nontrophic interactions, biodiversity, and ecosystem ...€¦ · functioning relationship. thus,...

vol. 171, no. 1 the american naturalist january 2008 �

Nontrophic Interactions, Biodiversity, and Ecosystem

Functioning: An Interaction Web Model

Alexandra Goudard1,2,* and Michel Loreau3,†

1. Biogeochimie et Ecologie des Milieux Continentaux, UniteMixte de Recherche 7618, Ecole Normale Superieure, 46 rued’Ulm, F-75230 Paris cedex 05, France;2. Universite Pierre et Marie Curie, 4 place Jussieu, 75252 Pariscedex 05, France;3. Department of Biology, McGill University, 1205 avenue DocteurPenfield, Montreal, Quebec H3A 1B1, Canada

Submitted December 19, 2006; Accepted August 20, 2007;Electronically published November 12, 2007

Online enhancements: appendixes, figure.

abstract: Research into the relationship between biodiversity andecosystem functioning has mainly focused on the effects of speciesdiversity on ecosystem properties in plant communities and, morerecently, in food webs. Although there is growing recognition of thesignificance of nontrophic interactions in ecology, these interactionsare still poorly studied theoretically, and their impact on biodiversityand ecosystem functioning is largely unknown. Existing models ofmutualism usually consider only one type of species interaction anddo not satisfy mass balance constraints. Here, we present a modelof an interaction web that includes both trophic and nontrophicinteractions and that respects the principle of mass conservation.Nontrophic interactions are represented in the form of interactionmodifications. We use this model to study the relationship betweenbiodiversity and ecosystem properties that emerges from the assemblyof entire interaction webs. We show that ecosystem properties suchas biomass and production depend not only on species diversity butalso on species interactions, in particular on the connectance andmagnitude of nontrophic interactions, and that the nature, preva-lence, and strength of species interactions in turn depend on speciesdiversity. Nontrophic interactions alter the shape of the relationshipbetween biodiversity and biomass and can profoundly influence eco-system processes.

Keywords: biodiversity, ecosystem functioning, nontrophic interac-tions, interaction modifications, mass balance, model.

* E-mail: [email protected].

† Corresponding author; e-mail: [email protected].

Am. Nat. 2008. Vol. 171, pp. 91–106. � 2007 by The University of Chicago.0003-0147/2008/17101-42291$15.00. All rights reserved.DOI: 10.1086/523945

The relationship between biodiversity and ecosystem func-tioning has emerged as a central issue in ecology in thelast decade. Human activities contribute to species ex-tinction, and biodiversity loss can cause loss of ecologicalservices (Pimm et al. 1995; Vitousek et al. 1997; Sala etal. 2000; Loreau et al. 2001, 2002; Kinzig et al. 2002;Hooper et al. 2005). Therefore, a better understanding ofthe effects of biodiversity on ecosystem properties is crit-ically needed.

The relationship between biodiversity and ecosystemprocesses has mostly been studied experimentally and the-oretically in plant communities. Theoretical models usu-ally predict that primary productivity increases with plantspecies richness but saturates at high diversity (Tilman etal. 1997; Loreau 1998, 2000). Controlled experiments con-ducted in different localities (Hector et al. 1999; Spehn etal. 2005) or over several years (Tilman et al. 2001) oftenexhibit the predicted pattern. The positive effects of speciesdiversity on ecosystem functioning have been explainedby two main mechanisms (Tilman 1999; Loreau and Hec-tor 2001): a complementarity effect, which emerges fromfacilitation or niche differentiation, and a selection effectarising from the dominance of species with particulartraits. These models, which focus on a single trophic level,are based on niche theory and plant competition for alimiting nutrient. Food web models with several trophiclevels, however, predict that plant biomass does not alwaysincrease with plant diversity and that changes in diversitycan lead to complex changes in ecosystem functioning(Thebault and Loreau 2003; Ives et al. 2005). Recent ex-periments (Jonsson and Malmqvist 2000; Downing andLeibold 2002; Duffy 2002; Paine 2002; Duffy et al. 2003,2005) have showed that trophic interactions can indeedprofoundly affect the relationship between biodiversity andecosystem functioning.

Thus, the relationship between biodiversity and biomassor productivity has been mostly studied in plant com-munities or in food webs. The only form of direct speciesinteraction considered in these studies is the trophic in-

92 The American Naturalist

teraction (exploitation competition for a shared resourceis an indirect effect of the consumer-resource trophic in-teraction). Some experiments, however, suggest that non-trophic interactions, such as facilitation, may play an im-portant role in ecosystem functioning (Mulder et al. 2001;Cardinale et al. 2002; Rixen and Mulder 2005). For in-stance, Rixen and Mulder (2005) showed that water re-tention increases with species diversity through increasingfacilitation and leads to increased productivity in arctictundra moss communities. Experiments suggest that dif-ferent kinds of species interactions do not act in isolationfrom each other in nature but co-occur within the samecommunity (Callaway and Walker 1997). Evidence for theimportance of indirect interactions is also accumulating.Habitat modification is one type of indirect interactionthat has been studied experimentally (Bertness et al. 1999;Mulder et al. 2001; Cardinale et al. 2002; Rixen and Mulder2005). For instance, Bertness et al. (1999) have shown thatalgal canopy reduces physical stress such as temperatureor water evaporation and thus has positive effects on or-ganism recruitment, growth, and survival in rocky inter-tidal communities, but habitat modification by algal can-opy can also have negative effects by increasing consumerpressure.

Although there is growing recognition of the signifi-cance of nontrophic interactions in communities and eco-systems, these interactions are still poorly studied theo-retically, and we still know little about general patternsand mechanisms. Therefore, an important current chal-lenge is to understand how nontrophic interactions affectthe relationship between biodiversity and ecosystem func-tioning. More generally, there is an urgent need to includenontrophic interactions in ecological theory (Borer et al.2002; Bruno et al. 2003). There are some models of mu-tualism (Goh 1979; Heithaus et al. 1980; Addicott 1981;Ringel et al. 1996; Holland et al. 2002), but these modelsare specific, as they consider only one kind of speciesinteractions. We lack general models of interaction websthat include all types of direct species interactions (inter-ference competition, mutualism, exploitation, commen-salism, amensalism) as well as their indirect effects. Simplemodels of mutualism based on Lotka-Volterra equationsalso have the unrealistic property of leading to explosivesystems because they do not respect the physical principleof mass conservation (Ringel et al. 1996). Mass balance iscrucial for understanding the functional processes of nat-ural ecosystems. Thus, it is necessary to construct inter-action web models that satisfy mass balance constraints.Arditi et al. (2005) recently made a first step in that di-rection by adding interaction modifications to 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 approach to

study the structural and functional properties of interac-tion webs and hence the relationships between biodiversityand ecosystem properties that emerge in complex ecosys-tems.

We need theories and models to provide generalizationson the role of nontrophic interactions in ecosystem func-tioning. It is therefore necessary to construct the mostgeneral possible model of an ecosystem—a model of aninteraction web that includes both trophic and nontrophicinteractions and that respects the principle of mass con-servation. Here, we present a theoretical model that meetsthis need. Despite its generality, our model is too complexto be analytically tractable. Therefore, we study it usingnumerical simulations that mimic a community assemblyprocess. This allows us to investigate the relationship be-tween biodiversity and ecosystem functioning under morerealistic constraints than would an analytical equilibriumstudy of special cases, in agreement with Loreau et al.’s(2001) recommendation to study the relationship betweenbiodiversity and ecosystem functioning with a dynamicapproach. Using this assembly model, we study the rela-tionship between biodiversity and ecosystem properties,such as the biomass and productivity of the various trophiclevels, in an interaction web in comparison with a foodweb. This comparison allows us to examine the effects ofnontrophic interactions on the biodiversity-ecosystemfunctioning relationship. Thus, our interaction web modelprovides a useful basis for reaching greater generality re-garding the impact of species diversity and species inter-actions on the functional properties of complex ecosys-tems.

Model and Methods

The Interaction Web Model

The model is an extension of a model developed by The-bault and Loreau (2003) for a nutrient-limited ecosystemwith three trophic levels containing an arbitrary numberof plants, herbivores, and carnivores. Plants take up a lim-iting nutrient in their rhizosphere, thus creating species-specific resource depletion zones and allowing plant co-existence under some conditions (Loreau 1996, 1998).These species-specific resource depletion zones may beviewed as physical soil volumes, but they may also beviewed in a more abstract way as different niche spacesavailable to different species.

Here, we add nontrophic interactions to this food webto construct an interaction web model that satisfies massbalance constraints. Nontrophic interactions are includedin the form of interaction modifications: each species canmodify the trophic interaction between any two species.

Nontrophic Interactions, Biodiversity, and Ecosystem Functioning 93

Figure 1: Interaction web model. Solid arrows represent nutrient flows.For clarity of the figure, flows of nonassimilated nutrient returned to thesoil nutrient pool during consumption by carnivores and herbivores arerepresented only on the left trophic chain, and flows of nutrient eitherrecycled or lost from the ecosystem following death are represented onlyon the right trophic chain. Dotted lines represent interaction modifica-tions. Only five examples of interaction modifications are representedhere for the sake of clarity. For instance, herbivore species Hz modifiesthe trophic interaction between herbivore species Hy and carnivore speciesCx, with a magnitude of interaction modification of mxyz. The modificationof the nutrient flow between plant species Pj and its species-specificresource depletion zone Lj corresponds to intraspecific competition orfacilitation.

The model is described by the following dynamic equa-tions and figure 1:

SdCi p q a m H C � u C ,� C C H C H j i C ii j i j idt jp1

SdHi p q a m P H� H H P H P j ii j i jdt jp1

S

� a m C H � u H ,� C H C H j i H ij i j i ijp1

dPi p a m L PP L P L i ii i i idt

S

� a m H P � u P ,� H P H P j i P ij i j i ijp1

dL Vi ip g R � L � a m PL ,i P L P L i ii i i i( )dt VR

SdR Vip I � l R � g R � L�R i( )dt Vip1 R

S S

� (1 � l )u P � (1 � l )u H� �P P i H H ii i i iip1 ip1

S

� (1 � l )u C� C C ii iip1

S S

� (1 � q )a m H C�� C C H C H j ii j i jip1 jp1

S S

� (1 � q )a m P H , (1)�� H H P H P j ii j i jip1 jp1

where

nn

mxyzm p m log (1 � X ) p (1 � X ) . (2)� �xy xyz z zzp1ezp1

Here, S is the number of species per trophic level, n pis the total number of species, and , , and are3S P H Ci i i

the nutrient stocks of plant, herbivore, and carnivore spe-cies i, respectively.

We assume the stoichiometric composition of each spe-cies to be constant; hence, its nutrient stock is proportionalto its biomass. Parameter is a per capita potential con-axy

sumption rate, that is, the intensity of the trophic inter-action between predator species and prey species inx ythe absence of interaction modification ( ). Param-a ≥ 0xy

eter is the nutrient uptake rate of plant species i. Eacha P Li i

herbivore or carnivore species may be more or less spe-x

cialist (if one of its potential consumption rates is muchaxy

higher than the others) or generalist (if all its potentialconsumption rates are of similar magnitude), with aaxy

preference for certain prey species. The constants andqH

are the conversion efficiencies of herbivores and car-qC

nivores, respectively (see app. A in the online edition ofthe American Naturalist for parameter values).

Nontrophic interactions are introduced into the modelby adding nontrophic modifications of trophic interac-tions: each species is allowed to modify the trophic in-zteraction between species and with an effect that de-x ypends on both its biomass and a magnitude ofXz

interaction modification of (fig. 1). Parameter ism mxyz xy

the nontrophic coefficient: it is the total nontrophic effectof all species of the community on the trophic interactionbetween species and . Thus, species consumes speciesx y x

with a realized consumption rate . The functiony a mxy xy

that describes nontrophic effects (eq. [2]) was chosen suchthat it satisfies several conditions. First, it is a strictly in-creasing function of both the magnitude of interactionmodification and biomass . Second, if eitherm Xxyz z

or , then , and speciesmxyzm p 0 X p 0 (1 � X ) p 1xyz z z

does not affect the trophic interaction between speciesz


and . Thus, in the absence of interaction modifications,x y, and the realized consumption rate is equalm p 1 a mxy xy xy

to its potential value . Third, the function for non-axy

trophic effects is strictly positive, so that the sign of therealized consumption rate does not change. Thea mxy xy

magnitude of interaction modification can be positivemxyz

or negative without changing the sign of . Whatevera mxy xy

the nontrophic effects of other species, the nutrient flowbetween species and is never reversed. Fourth, whilex ythe magnitude of interaction modification comes asmxyz

the exponent of a power function to keep the functionpositive, we chose a linear dependence on biomass :Xz

preliminary results showed that a power dependence onbiomass is strongly destabilizing. A positive leads tomxyz

multiplication of the potential consumption rate by aaxy

factor , whereas a negative leads to di-Fm Fxyz(1 � X ) mz xyz

vision of by this factor.axy

In the presence of interaction modifications, each spe-cies can affect any other species by modifying one orxseveral trophic interactions that involve species x andhence increasing or decreasing the population growth rate

of species x. In the absence of interaction modi-dX /dtx

fications, the only direct species interaction is 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 ( for species ). Our model web is then am ( 0 zxzz

full 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. Interaction mod-ifications change the material flow between a resource anda consumer by multiplying it by some factor, but there ismass conservation overall.

The model also includes nutrient cycling. The constantis the loss or death rate of species , and is theu x lx x

nonrecycled (lost) proportion of nutrient coming fromspecies . The variable R is the nutrient mass in the soilxnutrient pool with volume ; is the nutrient mass inV LR i

the set of species-specific resource depletion zones, withtotal volume , of plants from species i. Nutrient is trans-Vi

ported between species-specific resource depletion zonesand the soil nutrient pool at a diffusion rate g per unittime. In our simulations, g was quite high ( ) tog p 10allow rapid soil homogenization and strong indirect plantcompetition for the limiting nutrient. Parameter I is the

nutrient input in the soil nutrient pool per unit time, andis the rate of nutrient loss from the soil nutrient pool.lR

Community Assembly

We constrained our interaction web model as little as pos-sible in order to explore its general properties. Accordingly,we randomly assigned the various biological parameters(potential consumption rates, intensities of interactionmodifications, death rates, nonrecycled proportions of nu-trient) to a regional pool of species from a uniform dis-tribution within appropriate intervals and let the local eco-system assemble spontaneously. The establishment of aspecies depends on both its intrinsic traits (parameter val-ues) and its interactions with the other species alreadypresent in the ecosystem.

The model was simulated numerically using C�� pro-gramming, and numerical integration of the dynamicequations was performed with a Runge-Kutta method 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 ecosystem results from an assembly pro-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 nutrient pool. Species were picked atrandom from a regional species pool with species richness

(S species at each trophic level) and introducedn p 3Sregularly to the community. Each successive introductionoccurred after a constant period irrespective of whether anew equilibrium was reached (thus, if the introductionperiod was 100 time steps, there were 1,000 introductionevents during a simulation).

Local species richness is the total number of species inthe locally assembled ecosystem (to be distinguished fromregional species richness). The total volume of the soil waskept constant irrespective of local species richness:

, where if species i was notS

V p V �� V V p 0soil R i iip1

present in the community. Thus, when a plant speciesbecame extinct, the volume of its species-specific resourcedepletion 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 resource depletionzones was created and subtracted from the volume of thesoil nutrient pool. This volume allocation rule respects theconservation of total soil volume while at the same timeallowing different plant species to occupy complementaryresource depletion zones in the soil. Complementarity be-tween plant species has both theoretical and empirical jus-tifications (Loreau 1998; Loreau and Hector 2001). To beconsistent, however, this allocation rule requires that the


total soil volume be greater than the sum of the volumesof the depletion zones of all possible plant species, whichwas the case in our simulations (app. A). Mass conser-vation was also satisfied upon extinction of a plant speciesby adding the nutrient stocks in its residual biomass andin its resource depletion zone to the soil nutrient pool and,conversely, upon introduction of a plant species by sub-tracting these nutrient stocks from the soil nutrient pool(app. B in the online edition of the American Naturalist).

Community and Ecosystem Properties

We examined the relationship between the structure andfunctioning of interaction webs and the impact of non-trophic interactions by analyzing the effects of regionalspecies richness, nontrophic connectance, and maximalnontrophic magnitude (independent parameters of the re-gional species pool) on various properties of the local eco-systems, that is, local species richness, species richness ateach trophic level, proportions of the various types of spe-cies effects and species interactions, interaction web con-nectance, total biomass (total biomass of all species in thelocal ecosystem), biomass of each trophic level, and pro-duction of each trophic level. When we varied regionalspecies richness, we kept equal numbers of species (S) atall trophic levels in the regional species pool.

Food web connectance of the regional species pool wasdefined as the number of realized trophic interactions di-vided by the number of possible trophic interactions andwas kept constant in all simulations. Its value was closeto 1 because consumers 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 small probability of being exactly0). Note that our definition differs from the conventionaldefinition of food web connectance because feeding linkswithin trophic levels and between plants and carnivoreswere not allowed in our model.

Interaction modifications were assigned in two steps:(1) there was a certain probability that ; (2) ifm ( 0xyz

, the magnitude of the interaction modificationm ( 0xyz

was chosen in a uniform distribution. We call the prob-ability that the nontrophic connectance of them ( 0xyz

regional species pool, which is the probability that a speciesmodifies the trophic interaction between any two species:

nontrophic connectance p

number of realized interaction modifications. (3)

number 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 interactionsbetween species at adjacent trophic levels in the pool, thatis, .2 23S(S � S � S)

We also varied the range of values of the magnitude ofinteraction modification. Parameter was randomlymxyz

taken between a maximum value called “maximal non-trophic magnitude” and a symmetrical minimum equal tominus the maximal nontrophic magnitude. The maximalnontrophic magnitude was then allowed to take on dif-ferent values. Thus, we explored the impacts of nontrophicinteractions by manipulating both the nontrophic con-nectance and the maximal nontrophic magnitude of theregional species pool.

We analyzed the community and ecosystem propertiesin the local ecosystems that 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 Eig

(including trophic and nontrophic effects) of species g onspecies i was measured by the partial derivative of thegrowth rate of species i with respect to the biomassdX /dti

of species g :Xg

�(dX /dt)iE p . (4)ig�Xg

If , the effect of species g on species i is a facilitation.E 1 0ig

If , it is an inhibition. If and , theE ! 0 E 1 0 E 1 0ig ig gi

interaction between species i and g is a mutualism. Ifand , species i and g are in competition. IfE ! 0 E ! 0ig gi

and , the species interaction is an exploitationE 1 0 E ! 0ig gi

(including nontrophic forms of exploitation). If E p 0ig

and , it is a commensalism. If and ,E 1 0 E p 0 E ! 0gi ig gi

it is an amensalism. Density-mediated indirect interactions(Abrams 1995), such as exploitative nutrient competition,do not enter into the calculation of . The species in-Eig

teractions thus defined are phenomenological net inter-actions, just as they are defined traditionally in ecology. Aphenomenological rather than a mechanistic definitionwas necessary to account for the wide variety of trophicand nontrophic effects in a simple unified framework. Thisallowed us to investigate the effects of regional speciesrichness, nontrophic connectance, and maximal non-trophic magnitude on the prevalence and strength of spe-cies effects and species interactions.

Interaction web connectance of the local communitywas measured by the proportion of nonneutral speciesinteractions among all possible species interactions, thatis, by the proportion of species interactions in which at


Figure 2: Temporal changes in local species richness (A) and biomass (B) during the ecosystem assembly process. Biomass and local species richnessare shown for the community as a whole (black dotted lines), all plants (black solid lines), all herbivores (black dashed lines), and all carnivores(gray solid lines). The gray dashed line represents the nutrient mass in the soil nutrient pool. Regional species , nontrophicrichness p 45

, maximal nontrophic .connectance p 0.2 magnitude p 0.2

least one of the two net species effects ( or ) is non-E Eig gi

zero:

interaction web connectance p

number of nonneutral species interactions

local species richness (local species richness � 1)/2

p 1 � proportion of neutral species interactions.

(5)

We call “mean value of facilitation” the mean value of allpositive net species effects, and we call “mean value ofinhibition” the mean value of all negative net specieseffects.

We calculated the proportions, mean values, and stan-dard deviations of species effects and the proportions ofspecies interactions both in the community as a whole andwithin each trophic level. We then considered only inter-specific species effects and interactions without taking intoaccount the effect of a species on itself ( ).Eii

The production of each trophic level was measured byits nutrient inflow (nutrient mass per unit time):

S S

carnivore production p q a m H C ,�� C C H C H j ii j i jip1 jp1

S S

herbivore production p q a m P H , (6)�� H H P H P j ii j i jip1 jp1

S

primary production p a m L P .� P L P L i ii i i iip1

We measured all these community and ecosystem prop-erties during the course of community assembly. After atransition phase, however, the ecosystem systematicallyreached a quasi-stationary regime (fig. 2), that is, a phasewhere aggregated variables such as local species richnessand biomass showed small variations around a constanttemporal mean value. Therefore, we compared the com-munity and ecosystem properties in the quasi-stationaryregime by calculating the temporal mean during the last


Figure 3: Local species richness in the quasi-stationary regime as a function of regional species richness in food webs (A; nontrophic, maximal nontrophic ) and in interaction webs (B; nontrophic , maximal nontrophicconnectance p 0 magnitude p 0 connectance p 0.2). We present the mean and standard deviation for local species richness (filled circles), plant species richness (unfilled circles),magnitude p 0.2

herbivore species richness (triangles), and carnivore species richness (squares).

10% of the simulation iterations. For each value of regionalspecies richness, nontrophic connectance, or maximalnontrophic magnitude, we performed eight simulationswith different random compositions of the regional speciespool, as is often done in experiments (Hooper andVitousek 1997; Hector et al. 1999; Knops et al. 1999; Til-man et al. 2001), and we calculated the mean and standarddeviation of the measured properties for these eight rep-licates. We present below figures for a relatively high re-gional species richness (45 species, i.e., 15 species pertrophic level), but the results are qualitatively similar what-ever the number of species. In the figures below that de-scribe the effects of regional species richness, both non-trophic connectance and maximal nontrophic magnitudewere set to 0.2.

Results

Local species richness and species richness at each trophiclevel in the quasi-stationary regime increased with regionalspecies richness in both food webs and interaction webs(fig. 3A, 3B). Therefore, the results obtained as functionsof regional species richness or local species richness werevery similar.

Impact of Nontrophic Interactions and Species Richness onthe Prevalence and Strength of Species Interactions

Diversity of Species Interactions in Interaction Webs. Non-trophic connectance has an important effect not only onecosystem properties but also on the nature of speciesinteractions. When nontrophic connectance increased, theproportion of 0 species effects decreased, while the pro-

portions of facilitation and inhibition increased up to aplateau (fig. 4A). In a food web, the only direct speciesinteraction is exploitation between trophic levels (fig. 4B,when nontrophic , and fig. 4F). In anconnectance p 0interaction web, the proportion of neutral interactions de-creased to 0 as nontrophic connectance increased, andhence interaction web connectance increased to 100%. Theproportions of commensalism and amensalism first in-creased and then decreased to 0 as nontrophic connectanceincreased, while the proportions of mutualism, competi-tion, and exploitation increased as nontrophic connectanceincreased (fig. 4B). Thus, interaction modifications creatednontrophic interactions between species (such as mutu-alism or competition).

Impact of Interaction Modifications on the Prevalence andStrength of Species Effects. As either nontrophic connect-ance (fig. 4A) or maximal nontrophic magnitude (fig. 4E)increased, the mean value of facilitation increased and themean value of inhibition decreased; that is, the mean ab-solute values of species effects increased, and the standarddeviation of species effects also increased. As maximal non-trophic magnitude increased, the proportion of 0 specieseffects first fluctuated and eventually increased (fig. 4E).Thus, both nontrophic connectance and maximal non-trophic magnitude had an impact on the prevalence,strength, and variability of species effects.

The increase in the mean value of species effects withnontrophic connectance is a consequence of the assump-tion that nontrophic effects act multiplicatively on con-sumption rates. Since the magnitude of interactionmodification is randomly taken from a uniform dis-mxyz

tribution, it has the same probability of being positive or

Figure 4: Proportions and strength of species effects (A, C, E) and proportions of species interactions (B, D, F) in the community as a whole inthe quasi-stationary regime as functions of nontrophic connectance (A, B; regional species , maximal nontrophic )richness p 45 magnitude p 0.2and regional species richness in interaction webs (C, D; nontrophic , maximal nontrophic ). E shows theconnectance p 0.2 magnitude p 0.2proportions and strength of species effects as functions of maximal nontrophic magnitude (regional species , nontrophicrichness p 45

). F shows the proportions of species interactions as functions of regional species richness in food webs (nontrophicconnectance p 0.2, maximal nontrophic ). A, C, and E show the mean and standard deviation for the proportions of neutral effectsconnectance p 0 magnitude p 0

(gray diamonds, gray lines, #100), facilitation (filled circles, #100) and inhibition (filled triangles, #100), the mean and standard deviation for themean value of facilitation (unfilled circles, #100), the mean value of inhibition (unfilled triangles, #100), and the standard deviation of specieseffects (unfilled squares, #10). B, D, and F show the mean and standard deviation for the proportions of mutualism (filled circles, #100), competition(filled triangles, #100), exploitation (filled squares, #100), commensalism (unfilled circles, #100), amensalism (unfilled triangles, #100), and neutralinteractions (gray diamonds, gray lines, #100). Dashed lines (gray-filled outlined diamonds, #100) represent interaction web connectance.


negative; thus, the potential consumption rate that it mod-ifies has the same probability of being multiplied by afactor or divided by the same factor (eq. [2]). On the otherhand, the net species effect of a species g on anotherEig

species i is proportional to the magnitude of the realizedconsumption rates, aijmij, that are affected by species g. Thedifference between the multiplicative effect of interactionmodifications on the magnitude of realized consumptionrates and the additive effect of realized consumption rateson the net species effect explains why, on average, non-Eig

trophic interactions tend to increase net species effects. Onaverage, 50% of the will be in the range 0–1 and 50%mxy

will be in the range 1–�. Therefore, with increasing non-trophic connectance, the arithmetic mean of the realizedconsumption rates will increasingly exceed , lead-a m axy xy xy

ing to an increasing average strength of both inhibitionand facilitation.

This property might be viewed as a limitation of themathematical formulation of our model, which involvesboth multiplicative and additive effects while there is nodistribution of numbers in which both their product andtheir sum are equal to 1. But it might be realistic, becausebiological rates do have a lower bound of 0 and no upperbound, so that nontrophic interactions may be expectedto have the effects predicted by our model. Ultimately, thisissue will have to be resolved using empirical data onnontrophic effects in natural ecosystems, but these dataare currently sorely lacking.

The Interaction Web Connectance Increases with SpeciesRichness. As regional species richness (and hence alsolocal species richness) increased, the proportion of 0 spe-cies effects decreased, while the proportions of facilitationand inhibition increased (fig. 4C). Accordingly, the pro-portion of neutral species interactions decreased to verylow values (fig. 4D), whereas the proportions of com-mensalism and amensalism first increased and then de-creased, the proportions of mutualism and competitionincreased, and the proportion of exploitation was un-changed. Thus, interaction web connectance increasedwith species richness to nearly 100%: the more numerousspecies are, the more they interact with other species.

In food webs, the proportion of neutral species inter-actions increased with regional species richness (fig. 4F)whereas the proportion of exploitation decreased. Thus,the increase in interaction web connectance with speciesrichness in interaction webs (fig. 4D) is due to the presenceof nontrophic links between species.

The increase in interaction web connectance with spe-cies richness is a general property of interaction webs thatcan be explained intuitively as follows: as regional speciesrichness increases, the number of trophic links of a givenspecies increases (as long as consumers are not strict spe-

cialists), which increases the probability for this species tohave at least one trophic link modified by any other speciesin the web. The fact that interaction web connectancetends to 100%, however, is due to the assumption thatconsumers are generalists in our model. Other food webconfigurations may lead to smaller upper limits.

The Strength of Species Effects Decreases with Species Rich-ness. As regional species richness increased (fig. 4C), themean value of facilitation decreased and the mean valueof inhibition increased; that is, the mean absolute valuesof species effects decreased. Thus, species effects tendedto be denser, that is, proportionally more numerous, andweaker as regional species richness increased in interactionwebs (fig. 4C). Weak average interaction strength probablybuffers the destabilizing effects of interaction web con-nectance and species richness. Note that average interac-tion strength also decreased with regional species richnessin food webs (the mean absolute value of inhibition de-creased; results not shown).

Species Effects and Species Interactions within the VariousTrophic Levels. Patterns within the various trophic levelswere very similar to those reported in figure 4 for thecommunity as a whole. The only difference concerned theprevalence of facilitation and inhibition, and hence of in-terspecific competition and cooperation, among plants asnontrophic connectance increased (fig. C1 in the onlineedition of the American Naturalist). In plants, the pro-portion of inhibition increased more rapidly than that offacilitation, whereas in the community as a whole (fig. 4A)or in other trophic levels (results not shown), the pro-portions of inhibition and facilitation varied in the sameway. In the same manner, the proportion of interspecificcompetition increased more rapidly than that of mutu-alism in plants, whereas these proportions varied in thesame way in other trophic levels or in the whole com-munity (fig. 4B).

Impacts of Nontrophic Interactions on Biodiversityand Ecosystem Properties

Nontrophic Interactions and Ecosystem Properties. Non-trophic connectance and maximal nontrophic magnitudehad an important effect on ecosystem properties. Localspecies richness decreased as either nontrophic connect-ance (fig. 5A) or maximal nontrophic magnitude (fig. 5B)increased, except at a low level of nontrophic parameters.Plant species richness decreased as either nontrophic con-nectance (fig. 5A) or maximal nontrophic magnitude (fig.5B) increased. In contrast, consumer species richness wasless affected by variations in nontrophic connectance; it


Figure 5: Local species richness, biomass, and production in the quasi-stationary regime as functions of nontrophic connectance (A, C, E; regionalspecies , maximal nontrophic ) and maximal nontrophic magnitude (B, D, F; regional species , non-richness p 45 magnitude p 0.2 richness p 45trophic ). We present the mean and standard deviation for total (filled circles) local species richness and biomass and plant (unfilledconnectance p 0.2circles), herbivore (triangles), and carnivore (squares) local species richness, biomass, and production (per unit time). Dotted lines represent thenutrient mass in the soil (diamonds).

decreased as maximal nontrophic magnitude increased,except at a low level of maximal nontrophic magnitude.

The biomass and production of each trophic level de-creased sharply as either nontrophic connectance (fig. 5C,5E) or maximal nontrophic magnitude (fig. 5D, 5F) in-creased. Interaction webs with high levels of either non-trophic connectance or maximal nontrophic magnitudewere systems in which the biological processes of pro-

duction and recycling were very low and inorganic flowsprevailed. Such high levels of nontrophic connectance andmagnitude are presumably absent in natural ecosystems.

The Impacts of Nontrophic Interactions on Ecosystem Pro-cesses Are Mediated by Changes in Realized ConsumptionRates and Species Interactions. The decrease in biomassand production at all trophic levels as nontrophic con-


Figure 6: Biomass and production in the quasi-stationary regime as functions of regional species richness in food webs (A, C; nontrophic, maximal nontrophic ) and in interaction webs (B, D; nontrophic , maximal nontrophicconnectance p 0 magnitude p 0 connectance p 0.2). We present the mean and standard deviation for total biomass (filled circles), total plant (unfilled circles), total herbivore (triangles),magnitude p 0.2

and total carnivore (squares) biomass and production (per unit time). Dotted lines represent the nutrient mass in the soil (diamonds).

nectance or maximal nontrophic magnitude increases canbe explained by the impacts of nontrophic interactions onrealized consumption rates and species interactions (fig.4). First, mean realized consumption rates increase withnontrophic connectance, as mentioned above, which con-tributes to a decrease in the biomass and production atthe next lower trophic level. These declines in the biomassand production of lower trophic levels cascade up the foodweb and lead indirectly to decreased carnivore biomassand production. Increasing nontrophic connectance alsoleads to more intense competition between consumers fortheir resource and a smaller resource-use complementarity,which contributes to a decrease in herbivore and carnivorebiomass and production.

Second, the proportions of inhibition and competitionincrease more than those of facilitation and mutualism inplants when nontrophic connectance increases. Thus, non-trophic interactions tend to make competition betweenplant species stronger, which may also partly explain thedecrease in primary production and plant biomass andhence indirectly in consumer production and biomass.

The Biodiversity–Ecosystem Functioning Relationshipsin Food Webs and Interaction Webs

Effects of Species Richness on Ecosystem Processes in FoodWebs and Interaction Webs. In both food webs and in-teraction webs, total biomass increased with regional spe-cies richness and hence with local species richness (fig. 6A,6B). Plant and carnivore biomasses increased in parallel,which suggests a bottom-up control of plants on carni-vores. In contrast, herbivore biomass was less affected byspecies richness, which suggests a top-down control ofcarnivores on herbivores. The soil nutrient concentrationdecreased as species richness increased, which shows abetter exploitation of the limiting nutrient by plants.

Production at all trophic levels increased with regionalspecies richness (fig. 6C, 6D). Primary production in-creased with species richness because of the better ex-ploitation of the limiting nutrient by plants; as a result,the increase in plant biomass was much higher than thedecrease in the soil nutrient stock, and primary productionincreased (eq. [6]; fig. 6A, 6B). Herbivore production de-pends on plant biomass and herbivore biomass (eq. [6]),


Figure 7: Relationships between local species richness and total biomassdriven by variations of different properties of the regional species pool:regional species richness in food webs (filled circles; nontrophic

, maximal nontrophic ), regionalconnectance p 0 magnitude p 0species richness in interaction webs (unfilled circles; nontrophic

, maximal nontrophic ), nontrophicconnectance p 0.2 magnitude p 0.2connectance in interaction webs (triangles; regional species richness p

, maximal nontrophic ), and maximal nontrophic45 magnitude p 0.2magnitude in interaction webs (squares; regional species ,richness p 45nontrophic ).connectance p 0.2

and herbivore biomass was top-down controlled; there-fore, herbivore production showed the same pattern asplant biomass (cf. fig. 6A, 6C and 6B, 6D). In the samemanner, carnivore production depends on herbivore andcarnivore biomasses (eq. [6]), but herbivore biomass wastop-down controlled; therefore, carnivore production fol-lowed carnivore biomass. Production (especially in plantsand herbivores) was generally high compared with theinorganic nutrient input, I. This results from the highrecycling efficiency of ecosystems in the quasi-stationaryregime, after a long assembly process. This high recyclingefficiency also explains why the amount of nutrient in thesoil often remained relatively high.

Differences between food webs and interaction webshighlight the role of nontrophic interactions in ecosystems.The increase in biomass and production with regional (fig.6) and local (fig. 7) species richness was less rapid ininteraction webs than in food webs. Differences betweenfood webs and interaction webs were greater at higherlevels of regional species richness (fig. 6A, 6B). The positiveeffect of nontrophic interactions on the average realizedconsumption rate as species richness increases explainswhy nontrophic effects are more important in species-richecosystems than in species-poor ecosystems in our model.Higher realized consumption rates make species more ef-ficient but also more competitive.

Biodiversity–Ecosystem Functioning Relationships. Westudied the relationship between total biomass and localspecies richness at the quasi-stationary regime thatemerged from variations in one of the parameters of theregional species pool, that is, regional species richness,nontrophic connectance, or maximal nontrophic magni-tude (fig. 7). Whatever the parameter driving variationsin local species richness, there was a positive relationshipbetween total biomass and local species richness. However,the shape of this relationship differed. The relationshipwas roughly linear in both food webs and interaction webswhen regional species richness varied, but it was nonlinearand concave-up when nontrophic connectance or maximalnontrophic magnitude varied in interaction webs. Thisconcave-up relationship between local species richness andtotal biomass is explained by the greater effect of non-trophic interactions on biomass than on species richness(fig. 5). Thus, our model predicts a positive relationshipbetween species richness and biomass in naturally assem-bled ecosystems, but with a strong impact of nontrophicinteractions on the shape of the diversity-biomass rela-tionship. Specifically, nontrophic interactions are expectedto decrease the magnitude of biomass and production, butchanges in their frequency or strength are expected toincrease the dependency of biomass and production onlocal species richness.

Discussion

Our theoretical model shows that species diversity andnontrophic interactions affect strongly and nonintuitivelythe community and ecosystem properties in communitiesthat have assembled through repeated colonization events.Our main results can be summarized as follows.

First, increasing regional species richness leads to in-creased species richness, biomass, production, and inter-action web connectance but decreased average interactionstrength in local communities. Thus, species-rich inter-action webs are expected to be more productive and moreconnected but to have weaker species interactions on av-erage. Their lower interaction strength is probably whatallows them to maintain a high diversity and connectance,in agreement with previous theory (May 1973; Kokkoriset al. 1999, 2002). Their high diversity in turn allows themto use the limiting nutrient more efficiently and hencehave a higher production and biomass, as predicted byexisting theory (Tilman et al. 1997; Loreau 1998, 2000;Tilman 1999).

Second, increasing the frequency and magnitude of non-trophic interactions in the regional species pool leads todecreased local species richness, biomass, and production.These counterintuitive effects result from the fact that non-trophic connectance and maximal nontrophic magnitudecontribute to an increase in the magnitude of trophic con-


sumption fluxes, and hence the strength of species inter-actions, on average. Increased average interaction strengthallows fewer species to coexist and imposes a higher mor-tality on lower trophic levels, which eventually deterioratesthe functioning of the whole ecosystem.

Third, as a consequence, interaction webs that includetrophic and nontrophic interactions are expected to havea lower local richness, biomass, and production than foodwebs that include only trophic interactions, all else (inparticular, regional species richness) being equal. A pos-itive diversity-biomass relationship emerges from the as-sembly of both food webs and interactions webs, but non-trophic interactions are expected to affect the shape of thisrelationship.

Thus, our results emphasize the need to take into ac-count nontrophic interactions in theoretical ecology. Oursimple, general model allows all types of species interac-tions to be incorporated by adding interaction modifica-tions to a food web. This mass-balance constrained modelof an interaction web provides a useful tool for studyingthe impact of diversity on the functional properties ofcomplex ecosystems with more realism.

The Biodiversity-Ecosystem FunctioningRelationship in Interaction Webs

Total biomass increases with regional species richness, andhence also with local species richness, in both food websand interaction webs. In both types of web, carnivore bio-mass is bottom-up controlled by plants, herbivore biomassis top-down controlled by carnivores, and total biomassdepends mostly on plant biomass. These control mecha-nisms are the same as in classical food webs, and our resultsconcerning food webs are in qualitative agreement withprevious theoretical studies (Thebault and Loreau 2003;Ives et al. 2005). Thebault and Loreau (2003), however,showed that in a two-level food web with generalist her-bivores, plant biomass and herbivore biomass increasenonlinearly with regional species richness and can evendecrease at a high level of species richness, whereas in ourfood web model, biomass does not decrease at a high levelof species richness. This difference is probably explainedby the presence of a third trophic level in our work andby the fact that, contrary to recent theoretical (Thebaultand Loreau 2003; Ives et al. 2005) and experimental(Downing and Leibold 2002; Duffy et al. 2003) studies inwhich species richness is controlled, our work provides adynamic approach to biodiversity and ecosystem func-tioning in which both factors result from an assemblyprocess. Our model predicts that biomass and productionare generally lower in interaction webs than in food webs.Nontrophic interactions are likely to generate strong con-straints on species coexistence. The species present in the

community have higher realized consumption rates onaverage, which makes them more efficient but also morecompetitive. Therefore, the probability of observing ex-treme resource exploitation and negative effects on eco-system properties is higher.

Species Diversity, Species Interactions,and Ecosystem Processes

Our work shows that species interactions depend on spe-cies richness: both the strength and the prevalence of theseinteractions are diversity dependent. In particular, themean absolute values of interspecific facilitation and in-hibition effects are expected to decrease with increasingspecies richness. These results agree with existing theory(Kokkoris et al. 1999, 2002). The analysis of natural foodwebs (Neutel et al. 2002) suggests that natural ecosystemsare characterized by a majority of weak interactions anda minority of strong interactions.

Our model also predicts that interaction web connect-ance increases with species richness and that the propor-tions of the various types of species effects and speciesinteractions depend on species richness. A higher speciesdiversity increases the probability for each species to in-teract with any other species. Although the relationshipbetween trophic connectance and species richness has beenwell studied in food webs (Martinez 1992; Montoya andSole 2003), we lack knowledge concerning nontrophicconnectance and interaction web connectance in ecosys-tems.

Species diversity is known to affect ecosystem function-ing through functional complementarity between speciesthat use different resources (Tilman et al. 1997; Loreau1998; Loreau and Hector 2001). Our model further sug-gests that it can affect ecosystem functioning in more sub-tle ways by changing the nature, prevalence, and strengthof species interactions. For instance, it can enhance eco-system processes by increasing the probability of facilita-tion (Cardinale et al. 2002). Recent experiments haveshown that a form of facilitation, whereby a species en-hances the access of other species to resources throughbiophysical modifications, affects the productivity and thediversity-productivity relationship in bryophyte commu-nities (Mulder et al. 2001; Rixen and Mulder 2005). Thus,species interactions, in particular nontrophic interactions,should be given more attention in studying the relation-ship between biodiversity and ecosystem properties.

Incorporating Nontrophic Interactionsinto Theoretical Ecology

Our work provides a consistent ecosystem model that in-corporates nontrophic interactions in the form of inter-


action modifications. Although interaction modificationswere regarded by Wootton (1994) as a class of indirecteffects, they may be viewed as either direct or indirecttrait-mediated interactions (Abrams 1995), depending onthe context. The interaction may be direct if, for instance,the interaction modifier directly affects the behavior of thetwo species whose trophic interaction is affected. But itmay also be indirect if, for instance, the interaction mod-ification occurs through habitat modification, which inturn affects the ability of the predator to detect or catchits prey. Also, the direct or indirect nature of trait-mediatedinteractions is not necessarily reflected in a model’s struc-ture (Abrams 1995). Therefore, our measure of the netspecies effect, , of species g on species i includes bothEig

trophic and nontrophic direct effects as well as, potentially,trait-mediated indirect effects. The ambiguity of currentterminology suggests that a reexamination of the concepts,definitions, and measures of direct and indirect effectsmight be useful.

Ecosystem engineers (Jones et al. 1994) are strong driv-ers of interaction modifications: by modifying their phys-ical environment, they create many nontrophic species in-teractions. Autogenic or allogenic engineers can modulateresource flows or abiotic parameters that influences re-source flows. Therefore, our modeling framework couldbe applied to the study of ecosystem engineers throughspecific nontrophic modifications of trophic interactionsor through modifications of abiotic parameters such asthose that govern the input, recycling, and loss of nutri-ents. Experimental studies show that ecosystem engineer-ing can induce changes in community structural prop-erties, such as species richness (Zhu et al. 2006), speciescomposition (Badano et al. 2006), and species interactions(Collen and Gibson 2001), and ecosystem functional pro-cesses such as primary production (Zhu et al. 2006). Ourmodel shows that interaction modifications of all kindscan profoundly affect ecosystem properties such as bio-mass and production at all trophic levels.

We view our interaction web model as a promising toolfor merging the community and ecosystem perspectivesin theoretical ecology. By incorporating nontrophic inter-actions in the form of modifications of trophic interac-tions, our model describes material flows in a consistentway and hence allows analysis of ecosystem properties. Atthe same time, it is a flexible and dynamic model thatallows all kinds of species interactions to occur, and hence,it allows analysis of community properties such as speciesdiversity and the connectance, prevalence, and strength ofspecies interactions.

Model Limitations

Despite its strengths and generality, our model also haslimitations. We represented nontrophic interactions in the

form of modifications of trophic interactions, but it wouldalso be possible to introduce them in the form of modi-fications of nontrophic parameters such as intrinsic deathrates and recycling rates. We assumed that nontrophic ef-fects act multiplicatively on potential consumption ratesbecause this led to the simplest functional form for in-teraction modifications that respects the direction of con-sumption fluxes between trophic levels. This assumptionmakes sense mathematically because biological rates havea lower bound of 0 and no upper bound; nontrophicmodifications of these rates may have asymmetric effects,depending on whether they are positive or negative. Butthis assumption indirectly drives many of the observedeffects of nontrophic interactions in our model, and em-pirical data to assess its validity are unfortunately sorelylacking. We also assumed that the various modificationsof a given trophic interaction by different species are ad-ditive, but interaction modifications might interfere witheach other and generate nonadditive effects. For instance,there could be a hierarchy of effects (one effect dominatesover the others) or a synergy of effects (the effect of onespecies can only be expressed in, or is modified by, thepresence of another species). Our linear approximation,however, is in line with the simplicity required for a generalecosystem model.

We assumed that there is a single limiting nutrient andthat the stoichiometric composition of each species is con-stant, although the stoichiometric composition of plantsis known to depend on factors such as environmentalconditions and the presence of consumers. This simpli-fication was made to avoid unnecessary complications ina 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-specific re-source depletion zone. This assumption was meant to favorplant species resource-use complementarity and coexis-tence to some extent, although strong nontrophic inter-actions led to species-poor communities despite this as-sumption. We expect our results to be qualitatively robustto relaxation of this assumption, as suggested by prelim-inary simulation results. A decomposer compartmentcould be added, but it is unlikely to change the resultsbecause we have already included nutrient cycling in ourmodel without detailing its mechanisms. Lastly, our cur-rent model considers an interaction web with three trophiclevels. It would definitely be interesting to model an in-teraction web with a more realistic and flexible trophicstructure by allowing consumers to feed on several trophiclevels and blurring the separation between distinct trophiclevels.

Our theoretical study suggests several hypotheses thatdeserve to be tested experimentally. It would be useful tostudy the influence of species richness on species inter-


actions: does the strength of species effects (facilitation,inhibition) depend on species richness in experimentalecosystems? Does their connectance increase with speciesrichness? It would also be useful to test the impacts ofnontrophic interactions on ecosystem processes and theirmechanisms: what is the influence of the connectance ormagnitude of nontrophic interactions on biomass and pro-duction at various trophic levels? Do nontrophic inter-actions generally increase resource consumption, possiblyleading to extreme resource exploitation?

Conclusion

We have developed a model of a full interaction web thatincludes both trophic and nontrophic species interactionsand that respects the principle of mass conservation. Thismodel shows the important role of species interactions,especially nontrophic interactions, in community and eco-system properties and in the relationship between biodi-versity and ecosystem functioning. The diversity-biomasspatterns obtained with our interaction web model are notstrikingly different from those shown in recent theoreticalstudies of food webs, but the mechanisms are far morecomplex. In particular, our model predicts that the nature,prevalence, and strength of species interactions changewith regional and local species richness, which makesthe mechanisms of the biodiversity–ecosystem functioningrelationship more complex in an interaction web. Spe-cies interactions, and especially nontrophic interactions,should be given more attention to improve understandingand predictions of the ecological consequences of biodi-versity loss.

Acknowledgments

We thank C. Fontaine, S. Kefi, N. Loeuille, E. Thebault,and four anonymous reviewers for constructive commentson the manuscript.

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Associate Editor: Sebastian DiehlEditor: Donald L. DeAngelis

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� 2007 by The University of Chicago. All rights reserved.DOI: 10.1086/523945

Appendix A from A. Goudard and M. Loreau, “NontrophicInteractions, Biodiversity, and Ecosystem Functioning: An InteractionWeb Model”(Am. Nat., vol. 171, no. 1, p. 91)

Table A1Parameter values

Parameter Values

Numerical integration:

Time step (Runge-Kutta of order 4) .01

Simulation duration 1,000 iterations, i.e., 100,000 numerical integrations

Introduction period (p) 100 time steps

Introduction biomass .01

Extinction biomass threshold .005

Regional species pool (randomly taken from a uniform distribution):

(death rate)ux , per unit timeGx � (1, … ,n) .1 ≤ u ≤ .5x

(nonrecycled proportion of dead organic matter)l x ,Gx � (1, … ,n) .1 ≤ l ≤ .3x

axy Potential consumption rate of speciesy by speciesx per time and mass unit

Carnivore potential consumption rate , per time and2G(x, y) � (1, … ,S) 0 ≤ a ≤ 0.01xy

mass unit

Herbivore potential consumption rate , per time and2G(x, y) � (1, … ,S) 0 ≤ a ≤ 0.01xy

mass unit

Nutrient uptake rate by plants , per time and massGi � (1, … ,S) 0 ≤ a ≤ 0.5L Pi i

unit

(carnivore conversion efficiency)qC .15 (15%)

(herbivore conversion efficiency)qH .15 (15%)

(magnitude of the nontrophic modification of the trophic interactionmxyz

between species and , by species )x y z ,3G(x, y, z) � (1, … ,S) �maximal nontrophicmagnitude≤ m ≤ �maximal nontrophicxyz

magnitude

mxy Nontrophic coefficient of speciesy on speciesx

a mxy xy Realized consumption rate of speciesy by speciesx per time and mass unit

Soil:

Vsoil Total volume of soil:V p 100soil

(total volume of species-specific resource depletion zones of plant speciesi)Vi if plant speciesi is present in the localV p 1i

ecosystem, elseV p 0i

(volume of soil nutrient pool)VR

S

V � � Vsoil iip1

g (diffusion rate) 10 per unit time

I (soil nutrient input) 100 per unit time

lR (soil nutrient loss rate) .05 per time and soil nutrient mass unit

R0 (initial nutrient mass in soil nutrient pool [randomly taken]) 1≤ R0 ≤ 10

1


Appendix B from A. Goudard and M. Loreau, “NontrophicInteractions, Biodiversity, and Ecosystem Functioning: An InteractionWeb Model”(Am. Nat., vol. 171, no. 1, p. 91)

Program Algorithm1. Choice of complexity parameters of the regional species pool: regional species richness, nontrophic

connectance, and maximal nontrophic magnitude.2. Creation of the regional species pool. The regional species pool consists of equal numbers of plant,

herbivore, and carnivore species, without omnivory. The parameters of the pool species are randomly taken froma uniform distribution within appropriate intervals (app. A): death rate , nonrecycled proportion of nutrientu lx x

coming from species , potential consumption rate , and magnitude of interaction modification . The initialx a mxy xyz

nutrient mass in the soil nutrient pool is also randomly taken from a uniform distribution.At each introduction event (3–5):3. Community and ecosystem properties are calculated and recorded (proportions, strength and variability of

species effects, proportions of species interactions, local species richness, species richness per trophic level, totalbiomass, biomass per trophic level, production at each trophic level, and mean consumption per species).

4. A species is randomly taken from the pool of those that are not already present in the community. Aspecies can thus be introduced, become extinct, and be reintroduced.

5. The species is introduced into the community. The initial biomass of the introduced species is subtractedfrom the soil nutrient pool to respect the principle of mass conservation. When a plant species is introduced, thevolume of its species-specific resource depletion zones is created, with a concentration equal to the soil nutrientpool concentration, and subtracted from the volume of the soil nutrient pool, and the nutrient mass of thisspecies-specific resource depletion zone is subtracted from the soil nutrient pool.

At each integration time step (6–7):6. Numerical integration of the dynamic equations (1) and (2) is performed with a Runge-Kutta method of

order 4.7. Species whose biomass is smaller than the extinction biomass threshold are removed. Their biomass is

added to the soil nutrient pool to respect the principle of mass conservation. When a plant species becomesextinct, the volume and the nutrient mass of its species-specific resource depletion zone are set to 0 and added tothe volume of the soil nutrient pool.

8. Repeat steps 6–7 for numerical integrations.1009. Repeat steps 3–8 for introduction events.100,000/100 p 1,00010. Repeat steps 2–9 for eight replicates.11. Repeat steps 1–10 for each value of the studied parameter (regional species richness, nontrophic

connectance, or maximal nontrophic magnitude).

1


Appendix C from A. Goudard and M. Loreau, “NontrophicInteractions, Biodiversity, and Ecosystem Functioning: An InteractionWeb Model”(Am. Nat., vol. 171, no. 1, p. 91)

Figure C1: Frequency and strength of species effects (A) and species interactions (B) within the plant trophiclevel in the quasi-stationary regime as functions of nontrophic connectance (regional species ,richness p 45maximal nontrophic ). A, Mean and standard deviation for the proportions of neutral effectsmagnitude p 0.2(gray diamonds, gray lines, #100), facilitation (filled circles, #100) and inhibition (filled triangles, #100), themean and standard deviation for the mean value of facilitation (unfilled circles, #100), the mean value ofinhibition (unfilled triangles, #100), and the standard deviation of species effects (unfilled squares, #10). B,Mean and standard deviation for the proportions of mutualism (filled circles, #100), competition (filledtriangles, #100), exploitation (filled squares, #100), commensalism (unfilled circles, #100), amensalism(unfilled triangles, #100), and neutral interactions (gray diamonds, gray lines, #100). Dashed line (gray-filledoutlined diamonds, #100) represents interaction web connectance.

nontrophic interactions, biodiversity, and ecosystem ...€¦ · functioning relationship. thus,...

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