediated nteractions(Abrams 1995), dependingonthe context. The interactionmaybe direct f, for instance,the interactionmodifierdirectlyaffects he behaviorof thetwo species whose trophic interaction is affected. But itmay also be indirectif, for instance,the interactionmod-ification occurs through habitat modification, which inturn affects the ability of the predatorto detect or catchits prey.Also,the director indirectnatureof trait-mediatedinteractions s not necessarilyreflected n a model's struc-ture (Abrams 1995). Therefore,our measure of the netspecies effect, Eg, of species g on species i includes bothtrophicandnontrophicdirecteffectsas wellas,potentially,trait-mediatedindirect effects. The ambiguityof currentterminology suggests hat a reexaminationof the concepts,definitions, and measures of direct and indirect effectsmight be useful.Ecosystemengineers(Joneset al. 1994) arestrongdriv-ers of interactionmodifications:by modifyingtheirphys-ical environment,they createmanynontrophic speciesin-teractions.Autogenicor allogenicengineerscan modulateresource flows or abiotic parameters hat influences re-source flows. Therefore,our modeling frameworkcouldbe applied to the study of ecosystem engineers throughspecific nontrophic modifications of trophic interactionsor through modifications of abiotic parameterssuch asthose that govern the input, recycling,and loss of nutri-ents. Experimental tudies show that ecosystemengineer-ing can induce changes in community structuralprop-erties, such as species richness (Zhu et al. 2006), speciescomposition (Badanoet al. 2006), and species nteractions(Collen and Gibson 2001), and ecosystemfunctionalpro-cesses such as primaryproduction (Zhu et al. 2006). Ourmodel shows that interaction modifications of all kindscan profoundly affect ecosystem propertiessuch as bio-mass and production at all trophic levels.We view our interactionweb model as a promisingtoolfor merging the community and ecosystem perspectivesin theoreticalecology.By incorporatingnontrophicinter-actions in the form of modifications of trophic interac-tions, our model describes material flows in a consistentway and hence allowsanalysisof ecosystem properties.Atthe same time, it is a flexible and dynamic model thatallows all kinds of speciesinteractions o occur,andhence,it allowsanalysisof community propertiessuch as speciesdiversityand the connectance,prevalence,and strengthofspecies interactions.

Model LimitationsDespite its strengthsand generality,our model also haslimitations. Werepresentednontrophicinteractions n the

form of modificationsof trophicinteractions,but it wouldalso be possible to introduce them in the form of modi-ficationsof nontrophic parameters uch as intrinsicdeathrates and recyclingrates.We assumed thatnontrophic ef-fects act multiplicativelyon potential consumption ratesbecause this led to the simplest functional form for in-teraction modificationsthat respectsthe direction of con-sumption fluxes between trophic levels.This assumptionmakes sense mathematicallybecausebiologicalrateshavea lower bound of 0 and no upper bound; nontrophicmodifications of these ratesmay have asymmetriceffects,depending on whetherthey are positive or negative.Butthis assumption indirectly drives many of the observedeffects of nontrophicinteractions n our model, and em-pirical data to assess its validity are unfortunatelysorelylacking. We also assumed that the various modificationsof a given trophic interactionby differentspeciesare ad-ditive, but interaction modificationsmight interferewitheach other and generatenonadditiveeffects. For instance,there could be a hierarchyof effects (one effect dominatesover the others) or a synergyof effects (the effect of onespecies can only be expressedin, or is modified by, thepresence of another species). Our linear approximation,however, s in line with thesimplicityrequired ora generalecosystemmodel.We assumed that there is a single limitingnutrient andthat the stoichiometriccompositionof eachspecies s con-stant, although the stoichiometriccomposition of plantsis known to depend on factors such as environmentalconditions and the presenceof consumers.This simpli-fication was made to avoid unnecessarycomplications na model designed to explore the role of nontrophic in-teractions. In our model, the soil is compartmentalized,each plant taking up nutrient in its species-specificre-sourcedepletionzone.Thisassumptionwasmeant to favorplant species resource-usecomplementarityand coexis-tence to some extent, although strong nontrophic inter-actions led to species-poorcommunities despite this as-sumption.We expectour results to be qualitatively obustto relaxation of this assumption,as suggestedby prelim-inary simulation results. A decomposer compartmentcould be added, but it is unlikely to change the resultsbecause we have already ncluded nutrientcyclingin ourmodel without detailingits mechanisms.Lastly,our cur-rent model considersan interactionweb with threetrophiclevels. It would definitelybe interestingto model an in-teraction web with a more realistic and flexible trophicstructureby allowingconsumersto feed on several rophiclevels andblurring he separationbetweendistincttrophiclevels.

Our theoreticalstudy suggests severalhypothesesthatdeserve to be tested experimentally. t would be useful tostudy the influence of species richness on species inter-


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Nontrophic nteractions,Biodiversity, nd EcosystemFunctioning 105actions:does the strength f specieseffects facilitation,inhibition)dependon speciesrichness n experimentalecosystems?oes theirconnectancencreasewithspeciesrichness? t wouldalso be useful o test the impactsofnontrophicnteractionsn ecosystem rocesses ndtheirmechanisms: hat s the influence f the connectancermagnitudefnontrophicnteractionsnbiomass ndpro-ductionat various rophic evels?Do nontrophicnter-actionsgenerallyncrease esourceonsumption, ossiblyleading o extreme esource xploitation?

ConclusionWehavedeveloped modelof a fullinteraction ebthatincludes othtrophicandnontrophicpeciesnteractionsandthatrespectsheprinciple f massconservation.hismodel showsthe important ole of species nteractions,especiallyontrophicnteractions,n communityndeco-systemproperties nd in therelationshipetweenbiodi-versityandecosystemunctioning. hediversity-biomasspatterns btainedwithourinteraction ebmodelarenotstrikinglyifferentrom hoseshown n recent heoreticalstudiesof food webs,but the mechanisms refarmorecomplex.nparticular,ur modelpredictshat henature,prevalence, nd strengthof species nteractions hangewith regionaland local speciesrichness,which makesthe mechanismsf thebiodiversity-ecosystemunctioningrelationshipmorecomplex n an interactionweb. Spe-cies interactions,nd especially ontrophicnteractions,shouldbegivenmoreattentiono improve nderstandingandpredictions f the ecological onsequencesf biodi-versityoss.

AcknowledgmentsWe thankC. Fontaine,S. Kdfi,N. Loeuille,E. Thebault,and ouranonymouseviewersorconstructiveommentson the manuscript.

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