e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378
avai lab le at www.sc iencedi rec t .com
journa l homepage: www.e lsev ier .com/ locate /eco lmodel
Topological keystone species in ecological interactionnetworks: Considering link quality and non-trophic effects
Vera Vasasa, Ferenc Jordanb,c,∗
a Department of Plant Taxonomy and Ecology, Eotvos University, Pazmany Peter s. 1/c, H-1117 Budapest, Hungaryb Institute of Ecology and Botany, Hungarian Academy of Sciences, Alkotmany u. 2-4, H-2163 Vacratot, Hungaryc Collegium Budapest, Institute for Advanced Study, Szentharomsag u. 2, H-1014 Budapest, Hungary
a r t i c l e i n f o
Article history:
Received 12 April 2005
Received in revised form 26 January
2006
Accepted 9 February 2006
Published on line 29 March 2006
Keywords:
Food web
Positive interactions
Keystone species
Interaction web
Network theory
a b s t r a c t
There is increasing evidence that non-trophic interspecific interactions play an at least as
important role in community dynamics as trophic relationships. More and more studies on
pollination, mutualism and facilitation are published but these effects are interpreted more
like alternative explanations than being synthesized with results of trophic analyses. Here,
we construct and analyze the interaction web of the well-studied Chesapeake Bay mesoha-
line ecosystem. By interaction web we mean a food web completed by a carefully selected
set of non-trophic links. We quantify the interaction structure of the web and the positional
importance of nodes by different network indices. We perform the suitable analyses for
different variants of the network: combinations of direction, sign and weights, as well as
considering also non-trophic links result in a set of webs of different information content.
We also create a semi-quantitative variant of the web, in which only the order of magnitude
of the mass flows are considered. The appropriate network indices for each web variant are
calculated and compared. Finally, however our paper is primarily of methodological nature,
we present some findings about the fish community of the Bay. We suggest that the mul-
tiple techniques presented here, adapted even from social network analysis, can help field
su
strength, sign and direction. The direction and sign structure
conservation efforts by
1. Introduction
The construction and analysis of food webs is a traditionalapproach to understanding the structure and functioning ofecological systems. Food webs depict the collection of pairwiseprey–predator interactions between species or their suitablydefined groups (Pimm, 1982, 1991; Pimm et al., 1991). Althoughthe information food web analysis provides is probably essen-tial, notorious methodological problems weaken the predic-tions appearing during all of the three basic phases of the
work (sampling during field data collection, right aggregationprocess during network construction, choosing sensible graphproperties during network analysis).
∗ Corresponding author.E-mail address: [email protected] (F. Jordan).
Apart of the methodological problems of how to constructa network showing who eats whom, a number of additionalissues are frequently claimed for: (1) trophic interactions arevery important but other pairwise (direct) non-trophic inter-specific interaction types are also of high importance (e.g.pollination: Memmott, 1999; mutualism: Bronstein, 2001; facil-itation: Turner, 1983; Callaway, 1995; see also Kareiva andBertness, 1997 and subsequent papers), (2) interactions couldbe characterized not only binarily (yes or no) but also by their
of a direct trophic interaction (a negative feeding effect intop–down direction and a positive food supply in bottom–updirection) is evident but still can be complemented by informa-
tion on the magnitude of energy flows (strength). Non-trophicinteractions are variable also in sign (e.g. the effects in bothdirections are positive in a mutualism) and direction (e.g. facil-itation is a one-way positive effect with no response). Thus,a complex interaction network, defined as a general exten-sion of a food web (but see Paine, 1980 for a different mean-ing), contains trophic and non-trophic, directed, signed andweighted effects between pairs of species. The combinationsof co-occurring pairwise effects give rise to indirect interac-tion modules (e.g. trophic cascade or apparent competition;see Menge, 1995, for a classification) embedded in community-wide interaction networks. Indirect chain effects do spreadin both bottom–up and top–down directions through trophiclinks and, as a result, may act also horizontally (Wootton, 1994;Menge, 1995; Abrams et al., 1996). If non-trophic interactionsare also considered, the network may also have direct hori-zontal links.
The role non-trophic interactions have in organizing acommunity has traditionally been considered more local,and their analysis mostly focused on species pairs. The cur-rent need for taking also non-trophic effects into accountwhile thinking within the network perspective is parallelwith the recognition of their less local nature (e.g. diffusemutualism; Jordano, 1987; Bronstein, 2001). We already haveplant–pollinator (Jordano, 1987) and competitive networks,even with weighted interactions (Paine, 1984). However, toour knowledge, there is no community-wide interaction net-work showing a variety of both trophic and non-trophic linksbetween species. An evident problem with weighting such anetwork is that there is no common currency, i.e. it is noteasy to define the common denominator of material trans-fers and a facilitation effect (it is no problem in case of binarywebs). Another problem is that mass-balance can comfort-ably be assumed in case of trophic flow networks but thereare lots of problems with assuming “interaction balance”. Thismight be one reason for studying more intensively the trophic-dynamic aspect (Lindeman, 1942) in the past. We also haveto note the problem of aggregation. It is frequently the casethat trophic effects are less specific, so the aggregation pro-cess seriously affects how the two kinds of interactions willcomplement each other within a single graph (also, differ-ent interactions might suggest different aggregation proce-dures). Nevertheless, the dual nature of interaction networkscould be helpful in better understanding, for example, thecommunity-wide answers to stress and disturbance (Bertnessand Shumway, 1993). An ultimate question to be addressed iswhether to take into account non-trophic effects or to mea-sure interaction strengths in a trophic network, if we wantto optimize our efforts in improving a traditional, binary foodweb.
Both ecological research and conservation practice claimfor quantitative, a priori approaches to characterizing theimportance of different species in ecosystems. Keystonespecies have been defined variously and a number of stud-ies report on their roles but their objective description isstill immature. One of the very few quantitative approaches
is their characterization based on the position they occupywithin food webs: topological keystone species have beendefined as being in key positions in trophic interaction net-works (Jordan et al., 1999; Sole and Montoya, 2001; Montoya
1 9 6 ( 2 0 0 6 ) 365–378
and Sole, 2002; Jordan and Scheuring, 2002). New graph the-oretical techniques have also been suggested for their finercharacterization (Allesina and Bodini, 2004; Jordan et al., 2006).Here, we wish to extend these techniques to directed, signedand weighted interaction networks including also non-trophiclinks. We have to note that the “keystone” term is strictlyused only for in “importance/biomass” context (Power etal., 1996), thus we should not use the term in the strictestsense. The topological importance indices could be easilycombined with biomass data but, in this paper, our pri-mary task is to compare different variants of the same web(and outline the methodological background of this problem),and our intention is to keep everything else as simple aspossible.
Our primary concern is to develop the methodology of eco-logical network analysis, in other words, we are more inter-ested in how to analyze such an “ideal” network than in how toconstruct it. The aims of our present paper are: (1) to constructan interaction network as a combination of a published foodweb and a collection of published data on non-trophic effects,(2) to construct the variants of this web according to differ-ent but only sensible combinations of link direction, strengthand sign, as well as whether including non-trophic links, (3)to apply several graph theoretical indices for (3) mapping thedirect and indirect interaction structure of these variants, and(4) determining the topological keystone species in the net-works, and finally (5) to compare the network variants fromthe perspective of the fish community. We present resultsconcerning the organization of the studied community butemphasize that this is more like illustration, since our paperis primarily of methodological nature.
2. Data base
Energy flows between the trophic components of the Chesa-peake Bay mesohaline community are well known (Baird andUlanowicz, 1989; see Table 1 for carbon flow data and Table 2for trophic groups) and have been analyzed extensively (seeBaird et al., 1995 for nitrogen and Ulanowicz and Baird, 1999for limiting nutrients). Since the Bay is well studied, there is alot of information, even if more sporadic, about the function-ally important and typical non-trophic effects between speciesinhabiting the Bay. We have collected a large number of non-trophic interspecific effects and selected the seemingly mostcharacteristic and unambiguous links in order to complementour food web such that an interaction network can be con-structed (Fig. 1). Table 3 shows the origin of non-trophic linksconsidered in our study. Of course, our selection is subjective,but not intentionally biased, and we emphasize that the anal-ysis of this interaction web is basically an illustration of ourmethodological investigation.
3. Methods
3.1. Network construction
Our task is to complete a food web by considering also theseemingly most important non-trophic relationships between
e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378 367
Table 1 – Carbon flow dataa in mg C/m2/year in the food web of the Chesapeake Bay ecosystem (after Baird andUlanowicz, 1989)
From To Annual flow From To Annual flow From To Annual flow
a We used data balanced by DATBAL, see Ulanowicz and Puccia (1990
pecies, which we supposed to be comparable in strength ofheir effect with the trophic ones. Also, since our interest isrimarily of methodological nature, we intend to incorporates many interaction types as possible. Tables 1 and 2 show theriginal food web data, while Table 3 shows the non-trophicffects and Fig. 1 shows our complete interaction network.ote that the food web is a subgraph of the interaction net-ork.
We mention that before the non-trophic links were incor-orated, we slightly modified the original data base. Only inter-ctions between living components were considered, since
e are interested in the more biological aspect of commu-ity organization and not in mass-balance and the thermo-ynamics of the system. This is a deviation from the classical
Table 2 – Components of the carbon flow network (after Baird a
10 Sea nettle 22 Ba11 Other suspension feeders 23 Me12 Mya 24 Sh
a Nodes #34–36 are only considered for the TUUU* variant and are not showrespectively).
14
traditions of “ecosystems ecology” but sounds more logicalfrom a “community ecology” point of view. As far as we see,it is imperative to bridge the gap between these schools andour work is aimed to contribute to this. The major prob-lem with non-living compartments, from a community-levelinteraction network viewpoint is that the interaction between“detritus” or “DOC” and a particular species as highly asym-metrical in many senses, i.e. very different from a classical“prey and predator” situation. If centrality is to be quantified,the directed nature of a couple of links produces serious arti-facts. Also, each species and living group are linked to the
detritus, so considering these “living–non-living” links wouldcause a serious artifact in mapping the topology of the inter-action network in the case of the undirected variants (detritus
nd Ulanowicz, 1989)
sters 25 Croakerher polychaetes 26 Hog chokerreis 27 Spotcoma spp. 28 White perchiofauna 29 Catfishstacean deposit feeders 30 Blue fishe crab 31 Weak fishh larvae 32 Summer flounderwife and blue herring 33 Striped bass
y anchovy 34 DOCa
nhaden 35 Suspended POCa
ad 36 Sediment POCa
n in Fig. 1 (DOC and POC mean dissolved and particle organic carbon,
368 e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378
Fig. 1 – The interaction network of the Chesapeake Bay ecosystem. Graph nodes represent species or suitably definedfunctional groups (for decoding, see Table 2). The thin links compose the food web graph (slightly modified after Baird and
mpthe
Ulanowicz, 1989, see explanation in text and also Table 1) coexplanation in text and Table 3). The network was drawn by
will inevitably be the “keystone species”, later we will illustratethe effect of this). So, dissolved and particle organic carbon
pools (DOC and POC, respectively) are out of our interest, sincethese groups do not interact living groups in the same wayas living entities do (detritus does not hunt for its “prey”).The links between groups #1 and #2 and between #2 and
Table 3 – Non-trophic effects considered: their interaction type,
lemented by selected non-trophic links marked bold (seeNetDRAW subroutine of UCINET (Borgatti et al., 2002).
#3 were deleted for the above reasons and the link betweengroups #3 and #19 was deleted following the secondary data
base in Ulanowicz and Puccia (1990). However, we have nottaken into account non-living components of the system, themethods make it very easy to incorporate them. For example,environment-mediated indirect interactions (Wootton, 2002;
sign and source are given
To Reference
23 Burkholder et al. (1992), Burkholder (1999)6 Burkholder et al. (1992), Burkholder (1999)
20 Burkholder (1999)6 Burkholder (1999)
25 Burkholder (1999)6 Burkholder (1999)
32 Burkholder et al. (1992), Burkholder (1999)6 Burkholder et al. (1992), Burkholder (1999)
33 Burkholder et al. (1992), Burkholder (1999)6 Burkholder et al. (1992), Burkholder (1999)
11 Bahr and Lanier (1981)14 Bahr and Lanier (1981)
4 Bahr and Lanier (1981)18 Eggleston et al. (1998)19 Eggleston et al. (1998)11 Key et al. (1997)
e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378 369
Fig. 2 – Relationships between the combinations of link qualities providing network variants. The theoretically possible butnot reasonable or illogical combinations of directedness, signedness and weightedness are marked by small, normal lettersand discussed in text. The flowchart of our interest shows the studied variants in large, bold letters. Solid arrows shows mpoi e on
BBofttfi
saapasiiDUrilviwetbihU(f
teps of increasing complexity, while dashed arrows show interaction network. The variant marked by an asterisk is th
runo et al., 2003) can simply be considered as direct links.efore analyzing our interaction network, we assume thatur construction is “perfect”: the nodes of the graph are theunctionally sensible units of the community, the links ofhe graph are the functionally sensible interactions betweenhem, and everything had been measured perfectly in theeld.
Based on the information on the direction, sign andtrength of interactions, and based on whether we take intoccount also non-trophic links, 16 variants of the same inter-ction network can be logically constructed (additionally, werovide both quantitative and semi-quantitative weighting inspecial case, explained later). The least information is pre-
ented by the pure topological food web (coded as TUUU,.e. Trophic, Undirected, Unsigned, Unweighted), while morenformation-rich webs are coded like TDSW (i.e. Trophic,irected, Signed, Weighted) or IUUU (Interaction, Undirected,nsigned, Unweighted). The theoretically most information-
ich web (i.e. IDSW) is not presented, since at the moment it ismpossible to express the weights on trophic and non-trophicinks in the same units (e.g. a common currency for food pro-ision by carbon transfer, substrate formation as a facilitativenteraction and helping recruitment in a mutualism). Eightebs of the 17 variants have been analyzed; the others were
xcluded for various reasons (see next paragraph). Fig. 2 showshe relationships between the studied eight webs and alsoetween the non-studied combinations. We note that ecolog-
cal networks in one of the above combinations of link quality
ave been analyzed by Ulanowicz and Puccia (1990, TDSW),lanowicz (1995, IDSW), Montoya et al. (2003, IDSW), Levine
1980, IDSU) and Jordan et al. (2003a, TUUU and TUUW) but, asar as we see, this comparative methodological approach was
ssible routes for increasing the information content of thely web containing non-biotic groups (like DOC).
still needed for seeing more clearly how data quality modifiesour predictions.
According to Fig. 2, there are many network variants con-structed by different combinations of direction, sign andweight of links as well as the presence and absence of non-trophic links, and only some of these are analyzed. We brieflymention the reasons for these decisions. It is not logical toconsider the signs of undirected links (in TUSU, TUSW andIUSU). It could be possible only if all of the links have sym-metrical signs, i.e. +/+ and −/− links are exclusive. Thus, thecombination of “U” on the second place (undirected) and “S”is forbidden. If links are not directed, it is illogical to weightthem related to the carbon flows (in TUUW). (But see Jordan etal., 2003a, for weighting related to the interaction frequency.)A prey–predator link may have a weight, of course, in bothdirections, but not the same. So, “W” without “D” is excludedfrom the space of reasonable network variants. However, thereare studies on weighting non-trophic links (like expressing therelative strengths of competition or mutualism); it is not pos-sible to weight all the trophic and non-trophic links in thesame unit. For a successful example for a special case, seeUlanowicz (1995). Thus, the combinations of “I” and “W” mustbe excluded. Considering the sign of interactions is possible bythe IMPACTS software but it is based on mass-balance models(until non-trophic links are built in).
3.2. Network analysis
The network perspective on ecology is to say that in orderto understand the behavior of the components of ecologicalsystems (e.g. species), the graph properties of ecosystem net-works must be analyzed. In other words, interaction networks
i n g
370 e c o l o g i c a l m o d e l l
are built up and influenced by their species but also con-strain the behavior of the constituent species. The topologyof these networks shows the possibilities for indirect inter-actions (e.g. trophic cascade), the species occupying criticallyimportant positions within the graph (i.e. topological keystonespecies) and the constraints the web itself puts on populationdynamics. Various graph indices characterize local, mesoscaleand global system properties. We note that because of similarinterest between ecological and social network analysis, themethodology of network analysis in these fields have com-mon roots in the past (Harary, 1961) and similar future interest(Jordan et al., 1999; McMahon et al., 2001; Luczkovich et al.,2003).
We use the following indices for the analysis of the Chesa-peake Bay interaction networks, the suitable ones for eachnetwork variant. These indices characterize and quantify theinteraction structure of the networks, and particularly, thepositional importance of graph nodes (representing the impor-tance of the components within the community). The indicescomplement each other in showing different types of central-ity, rather than giving competitive results, and their usefulnessin network analysis depend on the question asked.
The indices range from very simple ones, dependent onlyon characteristics very local to the focal node, to those thatinclude information on the widest web features. Throughoutthe paper, we define N as the number of nodes in a network.
The index that is most local and most widely used but prob-ably least informative about network topology is the degree ofa node (D). This is the number of adjacent nodes (in a foodweb, the sum of prey and predator species; Wassermann andFaust, 1994). Several analyses on topological key species havebeen focusing on the number of neighbors in food webs, i.e.the degree of nodes and their distribution in trophic networks(Sole and Montoya, 2001; Dunne et al., 2002; Montoya and Sole,2002; Williams et al., 2002). Degree considers only the linksdirectly connected to a node. We also consider network indicesreflecting short indirect effects, i.e. the neighbors of neighbors.We call these indices mesoscale indices, in contrast to the localnature of degree, and to the global nature of some indices char-acterizing the whole network (see later).
A mesoscale index quantifying how frequently a node i ison the shortest paths between every pair of nodes j and k iscalled betweenness centrality (BC) and used routinely in socialnetwork analysis (Wassermann and Faust, 1994). We calcu-lated it using the UCINET VI programme (Borgatti et al., 2002).The standardized index for a node i (BCi) is:
BCi =∑
j<kgjk(i)/gjk
(1/2)(N − 1)(N − 2)
where i �= j and k. gjk is the number of equally shortest pathsbetween nodes j and k, and gjk(i) is the number of these short-est paths to which node i is incident (of course, gjk may equalone). The denominator is the number of pairs of nodes with-out node i. This index thus measures how central a node is,in the sense of being incident to many shortest paths in thenetwork. If BC is large for node i, it means that this group has
i
an important role in mediating many rapidly spreading effectsin the web.
The ego-betweenness of a node is the betweenness valuewithin its own ego-network, i.e. the subgraph composed of
1 9 6 ( 2 0 0 6 ) 365–378
a particular node, its neighbors, and the connections betweenall of them. We use the normalized version of it, thus we cancompare the values that the nodes have. If a node has a highego-betweenness value, this means that the closest subcom-munity around it strongly depends on its presence, i.e. it is akey mediator of various effects at a mesoscale.
We also calculate the information centrality index (IC), whichconsiders all paths (including the shortest), weighted by pathlength (for detailed explanation, see Wassermann and Faust,1994, calculated by UCINET VI, Borgatti et al., 2002), becausethe effects spreading through the shortest path may not bethe most important ones.
Closeness centrality (CC) quantifies how short are the mini-mal paths from a given node to all others (Wassermann andFaust, 1994) and is again calculated using UCINET VI (Borgattiet al., 2002). The standardized index for a node i (CCi) is:
CCi = N − 1∑N
j=1dij
where i �= j, and dij is the length of the shortest path betweennodes i and j in the network. This index thus measures howclose a node is to others. The larger is CCi for node i, the morerapidly will its effects spread to other groups.
In contrast to the mesoscale indices, the global ones pro-vide information of the whole network. The network central-ization indices for degree, closeness, betweenness and informationcentrality are calculated according to Wasserman and Faust(1994, p. 175 and for details, see also pp. 180, 186, 191 and197). For the other indices, we define network centralizationas the coefficient of the variation of the species’ importancevalues.
Another measure we use, the keystone index (K; Jordan et al.,1999) derives predominantly from the application and modifi-cation of the “net status” index in sociometry (Harary, 1959)used also in ecology (Harary, 1961). The keystone index ofspecies i (Ki) is defined as:
Ki = Kbu,i + Ktd,i = Kdir,i + Kindir,i
=n∑
c=1
1dc
(1 + Kbc) +m∑
e=1
1fe
(1 + Kte),
where n is the number of predators eating species i, dc thenumber of prey of its cth predator and Kbc is the bottom–upkeystone index of the cth predator. And symmetrically, m isthe number of prey eaten by species i, fe the number of preda-tors of its eth prey and Kte is the top–down keystone indexof the eth prey. For node i, the first sum in the equation (i.e.∑
1/dc (1 + Kbc)) quantifies the bottom–up effect (Kbu,i) whilethe second sum (i.e.
∑1/fe (1 + Kte)) quantifies the top–down
effect (Ktd,i). After rearranging the equation, terms includingKbc and Kte (i.e.
∑Kbc/dc +
∑Kte/fe) refer to indirect effects
for node i (Kindir,i), while terms not containing Kbc and Kte
(i.e.∑
1/dc +∑
1/fe) refer to direct ones (Kdir,i). Both Kbu,i + Ktd,i
and Kindir,i + Kdir,i equals Ki. The keystone index emphasizesvertical over horizontal interactions (e.g. trophic cascades as
opposed to apparent competition). Its conceptual counter-part is the trophic field of a species (Jordan, 2001) and hasbeen applied several times in network analysis (e.g. Jordan,2001). Its important feature is the sensitivity to both dis-
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ance and degree: it quantifies positionality at an intermediatecale rather than giving very local or very global informa-ion (Jordan and Scheuring, 2002). We calculated the keystonendices of trophic groups by the FLKS 1.1 programme (availablen request).
We must note that the keystone index is useful only for aingle web type (TDUU), i.e. there is no possibility for compar-sons. The other four indices presented above (degree, between-ess, information centrality and closeness) are insensitive to theirection, sign and strength of links, so we use them to analyzend compare TUUU and IUUU. Unlike them, one can calculatehe importance of a node by summing the effects it has onach other nodes. The following two indices using this methodssume that the indirect chain effects are multiplicative anddditive.
An index useful for undirected, unsigned and unweightedetworks is called topological importance index (TI). We use it
or characterizing long indirect effects (cf. Yodzis, 2001) whilet is not biased for vertical interactions, i.e. takes into accountlso exploitative and apparent competition. The index itselfs the extension of an earlier one proposed for the analysis ofwo-steps long, horizontal, apparent competition interactionsn weighted host–parasitoid networks (Muller et al., 1999). Inn unweighted network, we define an,ij as the effect of i on jhen j can be reached from i in n steps. The simplest modef calculating an,ij is if n = 1 (i.e. the effect of i on j in 1 step):
1,ij = 1/Dj, where Dj is the degree of node j. When the effect oftep n is considered, the effect received by species i from allpecies in the same network is equal to 1 (i.e. each species isffected by the same unit effect.). Furthermore, we define the-step effect originated from a species i as:
n,i =N∑
i=1
an,ij
hat may vary among different species (i.e. effects originatedrom different species maybe different). Here, we define theopological importance of species i when effects “up to” n stepre considered as:
Ini =
∑n
m=1�m,i
n=
∑n
m=1
∑N
j=1am,ij
n
hich is simply the sum of effects originated from species i upo n steps (one plus two plus three . . . up to n) averaged overy the maximum number of steps considered (i.e. n).
In our study, we extend the TI index to directed networks.n this case, we define the direct effect i upon j as
1,ij = ˛ij∑D
i=1˛ij
here ˛ij = 1/Dj,out if i is a predator and ˛ij = 1/Dj,in if i is a preyf j. In the equations Di,in is the number of i’s preys, Di,out ishe number of i’s predators and D is the sum of them (degree),hile the other calculations remains the same. The method
ssumes that the effect each species receives from its preda-ors and preys are equal. Further, each prey and predator has
qual effects, respectively, which is normalized (the sum ofhem is defined as 1).
For a weighted network, all effects are defined in the sameay as above with the exception of calculating ˛ij, which is
6 ( 2 0 0 6 ) 365–378 371
defined as:
˛ij = εij
�j,
where �j is the sum of strength of links pointing to j, if i is aprey of j or the sum of strength of links originated from j, ifi is a predator of j. εij is the strength of the link connecting iand j. In this study, we define εij as the absolute value of thestrength calculated by the IMPACTS program (see next para-graph). The weighted approach of calculating two-step effects(i.e. a2,ij) was originally developed by Godfray and coworkersfor assessing apparent competition in a host–parasitoid com-munity (Muller et al., 1999; Rott and Godfray, 2000).
We also use the IMPACTS program developed by Ulanowiczand Puccia (1990) for measuring direct and indirect trophicimpacts from trophic flow data. The dietary coefficient (gij),quantifies the effect of prey i upon predator j (defined as theproportion of i within the menu of j). The effect of predator jon its prey i (fji) is measured by the fraction of net output (res-piratory output is excluded) consumed by predator j. The netimpact of i upon j equals gij − fji and defined as the one-step(direct) effect of i on j. Its values range from −1 to +1. Takingevery pair of N nodes, we calculate the above-defined directnet impacts and constitute the N × N net impact matrix, [Q].The total (direct and indirect) effects are calculated by sum-marizing the all integer powers of [Q]:
[M] =∞∑
h=1
[Q]h.
Using the following equation known from the input–outputtheory (Hannon, 1973):
[M] = {[I] − [Q]}−1 − [I],
where [I] is the identity matrix. Summing the rows of [M]equals the IMPACTS based importance. We use the summedabsolute values of effects (later called IMA) in order to mea-sure the interactive power of a group: keystones can be of highimportance because of both positive and negative effects onothers (otherwise mixed strong negative and strong positiveeffects might result in an effect around zero). Using the abso-lute of the resulting values does not lead to the loss of theinformation on sign structure. We use this method also forthe TDSU variant. In this case, we calculate the direct, pair-wise impacts as if each prey and predator would have hadequal effects.
The reader can see that the method of IMPACTS is anal-ogous to the calculation of the topological importance index.The latter is calculated for undirected and unsigned networks,by taking the average of the matrices while IMPACTS han-dles directed and signed graphs by summing them up. Asexpounded above, we extend these methods for all types offood webs analyzed in this study and used them to comparethe different trophic variants of the network. Because of thesemethods suppose the additivity of the effects, which is not the
case for interaction webs, they are not used. Note that whilethe calculation of the direct impacts with the IMPACTS methodcontains the flow to the non-living compartments as well, theTI method does not.
i n g 1 9 6 ( 2 0 0 6 ) 365–378
Fig. 3 – The network centralization index for degree,closeness and betweenness, calculated for TUUU (grey) and
372 e c o l o g i c a l m o d e l l
4. Results and discussion
Our results quantify the interaction structure of the networksand the positional importance of different nodes. First, wepresent the results of these quantifications, and then the com-parative results. We are interested in: (i) comparing the resultsbased on the food web to those based on the interaction weband (ii) comparing the values of the same network index inseveral network variants. Finally, (iii) we will present moredetailed results on the structure of the fish community: sincethe level of aggregation is much more similar in the case of fishspecies, the value of comparative approaches is larger here.
4.1. Species importance in network variants
The importance ranks of graph nodes in different networkvariants, according to the different sensible network indicespresented above are shown in Appendix A. (Note that the vari-ant TUUU*, the web with non-living components as well havethe detritus as most important group.) Group #8 (zooplank-ton) is the leader in both the simplest (D in TUUU) and themost complex (TDSW and IDUU) calculations. Then, depend-ing on the type of the web, groups #3 (bacteria in sediment),#1 (phytoplankton) and some six more groups appear amongthe key groups. The importance of the groups #8 and #3 areexplained that they are the obligate prey of many species. Thelight assimilated by the phytoplankton is the energy sourcefor the planktonic part of the food web and similarly, the mostimportant benthic producers are the bacteria in sediment.Comparing the direct and indirect effects measured by TI andIMA, the observation is that the group #8 has rather direct,while the group #1 has rather indirect role in regulating thecommunity. Bay anchovy (#22) is located in the centre of thefood web, so it has an important role in transmitting impacts(betweenness and information centrality). It has the smallest aver-age distance to the other nodes (closeness centrality). Accordingto the top–down component of the keystone index, striped bass(#33) has an important role in the top–down control of the web.
4.2. Comparison of the food web (TUUU) and theinteraction web (IUUU)
Different indices are not similarly sensitive to adding non-trophic links to the food web. In general, one can say that theadding of non-trophic links has a notable effect on the impor-tance ranks of the species, suggesting that they may have animportant role in organizing whole communities and do nothave only local significance as thought before. For investigat-ing the network’s more local features, we calculate the ego-betweenness of the nodes. It increases only for group #13, theoysters (and very slightly to #1, phytoplankton) when consid-ering the interaction web (see Appendix A). Thus, we can saythat non-trophic effects are patterned in a way emphasizingmore the importance of this group in its local subcommunity,outlining the subcommunity-organizing role of the oysters.
Analyzing the network from a global viewpoint, we findthat if non-trophic links are taken into account, this increasedcomplexity reduces the differences between the species’importance. First of all, the network centralization index is cal-
IUUU (white), showing that considering the non-trophicinteractions as well makes the web more homogenous.
culated for the three basic topological measures (D, CC andBC) and always found lower in the interaction than in the foodweb (Fig. 3). Secondly, the observation that the most importantnodes according to the different indices are approximativelythe same in the trophic web (#2, #7, #8 and #22) but different inthe interaction web outlines the more diverse roles of speciesin the latter. Finally, the fact that the ego-betweenness values donot change or decrease for all but two groups means that theinbuilt non-trophic links make the network more redundant,with more alternative pathways.
All these are suggesting that the community is more bal-anced than thought before, judging only by the food web(increasing stability, sensu MacArthur, 1955). The non-trophiclinks often represent interactions between the more or lessindependent parts of the trophic web, i.e. the plankton, thebenthos and the nekton and besides, the seemingly mostimportant non-trophic links belong to the trophically lessimportant groups (cf. the rank positions of group #6 inAppendix A).
4.3. Comparisons of the different variants of thefood webs
If only the order of magnitudes of the flows are known (TDSS),the results calculated by IMPACTS are very similar to thoseof TDSW (with exact flow data; Fig. 4). If only direct interac-tions are considered, there are only few changes in the impor-tance ranks of the species, for example that the effect of #1 isunderestimated. In the case of the overall (direct and indirect)effects, the differences between TDSW and TDSS are bigger,for example the importance of #3 is overestimated, but stillvery similar. TDSU considering no interaction strength showmuch more different results. The ranks of TDSS and TDSW arealso much more similar to each other than to TDSU: groups#2, #19 and #33 decrease and groups #9 and #14 increase insignificance as the quantitative weights are considered. TDSSalready indicates these changes.
If we are examining only the direct interaction structure ofthe food web, the importance ranks of TDSW and the TDUWare nearly the same. This is showing that the two methodsfor calculating the effect of the direct interactions, i.e. topo-
e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378 373
Fig. 4 – The IMA values of groups calculated for the TDSUand TDSS web variants plotted against the IMA indicescalculated for TDSW web. Semi-quantitative weighting(TDSS) gives a much better approximation to thequantitative weighting (TDSW) than the unweighted index(
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Fig. 5 – The coefficients of variation for: (a) TI 10 calculatedfor TUUU*, TUUU, TDUU and TDUW and (b) IMA in TDSU,TDSS and TDSW. In both cases, the complexity of thenetworks leads to larger differences between the relativepositional importance of groups. The keystone pattern is
TDSU).
ogical importance for unsigned and IMA for signed networks,re actually consistent. Examining the indirect effects as well,eglecting the signs of the interactions cause smaller differ-nces in the most important (#8, #3 and #1 remains the threeost important species, although in different order) and big-
er differences in the least important species.Summarizing the above-demonstrated results, our conclu-
ions are the following. Despite the differences between theost and the least information-rich web’s importance ranks,
he most robust results, as it is the zooplankton’s (#8) keystoneule, can be shown only by analyzing the simplest topologicaleb. For a more detailed picture of the keystone structure,
he web must be weighted and if indirect effects are takennto account it has to be signed. We must note that the longerndirect effects are considered, the more accurate data (bothn weight and sign) is needed. The one of the most notableesult for practice is that weighting links is important but aemi-quantitative approximation is accurate enough.
We characterize the centralization of the food web vari-nts by the coefficient of variation of the importance values ofraph nodes (Fig. 5). It is evident why the TUUU* web is moreeterogeneous then the TUUU web: it is an artifact causedy the non-living nodes’ high importance. The centralizationf the web is decreasing as we are losing information fromDSW through TDSS to TDSU and from TDUW through TDUU
o TUUU. This is the case whether considering only direct orirect and indirect effects as well. The decrease of the coef-cient of variation is caused by the decrease in the variationf the data analyzed. This indicates that comparing the cen-ralization of different networks is only appropriate if the dataases contain the same quality of information.
.4. Fish community structure
e analyzed the fish community (nodes #21–33) for two practi-
al reasons. First, this part of the web is uniformly aggregated,.e. the nodes represent one or two species. Second, there iseavy fisheries in the Bay, so there is a strong need to under-tand the roles fish play in the community.
more characteristic and keystone species are easier to beidentified if we have more data on the network.
The most important fish, one of the keystone species ofthe community is Bay anchovy (#22). It’s always among thefirst two species in the ranks for TUUU and IUUU. However,if the trophic web is directed, signed or weighted, it is rankedlower. As mentioned before, anchovy, consuming the plank-ton and consumed by most of the other fishes, has essentialrole in transmitting effects across trophic levels. That is whyit is a key species in the food web considering betweenness andinformation centrality indices. Because of its central position, ithas the smallest average distance to the other nodes (close-ness centrality). The topological importance index is showing thiseffect as well, while assigning it the second highest impor-tance value, even whether direct or all impacts are taken intoaccount. Although anchovy is really common in the Chesa-peake Bay today, its keystone role in the community suggeststhat it should not been neglected when setting the conserva-tion priorities.
According to the top–down component of the keystone index,striped bass, one of the top predators (#33) is the most impor-tant top–down controller of the web and bluefish (#30) is thethird most important. If we investigate the strength of pair-
wise interactions between it and the other species by consid-ering the suitable component of the IMA index, the surprisingresult comes that it is mostly influenced indirectly, by group#8 and not by any of their direct neighbors. This is interesting
374 e c o l o g i c a l m o d e l l i n g 1 9 6 ( 2 0 0 6 ) 365–378
Fig. 6 – The relative positional importance of fish species in different network variants. From the simplest (a) to morecomplex (c and d) cases the keystone pattern is more characteristic. This result comes already if weighting is
portonsid
semi-quantitative (TDSS). Species #22 is always the most imalso becomes visible as soon as non-trophic links are also c
to be considered in analyses, where the number of neighbors(D) is the only parameter characterizing a graph node. Also,in the first case, where top-species are mostly influenced bya bottom species, consequences might be drawn consideringthe possible relationships between fishing down and harm-ful algal blooms, especially because both are global, trendlikeevents.
Fig. 6a informs us about the relative positional importanceof fish species in the TUUU web variant, according to the TIindex for 10 steps long effects. If we consider more informa-tion (e.g. TDSW) and calculate the IMA values, we have a muchless homogeneous distribution resulting in the more charac-teristic keystone pattern (Fig. 6c; IDUU in Fig. 6d is somewhatsimilar). This is still the case when only the order of magnitudeof the mass flows are known (TDSS) (Fig. 6b). The relativelymost important fish is always #22 but the relative importanceof some others changes sensitively according to the differentkinds of information the web variants represent.
5. Conclusions
We have quantified the importance of the different trophicgroups composing the interaction network of the ChesapeakeBay mesohaline ecosystem. Different indices were used anddifferent variants of the same web were studied, according tothe combinations of whether to consider link direction, link
sign, link strength and non-trophic links. Only the sensiblecombinations were studied and for each network variant onlythe sensible network indices were calculated. Our basic inter-est was whether and how do the positional importance rank
ant player in the community but the importance of othersered (e.g. #25).
of trophic groups differ. Each rank was different. In contrastwith previous ecological network analyses, we emphasized (1)the importance of mesoscale indices describing neither localnor global interactions and (2) analyzed a web containing bothtrophic and non-trophic interactions simultaneously. The firstis important because of the dampening spread of indirecteffects in food webs, while the second is timely because ofrecognizing the typically diffuse nature of mutualisms.
We emphasize that here we investigated an ‘effect’ net-work which is parallel in many ways with the ‘energy’ or‘matter’ networks analyzed from the flow-storage perspec-tive, also common in the literature (e.g. Borrett et al., 2006;Gattie et al., 2006; Schramski et al., 2006). The flow-storageanalysis handles ecosystems as ‘structural, functional unitsthat import, process and export energy, material and informa-tion’ (Gattie et al., 2006), hence gives insights into the originand the fate of these three mediums. In this sense, Pattenhas shown that indirect effects tend to dominate direct ones(Patten, 1995; Borrett et al., 2006) and Fath and coworkers(Fath, 2004; Schramski et al., 2006) concluded that the control,defined as the extent to which elements contribute to the sys-tem’s overall flow-storage pattern, is widely, but not equallydistributed among the components. However, these studiesdeal with only positive flows of medium, thus, the negativeeffects of predation fall out of their scope (Ulanowicz, 2004). Incontrast, we were interested in how interacting species affecteach other and how these effects spread through the network,
using carbon flows as only the basis of effects. Hence, the‘indirect effects’ and ‘control’ terms from the perspective offlow-storage analysis are related to the bottom–up effects inour terminology. Although there are many methodically sim-
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lar points in the analysis of interaction and flow-storage net-orks, several questions addressed are clearly different and a
ynthesis is needed if ecosystems are to be understood.Our analysis is basically a methodological work and an
llustrative case study, thus, our results and conclusions aretatistically not sensible and one of the most important futureasks is to extend this analysis to further networks (for somexamples, excellent trophic network data bases and additionalon-trophic data exist for the Benguela upwelling system, theorth Atlantic, or the Ythan River). This study is more like an
llustration of some technical developments. Also, for testinghe robustness of the results, i.e. their sensitivity to method-logical noise, perturbation studies should also be welcome
cf. Jordan et al., 2003b). As a further theoretical development,he sign structure of interaction networks could also be takennto account, providing possibly helpful information aboutoop balance and structural stability.
We emphasize that more connections between ecologistsnd sociologists could be very helpful in both fields, eithernly for methodological developments or for better under-tanding comparative community dynamics within the net-ork context. The common roots vary richly from conceptual
o methodological crossroads (see Harary, 1959; Harary, 1961;ordan et al., 1999; McMahon et al., 2001; Williams et al., 2002;uczkovich et al., 2003; Krause et al., 2003).
Particular problems need different approaches suggest-
ng basically the same but sometimes also different rolesssigned to functional groups of species. Thus, the furtherevelopments in this field are proposed to help functional and
problem-centric ecology and conservation biology. We believethat this study may contribute to the theoretical developmentof how to set conservation priorities and how to optimize datacollection. We suggest that it is imperative to make differencebetween the usefulness of different data types. Whether tomake more effort on mapping non-trophic interaction path-ways or measuring link weights is a practical question. The-oretically, both give important information but conservationbiology is typically in decision situation. As for the more aca-demic part of our work, it is a fundamental question how toquantify the relative importance of species and whether theproposed methods might consider only the trophic structureof communities or emphasis must be set also on non-trophiceffects.
Acknowledgements
Wei-chung Liu, Istvan Scheuring and two anonymous review-ers are kindly acknowledged for helpful comments on themanuscript. FJ was supported by the OTKA T 37726 grant ofthe Hungarian Scientific Research Fund and also by the BrancoWeiss Fellowship from Society in Science, ETH, Zurich.
Appendix A
The positional importance rank of trophic groups (in bold)according to the values (in normal) of different indices (initalics) characterising different network variants (in bold). Seeexplanation in text.
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