Defining and Measuring Trophic Role Similarity in Food Webs Using ...

304J. J. LUCZKOVICH ET AL.formally defined. Elton (1927) first defined theniche as the fundamental role of an organism inits community F largely in its relation to foodand to enemies. In defining a species’ role in thecommunity, Elton drew an analogy with socialroles in human communities. Since that earlydefinition, numerous authors have discussedideas related to trophic role, defining and usingdifferent terms such as trophic level (Lindemann,1942), trophic niche (Hutchinson, 1958), ecologicalguild (Root, 1967), and trophospecies(Yodzis, 1988). Although each of these terms insome way defines a species’ trophic role in anecosystem, each concept differs slightly, andnone is completely satisfactory in capturing thetotal feeding relations of a species in a food web(Cousins, 1985, 1987; Perrson et al., 1996). Here,we formally define the trophic role of a speciesusing a concept drawn fromsocial role analysiscalled regular equivalence, and present examplesusing this approach to measuring similarity inthe trophic role of species in food webs.Recently, some new approaches to the definitionand measurement of the trophic role of aspecies in a food web have been proposed, inwhich both predator and prey relations areconsidered (Goldwasser & Roughgarden, 1993).Yodzis & Winemiller (1999) used Jaccard’s(1900) similarity coefficient to operationallydefine trophospecies, first computing predatorrelations, then prey relations, and finally bothsets of relations at the same time. Given thematrix of species-by-species Jaccard similaritycoefficients, they used clustering algorithms toidentify classes of species of fishes in an aquaticfood web in a Venezuelan floodplain. Theyconsidered two separate sets of a species’ linksto other species (using both topological andenergy flow links in separate analyses), firstcomputing the similarity of predator links, thencomputing the similarity of prey links, thenmultiplying the similarity measures F what theycalled the multiplicative similarity. They alsoanalysed trophic similarity by ignoring thedistinction between predator and prey, simultaneouslycomputing similarity in the dual roleeach species played as a consumer and a resourceF what they called the additive similarity. Theyconcluded that the additive similarity measureprovided the best measure of trophic similaritybased on predator and prey relations. TheYodzis–Winemiller approach is an importantadvance because it formalizes a fundamentalconcept (providing all the associated benefits ofprecision and measurability), and because itincorporates connections up and down the foodweb. However, it also retains an importantweakness in the classical conceptualization oftrophic role: it fails to detect the equivalence ofroles of species that do not share the samepredators and prey, but that may be similar intheir trophic position.As an example, consider two species of insectthat serve as prey for two different species ofcongeneric birds, and which consume verysimilar, congeneric species of plants. In theYodzis–Winemiller approach, these two insectspecies would be placed in separate trophicgroups because they share no predators and noprey. Yet this measure misses the similarity at ahigher level of trophic organization: both specieshave a similar trophic position within the foodweb. Both insects are functional herbivores, andthey are eaten by very similar species, but thereis no way for the Yodzis–Winemiller approachto detect that these two have any more incommon than any random pair of species thatalso share no prey or predators. One solutionwould be to modify the Yodzis–Winemillerapproach to measure similarity not at the specieslevel but at a more aggregate level. For example,in computing the shared prey coefficient for eachinsect species, we could hypothetically count nothow many species they have in common, buthow many genera they eat in common. Similarly,in computing the shared predator coefficient, wecould count the number of shared bird generathey have in common. The obvious problem withthis is that we do not know that the genus is theright level of aggregation, nor do we have anybasis for choosing phylogenetic affiliation as thebasis for grouping. However, the general ideais right: to evaluate the trophic similarity of twospecies, we want to know whether they havesimilar trophic ties to the same types of species,but not necessarily the same species. Thequestion is how to define these types. Ideally,we would use trophospecies F groups of speciesthat are trophically equivalent. But this wouldseemto require knowing the answer in order to

DEFINITION OF TROPHIC ROLE SIMILARITY 305construct the answer. We are trying to groupspecies into trophically equivalent sets, yet to dothis for any given pair of species, we would liketo know which trophic equivalence set all of theother species belong to, which is circular.Surprisingly, there is a solution, which isrecursive in nature; and, it is drawn fromthesocial sciences, where the concept of social roleof individuals within social systems has beenextensively analysed.The social sciences have paid much attentionto the importance of structure and position inthe study of social roles. Social scientists havedevoted a considerable amount of effort toformalizing the notion of role in the contextof a systemof social relations (Nadel, 1957;Merton, 1959; Homans, 1961; Goodenough,1969; Mayhew, 1980). For example, in thesystemof social relations in a hospital, whatdefines a person as playing the role of doctor isthe characteristic set of relations they have withpersons who are playing related roles such aspatients, nurses, medical record keepers, pharmaceuticalsales people and receptionists. Earlysocial network research modeled kinship systemsusing algebraic approaches (White, 1963; Boyd,1969), followed by Lorrain & White’s (1971)‘‘structural equivalence’’ concept that formallydefined roles and reduced the complexity of anetwork of relations to a set of simplified rolestructures. In this structural equivalence model,two individuals are seen as occupying the sameposition to the extent they have the same kindsof social ties to the same third parties. This workin turn paved the way for the development of amore general concept known as regular equivalence(White & Reitz, 1983), in which twoindividuals are seen as playing the same role tothe extent they have similar ties to analogousindividuals (i.e. those playing correspondingroles). Differences between structural and regularequivalence approaches were discussed byBorgatti & Everett (1989), Everett & Borgatti(1991) and Everett & Borgatti (1994).As an extension of Elton’s (1927) ‘‘role inrelation to food and enemies’’ paradigm, we willnow apply the theory of social role analysis andthe concept of regular equivalence to ecologicalfood webs and formalize the measurements oftrophic role similarity in detail. We will apply theregular equivalence approach to the analysis oftrophic role structure in two food webs, one abinary or topological network and the othera carbon-flow network. We will also compareperformance of the regular equivalence approachwith the structural method proposedrecently (Yodzis & Winemiller, 1999) in creatingtrophic aggregations. Finally, we will present animage graph of each food web, based on theregular equivalence approach, which provides areduced-complexity view of the entire food web.2. Theory of Role Analysis2.1. GRAPH-THEORETIC TERMINOLOGYWe represent community food web data asa directed graph, or digraph, G(V,E), whichconsists of a set of nodes V (also known asvertices) representing species or compartments,and a set of directed ties (also known as edges orarcs) E which represent predation, parasitismorany other trophic relationship. The notation(a,b) A E indicates the presence of a tie from a tob. Similarly, (b,a) A E, indicates a tie from b to a.Optionally, we can also define a real-valuedfunction F on E which assigns a value (such as aquantity of flow) to each tie, so that f (u,v)¼ 0.27 would indicate a flow of 0.27 units fromu to v. For simplicity of exposition, we shallusually assume here that ties are simply presentor absent rather than valued. However, themathematics generalizes straightforwardly tovalued data, and we will use valued energy flowdata in our empirical examples.The set of nodes adjacent to a given node v iscalled the neighborhood of v and denoted N(v).In a directed graph, a node’s neighborhoodconsists of two parts: an out-neighborhood,defined as the set N o (v) ¼ { p| (v, p) A E}, andan in-neighborhood, defined as the set N i (v) ¼{q|(q,v) A E}. In the case of food web predationdata, the out-neighborhood N o (u) is the set ofspecies that are predators of species u, and thein-neighborhood N i (u) is the set of species thatare prey of species u.An equivalence relation defined on a set ofnodes is a binary relation that is reflexive,symmetric and transitive. An example of anequivalence relation is the relation R induced bya partition of nodes into mutually exclusive

306J. J. LUCZKOVICH ET AL.classes so that two nodes are related by R if andonly if they belong to the same class. Thenotation (u,v) A R is used to indicate that node uis equivalent (i.e. in the same class as) to node v.Alternatively, we can write u v to indicate theequivalence of the two nodes. The equivalenceclass of a node u is denoted C(u). Below, wedefine two particular equivalence relations:structural equivalence, in which nodes areequivalent if they are connected to the sameother nodes; and regular colorations, in whichequivalent nodes are connected to equivalent(but not necessarily the same) nodes.2.2. STRUCTURAL EQUIVALENCEAn equivalence relation R is called structural(Lorrain & White, 1971; Everett & Borgatti,1991) if, for all nodes u and v, (u,v) A R if andonly if N o (u) ¼ N o (v) and N i (u) ¼ N i (v). Inother words, a structural equivalence F alsoknown as a structural coloration (Everett &Borgatti, 1991) F equates nodes that haveincoming and outgoing ties to exactly the sameothers. Figure 1(a) illustrates a structuralequivalence by painting equivalent nodes thesame color. In contrast, the equivalence depictedin Fig. 1(b) is not structural because nodes 5 and6 are colored the same, yet node 9 has anoutgoing tie to node 5, and node 6 does not. Inaddition, node 5 has an outgoing tie to node 4,and node 6 does not. In structural equivalence,nodes 5 and 6 could not be equivalent. Forvalued data, such as energy flows, we wouldrequire that fully equivalent nodes have equallyvalued ties to the same others.As defined above, structural equivalence is anidealized mathematical concept in which nodesare either equivalent or they are not. When usingthe concept to analyse empirical data, we usestructural equivalence algorithms (Borgatti et al.,1999) that measure the extent of structuralequivalence between every pair of nodes. Whenthe appropriate measure of pattern similarityis chosen (such as the Jaccard coefficient), thelatter approach is almost identical to recentapproaches used by ecologists for identifyingtrophospecies (Yodzis & Winemiller, 1999).Fig. 1. (a) A structural coloration (left), (b) a nonstructuralcoloration (right), (c) a regular coloration, (d) anon-regular coloration, (e) disconnected graphs representingdifferent webs. Color codes: solid circles, black; opencircles, white; angled hatching, green; vertical hatching, red;cross hatching, blue; stipled, yellow.2.3. REGULAR EQUIVALENCEStructural equivalences are members of afamily of equivalence relations called regularequivalences or regular colorations (White &Reitz, 1983; Borgatti & Everett, 1989; Everett &Borgatti, 1991). An equivalence relation R of adigraph G(V,E) is regular if, for all nodes u, v AV, u v implies that if there exists a tie (u,y) A E,then there exists a node z such that (v,z) A E andy z, and if there is a tie (p,u) A E, then thereexists a node q such that (q,v) A E and p q. Inother words, in a regular equivalence, if nodes uand v are equivalent, then if one has a tie to athird party, then the other one has a correspondingtie to an equivalent third party (but notnecessarily the same one). Applied to food webs,this means that regularly equivalent species preyon equivalent species and are preyed upon byequivalent species. It should be noted that a

DEFINITION OF TROPHIC ROLE SIMILARITY 307graph may have more than one regular equivalence.Since our goal is usually to seek thesimplest model, our discussion will center onthe maximal non-trivial regular equivalence (i.e.the one with the fewest equivalence classes).Consequently, when we write ‘‘regular equivalence’’it is understood that this means the maximalregular equivalence unless otherwise stated.An example of a regular equivalence is shownin Fig. 1(c), again using same color and fill toindicate equivalence. To determine whether itis regular, we check each pair of nodes that iscolored the same against the definition. Forexample, nodes 2 and 3 are colored the same,indicating they are equivalent. According to thedefinition, their out-neighborhoods (and their inneighborhoods)must contain equivalent nodes(i.e. the same colors). Node 2 has incoming tiesfromcross-hatched nodes and vertical-linednodes and no other colors. Node 3 is the same.Node 2 has outgoing ties to stippled nodes andcross-hatched nodes. Node 3 is again the same(recall that the definition says nothing about thenumber of nodes of each type that a node maybe connected to). Hence this pair satisfies thedefinition. If all pairs do, the equivalence isregular. In contrast, Fig. 1(d) gives an exampleof a non-regular equivalence. One reason it isnot regular is that node 4 is colored the same asnode 2, yet node 4 has no tie to a vertical-linednode, while node 2 does.It is useful to note three desirable features ofregular equivalence. The first is that cycles, suchas the one connecting nodes 2, 3 and 11 inFig. 1(c), pose no difficulty. The second is thatomnivory also provides no particular problem,as nodes 10 and 1 feed at the same two levels ofthe food chain (and are equivalent) whereasnodes 4 and 2 feed at different levels (and are notequivalent). The third is that two species can beregularly equivalent without having any prey orpredators in common, as is the case with nodes 5and 6. Thus, regular equivalence captures theidea that species can occupy analogous positionsin the food web, without the limitation found inthe Yodzis–Winemiller approach that requiresspecies assigned to the same class to interacttrophically with the very same species. Animplication of the last feature is that regularequivalence can be used to class together speciesthat are members of different food webs, yetoccupy analogous positions. An example is givenin Fig. 1(e) in which nodes 12 and 2 are regularlyequivalent, yet obviously belong to separatewebs. In this way, regular equivalence may beused to improve upon the practice of comparingwebs by counting numbers of species by basal-,intermediate- and top-level species (Cohen &Newman, 1985).2.4. IMAGE GRAPHSThe image graph induced by an equivalencerelation R on G is defined as the digraphG 0 (C(V),E 0 ) in which the nodes are the equivalenceclasses of R, and two nodes (i.e. classes) xand y are adjacent in G 0 if there exist adjacentnodes u and v in G such that C(u) ¼ x and C(v)¼ y [recall that C(u) denotes the equivalenceclass of node u]. In this sense, an equivalencerelation induces a mapping, known as a graphhomomorphism, from the original network to areduced model of the network. Figure 2(a) showsthe image graph for the structural equivalence inFig. 1(a). It should be noted that in a structuralequivalence, the image graph is a completedescription of the entire systemof relations,Fig. 2. (a) Image graph for the structural coloration inFig. 1(a); (b) image graph for the regular coloration inFig. 1(c); (c) a graph of a hypothetical food web showingthe regular equivalence of feeding relations. Directed tiesare depicted by arrows that show the predator’s consumptionof a prey. Color codes are the same as in Fig. 1.

308J. J. LUCZKOVICH ET AL.without loss of relational information. If twocolors blue and red are mutually connected inthe image graph, then, in the original graph,every blue node has a connection to every rednode, and every red node has an incoming tiefromevery blue node.The image graph of a regular equivalence alsoprovides a reduced model of the network(s) as awhole. Unlike structural equivalence, however,with regular equivalence there is some loss ofrelational information, although the extent andtype of loss are strictly controlled. Figure 2(b)gives the image graph associated with the regularequivalence in Fig. 1(e). Given the rules of aregular equivalence, we are guaranteed that if, inthe image graph, there is a tie from a verticallinednode to a cross-hatched node, then in theoriginal graph, every cross-hatched node has anincoming tie from at least one vertical-linednode, and every vertical-lined node has anoutgoing tie to at least one cross-hatched node.Thus, the image graph from a regular equivalenceprovides a simplified graph of the patternof connections among individual species; it is adisciplined way of representing complex foodweb structure in which some detailed informationis lost, but the fundamental pattern orstructure of ties is preserved.A look at Fig. 2(c) suggests that in addition tocapturing the notion of species trophic role,regular equivalence has some relationship to thenotion of trophic position. Node 3, which iscross-hatched, is two links above a primaryproducer, and one link below a top predator. Allthe other cross-hatched nodes are positioned thesame. Note that nodes 7 and 8 are very much likethe cross-hatched nodes: they have an outgoingtie to the top predator, and an incoming tie froma vertical-lined prey. But unlike the crosshatchednodes, they also have a tie directly tothe primary producers (white nodes), i.e. they areclassic omnivores. As a result, they are placed ina different class, which corresponds to a differenttrophic position. Similarly, node 6, which also isprey for the top predators, is in a different classfromthe vertical-lined nodes, the cross-hatchednodes and the gray nodes because it feedsdirectly and exclusively on a primary producer.Note that this means that node 11, the primaryproducer connected to node 6, cannot be in thesame class as the other producers 12 and 13. Inthis sense, regular equivalence makes finerdistinctions than do most other measures oftrophic position, because the producers will begrouped into separate classes with differingrelations to the herbivore and omnivore classes.Consequently, we can think of regular equivalenceas defining classes of isotrophic species,rather than trophic levels of species.The relationship between regular equivalenceand trophic position is clearest when the foodweb contains no cycles. In such webs, two speciesthat are colored the same are equally distantfromcorresponding top predators and producers.For example, if a b, then if a is positionedtwo links above a given producer species c, thenb is also two links above either c or some otherproducer d that is regularly equivalent to c.Thus, under such circumstances, nodes in thesame class are necessarily at the same trophicposition. In food webs with cycles, the correspondencenoted above can break down. However,there is a simple solution. In graph theory,the length of the shortest path between twonodes is known as the geodesic distance betweenthem. By calculating the geodesic distancebetween all pairs of nodes, we can construct anode-by-node geodesic distance matrix. Applyingregular equivalence to this matrix instead ofthe raw adjacency matrix generates a regularequivalence that necessarily preserves trophiclevels, as two nodes that are colored the samewill be the same geodesic distance from equivalentothers. This approach can also be applied toflow data by applying regular equivalencesimultaneously to both the flow matrix and thegeodesic distance matrix, but the details of thistechnique are beyond the scope of this paper.While regular equivalence classes are homogeneouswith respect to trophic position, asexplained above, they produce classes that differfromthe groupings that result using the classicdefinition of trophic level. One reason is thattrophic-level concept considers only distancefromthe producers, and does not include bothdownward and upward trophic paths. Forexample, if two nodes a and b are both twolinks above the producer, then they are notalways regularly equivalent, because they maydiffer in the number of predator links above

DEFINITION OF TROPHIC ROLE SIMILARITY 309themin the web. In addition, some approaches,such as Adams et al. (1983), are sensitive to therelative number of prey at different levels, whichis not the case with regular equivalence ingeneral, although it is true of a subset of regularcolorations known as exact regular colorations(Everett & Borgatti, 1996). For example, inFig. 2(c), nodes 1 and 2 are placed at the samelevel by a regular coloration, even though node 1preys on two cross-hatched species while node 2preys on only one.A distinctive feature of the definition ofregular equivalence is its recursive or implicitquality. That is, to determine whether two nodesare equivalent, one needs to know the colors ofall the other nodes, but to determine their colorsone needs to know the colors of all nodesincluding the two nodes one started with, and soon. This may appear to be computationallyintractable, but in fact is not. There are severalefficient algorithms available for regular equivalence(Borgatti et al., 1999), which vary in thetype of output they produce (e.g. pairwisecoefficients giving the degree of equivalence vs.approximate discrete classes), type of data theycan handle (e.g. valued or binary), and othervariables.that are attracted to a Malaysian pitcher plantand become drowned or are preyed upon byother insects that live in the plant. In a sense, it ismerely a sub-web of a larger food web, becausethe insects are consuming energy elsewhere andbringing it to the pitcher plant. Thus, the role of‘‘producers’’ is filled by the drowned insects andlive insects (ants) that visit the pitcher plant,which are really consumers in a larger food web.In the original presentation (Beaver, 1985), sometaxa were grouped into ‘‘trophic types’’, which infact were structurally equivalent, because theyhad the exact same links to other taxa in thefood web. Stiling (2002) disaggregated thesetaxa, and we did as well, so as to start with acompletely disaggregated food web at the specieslevel (Fig. 3).3.3. ST. MARKS, FLORIDA SEAGRASS CARBONFLOW WEBThe food web data were obtained fromdirectsampling and literature surveys of the St. Marksseagrass ecosystem(Baird et al., 1998; Christian& Luczkovich, 1999, Luczkovich et al., in press).3. Methods3.1. EMPIRICAL APPLICATIONSIn the last section we laid out the mathematicaland computational underpinnings of the regularequivalence model. Now we illustrate the empiricalapplication of these concepts using twoempirical food web datasets, namely the Malaysianpitcher plant insect web (Beaver, 1985) asreproduced and discussed in Stiling (2002), andthe St. Marks, Florida seagrass food web (Bairdet al., 1998; Christian & Luczkovich, 1999;Luczkovich et al., in press). The insect datasetis a topological web consisting of predation links,while the St. Marks, Florida seagrass ecosystemis a dynamic web, with data consisting ofestimated carbon flows between compartments.3.2. MALAYSIAN PITCHER PLANT WEBThis is one of several food webs that have beendescribed by Beaver (1985), involving insectsFig. 3. A simple food web diagram of the insects in thepitcher plant Nepenthes albomarginata in West Malaysia[modified from the diagram presented by Stiling (2001),based on orginal data fromBeaver, 1985]. Appendix A liststhe species’ numerical codes for each node. Predators arepositioned higher than their prey, and lines representtrophic linkages.

310J. J. LUCZKOVICH ET AL.In order to obtain a species list for the food web,collections of producers and various consumerswere conducted along the study area transects(for more detailed methods see Baird et al., 1998;Christian & Luczkovich, 1999; Luczkovich et al.in press). In order to determine the structure ofthe diet matrix required for the network foodweb model, diet data were summarized based onthe existing literature on seagrass invertebratesand birds, and stomachs were examined directlyfor fishes in the study areas. A 51-compartmentdiet matrix for this study was prepared as acarbon-flow matrix taken directly from thenetwork model based on the approach describedby Ulanowicz (1986).3.4. STATISTICAL ANALYSISTo analyse the data, we used the REGEalgorithm implemented in the UCINET 5.8network analysis software package (Borgattiet al., 1999). The algorithmtakes any real-valuedN N (species-by-species) matrix X as input,and returns a species-by-species matrix R ofcoefficients (ranging from0 to 1) which records,for each pair of species, the extent of (maximal)regular equivalence. The essence of the algorithmisas follows:(0) Set r ij ¼ 1 for all i and j (i.e. let all speciesbe 100% equivalent to start).(1) For each species i and j,(1A) For each species k eaten by i, find speciesm eaten by j that is most equivalent to k andwhich is eaten in the most similar proportion ask is eaten by i, in other words which maximizesthe quantity z k ¼ r km Min(x ik ,x jm )/Max(x ik ,x jm ).(1B) For each k which eats i, find species mthat eats j that is most equivalent to k and whicheats j in the most similar proportion as k eats i,in other words which maximizes the quantityy k ¼ r km Min(x ki ,x mj )/Max(x ki ,x mj ).(1C) Set r ij and r ji ¼ S z k +S y k .(2) Repeat Step 1 until no more changes in r ijor maximum iterations exceeded. The maximumiterations ¼ N species or compartments.The resulting coefficients r ij have ordinalproperties. Details of the algorithmare discussedin Borgatti & Everett (1993). For comparison,we also calculate the additive similarity measureof Yodzis & Winemiller (1999) for the Malaysianpitcher plant insect food web, which is Jaccard’ssimilarity measure applied to all trophic links,whether predator or prey of a species. Tovisualize patterns of similarities emerging fromboth the REGE procedure and the Yodzis–Winemiller procedure, the similarity matriceswere submitted to a non-metric multi-dimensionalscaling (MDS) procedure in order torepresent similarities as distances in two-dimensionalspace (UCINET 5, Borgatti et al., 1999).The coordinates fromthe MDS procedure werethen used as input to the Pajek (Batagelj &Mrvar, 1999) network drawing software.In addition, in order to simplify interpretationof the results, a hierarchical clustering of theoutput matrix R fromthe REGE algorithmwasalso performed, yielding a dendrogram and setof nested partitions. Because it is appropriate forordinal data, we used Johnson’s hierarchicalclustering (Johnson, 1967). For display purposes,one partition within the hierarchicalclustering was selected to classify compartments.By definition, all partitions within a Johnson’shierarchical clustering are equally valid, representingdifferent levels of resolution rather thanalternative theories (Borgatti et al., 1990). Theparticular choice of partition was based on aseries of regressions designed to measure clusteradequacy. Since an ideal clustering of theR matrix would locate the largest values of Rwithin clusters and the smallest values of Rbetween clusters, we can measure the extent towhich a given clustering is optimal via ananalysis of variance in which the cases are pairsof nodes, the dependent variable is the REGEcoefficient for each pair, and the independentvariable is a dummy variable coded 1 if the pairare in the same cluster and 0 if they are indifferent clusters. The resulting R 2 (or eta-squareas it is called in the ANOVA context) can beinterpreted as a measure of cluster adequacy. Byplotting R 2 against the number of clusters, weobtain a scree plot, which can be examined forinflection points. A clustering with k classes ischosen if it provides a sizeable increase in R 2over the next simplest clustering (i.e. with k 1clusters), yet explains nearly as much varianceas the next most complicated clustering (k+1

DEFINITION OF TROPHIC ROLE SIMILARITY 311clusters). Of course, the researcher is free tochoose a cluster solution with higher resolutionif the R 2 is not substantially different.4. Results4.1. MALAYSIAN PITCHER PLANT INSECT FOOD WEBThe 19 taxa in the topological food web(Fig. 3) collected by Beaver (1985) and discussedin Stiling (2002) were analysed via the REGEalgorithm, generating the matrix of coefficients(available upon request fromthe authors andpublished on-line at http://drjoe.biology.ecu.edu/regefoodweb). Appendix A lists the namesand numerical identification codes for thecompartments in Beaver (1985) that are discussedhere. It is clear that the REGE algorithmdoes a very good job of detecting isotrophicspecies in the Malaysian pitcher plant insect foodweb. Based on the increase in R 2 of theregression analysis at four clusters and the rapiddrop after five clusters, as can be seen in thecluster adequacy scree diagram(Fig. 4), weidentified five isotrophic classes (with onesingleton class), which are identified by colorsin the cluster dendrogram(Fig. 5).Figure 6(a) shows a two-dimensional multidimensionalscaling (MDS) of the REGE coefficientsfor this web, with the color classes fromthe cluster analysis used to code the classes. ThisMDS displays the food web structure andrelative position in the network of the basal,intermediate and top consumers. One group ofbasal species (brown) included the drownedinsects (18) and organic debris (19). Anotherbasal group (singleton class) includes the liveinsects (white) (17). Both basal groups have noincoming ties F they function as the energysource or ‘‘producers’’ in this food web. TheREGE algorithmseparates basal species intotwo classes (brown and white), based on therelations of those nodes to the various colorclasses of their consumers: whereas brown classmembers are preyed upon by members of thegreen and red classes, the white class membersare only preyed upon by members of the yellowclass. The intermediate group (green) includesthe bacteria and protozoa (16), aquatic larvaeof dipteran insects [Megaselia sp. (5), Endonepenthiaschuitemakeri (6), Triperoides tenax (7),Triperoides bambusa (8), Dasychelea nepenthicola(9), Culex curtipalpis (12), Culex lucaris (13) andLestodiplosis sp. (4)]. These intermediate aquaticinsect larvae (green) are positioned midway fromtop to bottomin the MDS plot, which indicatesthat they occupy a position midway betweenfood sources and sinks. There were two topredyellowgreenbrownwhite111014152314131287956161819171.0 0.9 0.8 0.7REGE CoefficientFig. 4. A screen plot of the R 2 as it varies with numberof clusters fromthe average linkage cluster analysis usingthe REGE coefficients derived fromthe Malaysian pitcherplant insect food web.Fig. 5. A single-linkage cluster analysis dendrogramofthe Malaysian pitcher plant insect food web using theREGE coefficient matrix as a similarity measure (Taxacodes listed in Appendix A). Colors were used to indentifyclass membership based on the R 2 regression analysis(see text).

312J. J. LUCZKOVICH ET AL.consumer groups: the red group includedscavenging dipterans Nepenthosyrphus sp. (10),Pierretia urceola (11), and two species ofAnotidae (14 and 15); whereas the predatoryinsects Misumenops nepenthicola (1), encyrtids(Trachinaephagus) (2) and Toxorhynchies klossi(3) comprised the yellow group. Both the yellowand red groups are sinks for energy in this web(they have no outgoing ties).Five pairs of species were exactly regularlyequivalent, with coefficients of 1.0 (5 and 6; 7and 8; 10 and 11; 12 and 13; and 14 and 15).These nodes are shown as slightly displaced fromone another in Fig. 6(a) to prevent themfromcompletely obscuring one another. These pairsof nodes have the same ties to the same classesof species, but not necessarily the same species. Inthis example, each pair is tied to the exact samespecies, considering both predators and prey, sothey are both regularly and structurally equivalent.The complexity of the feeding relations in thisfood web network at the level of species can nowbe simplified from the original 19 nodes into fivecolor classes. After pooling nodes in the samecolor class into a single node and redisplayingthe network of these pooled nodes, we createda regular-equivalence image graph. The imagegraph of the pitcher plant food web shows thenetwork structure in simplified form [Fig. 6(b)].For comparative purposes, we also haveanalysed the same food web data using Yodzis& Winemiller’s (1999) preferred similarity measure,the additive trophic similarity measure,which is a measure of structural equivalence(Fig. 7 F similarity data available from theauthors or at http://drjoe.biology.ecu.edu/regefoodweb).The two types of similarity measures,structural (additive similarities) and the regular(REGE coefficients) equivalence, produced twovery different views of similarities among thespecies (compare Figs 5 and 7). Whereas thehierarchical clustering based on REGE coefficientsgrouped the basal producers into the sameclass (drowned insects 18, organic debris 19) atan early step, in the additive similarity clusteringthey are not joined until the next to the lastagglomeration step (Fig. 7). According to theadditive similarity measure, the node for organicdebris (19) is more similar to the top predatorM. nepenthicola (1) than it is to drowned insects(18). The additive similarity measure produces anon-intuitive grouping of nodes in the food web.We can visualize the structural equivalence ofthe network if we compute the MDS coordinatesin two dimensions using the additive similaritymeasure and display these with the networktrophic links [Fig. 6(c)], as we did above with theREGE coefficients [refer to Fig. 6(a) for comparison].The food web depicted with nodespositioned according to the MDS coordinatesobtained from the additive similarity measuredoes not describe the flow of energy or materialsfromsources to sinks, as it did when REGEcoefficients were used. Indeed, the basal nodes(17–19) are not particularly close to one another,and they cannot be arranged in such a way thatthey all lie near one end of the graph. The toppredators nodes (1–3) are close to one anotherand on one side of the plot, but are also in closeproximity to one of the basal nodes (18). This isa characteristic of the additive measure, whichignores the direction of the ties. In fact, thearrows representing energy flow among nodes inFig. 6(c) do not all point in the same generaldirection, rather they cross back over oneanother, so there is no approximation of trophicposition that can be uncovered in their arrangementby MDS. A few pairs of nodes appear togroup together: these are species with identicalyellowmagentagreenredblue111017568712139141519163142181.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2Additive SimilarityFig. 7. An average-linkage cluster diagramof theMalaysian pitcher plant insect food web using the additivesimilarity measure of Yodzis & Winemiller (1999).

DEFINITION OF TROPHIC ROLE SIMILARITY 313predator and prey relations, which thus haveadditive similarities of 1.0 and identical MDScoordinates [nodes 5 and 6; 7 and 8; 10 and 11;12 and 13; 14 and 15 F all are shown slightlydisplaced fromtheir coordinate positions so thatthey do not overlap one another in Fig. 6(c)].These species are the same pairs that were notedto be exactly regularly equivalent above. Outsideof these species with identical ties in the network,the additive similarity measure did not uncoverany nodes with similar network ties or positionwithin the network, as the REGE coefficientsdid. The additive similarity measure groupedtaxa independent of the direction of the ties F amajor drawback. In addition, the top predatorsin the web M. nepenthicola (1) and T. klossi (3)are joined only after the fourth agglomerationstep (Fig. 7), although they both clearly feed onaquatic insects living in the pitcher plants(Fig. 3). They feed on the same kinds of insects,but not the exact same species of insects, so theyreceive a low similarity in the additive similaritymeasure. But they do feed on the same trophicrole class of insects (green), as can be seen in theMDS plot [Fig. 6(a)]. Also, (1) and (3) are joinedat the same step as the bacteria and protozoa(16), yet these insects are the predators on theaquatic insect larvae group (4–9 and 12,13),whereas bacteria and protozoa are the prey ofthe aquatic larvae. Again, the direction of theties is ignored by the additive similarity measure,which results in the same level of similarity forthese species that occupy very different trophicroles.Thus, the additive similarity measure appearedto do a poor job at uncovering trophicroles of insects in the pitcher plant food web,except in the cases where species had exactly thesame predators and exactly the same prey. Incontrast, REGE was far superior in findingsimilarity among species’ trophic roles, even ifthe species consumed different prey and wereconsumed by different predators.4.2. ST. MARKS, FLORIDA SEAGRASS CARBONFLOW WEBWe now turn to another example, using thesesame procedures to illustrate the method toaggregate trophically similar taxa in a complexfood web. The 51 compartments in the food webcollected by Baird et al. (1998) were analysed viathe REGE algorithm, generating a matrix ofcoefficients (available fromthe authors andon-line at http://drjoe.biology.ecu.edu/regefoodweb).Appendix B lists the names and numericalidentification codes for the compartments thatare discussed here; see Christian & Luczkovich(1999) and Luczkovich et al. (in press) for thecomplete species list for each compartment. TheREGE coefficients were submitted to hierarchicalclustering and R 2 values were computed foreach partition. For display purposes we selecteda partition with ten clusters, which achieveda much better R 2 than partitions with fewerclusters and explained nearly as much varianceas partitions with more clusters (see scree plot,Fig. 8).We used colors to identify the ten isotrophicclasses of consumers in the MDS plot [Fig. 9(a)]:the lime green group, which was the mainproducer, the seagrass Halodule wrightii (2); thedark green group, which included other benthicproducers [epiphytic algae (3) benthic algae (5)]and benthic bacteria (12); the cyan group, whichincluded phytoplankton (1) and bacterio-plankton(6); the white group, which includedzooplankton (8); the yellow group, whichincluded various benthic invertebrates (nodes 7,9, 13, 15, 17, 19, 21, 22, 25, 27, 28), some fishesR squared0.50.40.30.20 5 10 15 20 25Number of ClustersFig. 8. A scree plot of the R 2 as it varies with number ofclusters fromthe average linkage cluster analysis using theREGE coefficients derived fromfeeding relations andcarbon-flow values of the St. Marks National WildlifeRefuge Seagrass EcosystemFood Web

314J. J. LUCZKOVICH ET AL.(31) and suspended POC (50); the blue group,which included the remaining benthic macroinvertebrates(nodes 10, 11, 16, 18, 20, 23, 24, 26,29, 30) and most fishes (nodes 32, 33, 35, 37–41);the purple group, which included the remainingfishes (nodes 34,36,42); the magenta group,which included herbivorous ducks (43); the redgroup, which included carnivorous birds (nodes44–48); and the brown group, which includedmeiofauna (14), and non-living groups [dissolvedorganic carbon (DOC) (49) and sediment particulateorganic carbon (POC) (51)]. Most producers(nodes 1–5) and non-living compartments(nodes 49, 51) appear on the right side of theMDS plot, the intermediate consumers (nodes 7–42) in the center of the plot, and the carnivorousbirds (nodes 44–48) at the lower left side of theplot. Herbivorous ducks (43) are at the far left,implying that they are a top predator (and theyare not preyed upon by other compartments inthis web), although they are separated fromother top carnivore birds. Thus, the plotproduces a food web that shows the approximatetrophic positions of all the taxa, moving fromlow trophic positions on the right to high trophicpositions on the left.Pairs of compartments that are close togetherin the MDS plot have similar trophic roles (e.g.they share high REGE coefficients), whichimplies that they have similar predators as wellas similar prey. For example, the pelagic andbenthic food webs can be separated: planktoniccompartments (1, 6, 8) occur in the upper rightand center portion of plot, and benthic species atthe lower right portion of the plot (3, 5, 12, 14,51). Because some consumer species are intermediatein their trophic roles (feed on benthicand planktonic species and are omnivores), theyappear in the center of the MDS plot. Thebacterial consumer compartments [bacterioplankton(6) and benthic bacteria (12)] aregrouped with the producers in the MDS, in partbecause they feed upon sediment POC and arefed upon by species that are otherwise detritivoresand herbivores. In this food web, thecarnivorous and herbivorous birds are not linkedto the detrital compartment (51), but all otherconsumers are, because the top carnivores’carbon has been assumed to migrate out of thesystem(Baird et al, 1998; Christian & Luczkovich,1999), consistent with the migratory natureof these birds. This similar migration pattern andabsence of connections to the sediment POC (51)caused all the birds to occur together on the leftside of the MDS plot.We collapsed the same-colored nodes andgenerated an image graph [Fig. 9(b)] that showsthe structure of the food web, while retaining therelationships among classes that define theisotrophic groupings. The isotrophic classeswe have identified are clearly associated withspecific trophic roles in the St. Marks food web.The lime and dark green groups included benthicproducers. The cyan group included planktonicproducers and the white group included planktonicconsumers (zooplankton). The yellowgroup included benthic invertebrates and fishesthat consumed benthic and planktonic producers,benthic bacteria, and other consumers(white, blue and purple groups). The yellowgroup consumed members of their own group(shown as a self-loop in the figure); they are inFig. 6. (a) A display of the two-dimensional non-metric multi-dimensional scaling of the REGE coefficients from theMalaysian pitcher plant insect food web using the network software Pajek (version 0.83). Arrows trace the path of energy inthe food web. Colors were used to indentify class membership based on the R 2 regression analysis (see text, Figs 4 and 5).Taxa which have identical links and thus REGE coefficients of 1.0 (5 and 6, 7 and 8, 10 and 11, 12 and 13, 14 and 15) havebeen plotted with a slight offset so that they do not obscure one another. (b) An image graph of the Malaysian pitcher plantinsect food web based on the regular equivalence relations shown in (a). (c) The food web network displayed as a twodimensionalnon-metric multi-dimensional scaling of the additive similarity coefficients of Yodzis & Winemiller (1999).cFig. 9. (a) A non-metric-multi-dimensional scaling of all nodes in the St. Marks seagrass ecosystem carbon-flow foodweb using REGE-derived isotrophic classes from the cluster analysis, and (b) the image graph of the reduced complexitynetwork of the same food web. Colors were used to indentify class membership based on the R 2 regression analysis (see textand Fig. 8). In the image graph, the codes for blue class {A} are {4,10, 11, 16, 18, 20, 23, 24, 26, 29, 30, 32, 33, 35, 37, 38, 39,40, 41} and yellow class {B} are {7,9,13,15,17,19,21,22,25,27,28,31,50}. See Appendix B for a complete list of compartmentidentification code names.

Fig. 6.Fig. 9.doi: 10.1006/jtbi.2003.3147J. J. LUCZKOVICH ET AL.

DEFINITION OF TROPHIC ROLE SIMILARITY 315turn preyed upon by the zooplankton (white),fishes and invertebrates in the blue and purplegroups, and birds in the magenta and redgroups. Yellow group members thus take partin many loops and cycles with blue, purple andwhite. In contrast, the blue group members(mostly fish and invertebrates) take part in acycle only with yellow and purple members. Bluegroup members feed upon planktonic producers,but only one class of benthic producers (Halodule,lime green group, and not the benthic algae,epiphytes or benthic bacteria, dark green group).Blue group members feed upon members of theirown class, as do yellow group members, but theydo not have members that are consumed byzooplankton (white), unlike yellow group members.Members of the blue group are consumedby members of the purple, magenta and redgroups and are thus somewhat similar to theyellow group members. The purple groupmembers are fishes that consumed the darkgreen group of benthic producers, but notplanktonic producers, nor the seagrass benthicproducers (lime green); they are different fromthe yellow and blue groups in this way. Inaddition, they do not have any feeding withintheir own group. The purple group also consumeswhite, yellow and blue group members.They are consumed only by the carnivorousbirds (red group), but not the herbivorous ducks(magenta). The herbivorous ducks (magenta) arein their own singleton class; no other bird groupfeeds on producers (but these ducks also takesome animal prey from the yellow and bluegroups, so they are not strictly speaking herbivorous).The red group (predatory birds) feedsfromthe yellow, blue and purple classes and hasone member that consumes another (raptorsconsume gulls). All compartments except for thebirds (red and magenta) are consumed by thebrown group, which includes detritus [sedimentPOC (51)].5. DiscussionA major goal in ecology is to develop acomprehensive model of the ecological niche orrole played by a species in a community orecosystem. One way that the ecologists haveworked toward this goal is through the analysisof trophic relations in food webs. Concepts suchas of trophospecies, trophic role, trophic group,trophic position, trophic level, guild and relatedconcepts pertaining to food webs have beendeveloped, all of which are attempts to reducetrophic complexity by identifying classes of‘‘isotrophic’’ species that occupy analogouspositions in the food web. Our contributionsto this endeavor are at two levels. First, wecontribute to the goal of comprehending trophicroles in ecosystems by providing a formalmathematical basis, expressing trophic role interms of equivalence relations and graph homomorphisms.In this respect, we extend the workof Yodzis & Winemiller (1999), who developedmeasures and algorithms for detecting structurein food webs, by locating their work within aframework of fundamental mathematical notionsof equivalence and isomorphism. In addition,we link their work to research in the socialsciences (Lorrain & White, 1971) and computersciences (Starke 1972), among other fields.Second, we introduce a specific model of foodweb structure F regular equivalence F whichwe believe captures the notion of trophic rolebetter than other formal models, such as thoseoffered by Yodzis & Winemiller (1999) andHirata & Ulanowicz (1985). The recursive natureof the definition might suggest that computationof the model would be difficult, but in fact veryefficient algorithms exist, such as REGE, whichprovide a measure of the degree of equivalenceamong all pairs of species, and these algorithmsare widely available (Borgatti et al., 1999). Wehave illustrated our approach here using twofood webs fromthe literature, namely theMalaysian pitcher plant insect topological foodweb and the St. Marks seagrass ecosystemcarbon-flow dynamic food web. We have alsoapplied the approach to rocky intertidal, desert,and rainforest food webs (Johnson et al. 2000).The regular equivalence model we introducehere contrasts with structural equivalence, inwhich nodes are equivalent to the extent theyhave the same kind of ties to the sameindividuals (rather than classes of individuals).We show that the additive trophic similaritymeasure recently developed by Yodzis & Winemiller(1999) is a measure of structural equivalence.Alternative measures of structural

316J. J. LUCZKOVICH ET AL.equivalence, based on euclidean distances andcorrelations, as well alternative algorithms, arewell known in the social sciences and areavailable in standard software packages. Structuralequivalence is ideally suited for certainanalytical tasks. Structurally equivalent speciesare literally are structurally indistinguishable,tied to the same others in the same ways.Consequently, for all structural ecologicalprocesses F by which we mean processes whoseoutcomes are principally determined by thestructure of the trophic network F we canexpect structurally equivalent species to havesimilar outcomes, or common fate (Homans,1950). Thus, in trophic impact studies, anythingthat happens to a species of predator or prey isexpected to happen to all other species in thesame structural equivalence class, because theyhave exactly the same prey or predators. In thisrespect, structural equivalence may have anadvantage over maximal regular equivalence, asit is unclear if or under what conditions thecommon fate hypothesis can be applied toregular equivalence classes.The disadvantage of structural equivalence isthat it cannot recognize the structural similarityof species that share no predators or prey.Consequently, species in clearly analogous positions,which are functionally equivalent, are seenas completely dissimilar. For example, in theMalaysian insect food web, food sources that aresimilar (organic debris and recently drownedinsects) were not regarded as similar by thestructural method of Yodzis & Winemiller(1999), yet they clearly are similar in that theyconsume nothing and are eaten by similarspecies. Regular equivalence solves this problemof not detecting similarity in trophic role whenspecies share no or very few prey or predators.Regular equivalence can be used to findspecies playing similar roles in spatially andtemporally separate food webs in which speciesdo not consume the exact same prey, possiblyopening the door for new kinds of comparativeresearch. For example, in Jackson et al. (2001),the authors proposed that humans have alteredcoastal ecosystems by fishing and huntingactivities that have caused long-lasting foodweb changes. In the Pacific kelp forest foodweb, when sea otters were extirpated, there was adecades-long lag response in the overgrazingeffect of sea urchins, their primary prey. The lagwas due to the fact that other unexploited speciessuch as abalones and spiny lobsters ‘‘y ofa similar trophic level assumed the ecologicalroles of overfished speciesy (emphasis ours)(p. 629)’’. Our regular equivalence analysiswould allow a quantitative assessment of thesimilarity of these species’ trophic roles, evenwhen they did not prey on the same species andwere preyed upon by different species. Thus, wecan create a simple model of trophically redundantrelationships when they occur in thesame food web at different points in time, oreven in spatially separate food webs. Whilestructural equivalence may also be used in suchan example, it would be better for studyingdirect trophic impacts and modeling directcompetition between species, whereas regularequivalence could be used to model the potentialfor species to overlap in trophic role, shouldtheir feeding habits shift as suggested by Jacksonet al. (2001).The approach we have outlined has someadditional benefits in quantifying trophic rolesand displaying reduced complexity food webs.Like the approach of Yodzis & Winemiller(1999), we have proposed a method that willfind similarity among species based on theirrelationships with predators and prey. This thenforms a new basis for aggregating species.The problemof how to aggregate nodes in anetwork of interactions so as to eliminate detailwhile retaining structure is universal in systemmodeling. In fact, this issue has been formallyapproached previously in ecological networkanalysis. A method for network aggregationwas proposed by Hirata & Ulanowicz (1985), inwhich a 17-compartment carbon-flow networkof a tidal marsh creek was aggregated into aseven-compartment system by minimizing theloss of mutual information that occurred duringa sequence of aggregation steps. Their graph ofthe seven-compartment ecosystem flow diagramwas essentially an image graph, but the aggregationswere based on minimizing the decreasein network mutual information, an empiricalstatistic they computed, which is based onnetwork redundancy measures. Although thegeneral approach is similar to ours, Hirata &

DEFINITION OF TROPHIC ROLE SIMILARITY 317Ulanowicz’s (1985) node aggregation approachis quite different in mathematical detail fromours, and we plan to compare them systematicallyin future work.Although we have emphasized the differencesbetween our maximal regular equivalenceapproach and the structural equivalence approachesof Yodzis & Winemiller (1999), it isimportant to remember that there is a fundamentalcommonality as well. As noted earlier,structural equivalence is a member of the regularequivalence family, and any species that arestructurally equivalent are necessarily regularlyequivalent, as we illustrated in the Malaysianpitcher plant food web example. In that example,four pairs of nodes with the exact same predatorand prey relationships were found to be structurallyequivalent, and they were also exactlyregularly equivalent. The similarity of trophicrelations between any two nodes could bedetermined using the structural methods alonewhen their nearest-neighbor links were considered;but when similarity among nodes wasestimated while considering links that were morethan a one link away fromthe nodes, suchsimilarity could be estimated using REGEcoefficients. If the equivalence approach provideslenses with which to view food webs, thestructural equivalence provides finer resolutionat the expense of the big picture, while regularequivalence provides the big picture at theexpense of only a few details.A key area for future research concerns thesignificance of isotrophic classes F whetherdefined by structural equivalence or regularequivalence F for the species that comprisethemand for the webs as a whole. Yodzis &Winemiller (1999) evaluate the relative merits oftwo different measures of trophic similarity byusing cophenetic correlations. But a copheneticcorrelation only measures the extent to which agiven similarity matrix can be nicely representedby a dendrogram(i.e. a hierarchical clusteringprocedure). It is not an external criterion ofvalidity. What is needed is empirical researchrelating membership in isotrophic classes tooutcome criteria, such as similar reactionsto environmental changes, similar effects onprey species, similar responses to the removalor addition of predator species, and so on. Onlythen can we really determine whether onedefinition of trophic grouping is more usefulthan other (and whether any kind of trophicgrouping is useful at all).This work was supported by a BiocomplexityIncubation grant fromthe National Science Foundation(SES0083508). The East Carolina UniversityDivision of Research and Graduate Studies supportedthe authors in the preparation of this paper andprovided publication costs. We thank P. Stiling fortaxonomic assistance and verification of the insect foodweb data. B. Patten is acknowledged for providinghelpful criticisms. We also thank R. Ulanowicz andR. Bernard for reviewing the manuscript.REFERENCESAdams, S. M., Kimmel, B.L.&Ploskey, G. R. (1983).Sources of organic matter for reservoir fish production:a trophic-dynamics analysis. Can. J. Fish. Aquat. Sci. 40,1480–1495.Baird, D., Luczkovich, J.J.&Christian, R. R. (1998).Assessment of spatial and temporal variability inecosystemattributes of the St. Marks National WildlifeRefuge, Apalachee Bay, Florida. Estuarine Coastal ShelfSci. 47, 329–349, doi:10.1006/ecss.1998.0360.Batagelj, V.&Mrvar, A. (1999). Pajek, version 0.83,available on-line at http://vlado.fmf.uni-lj.si/pub/networks/pajek.Beaver, R. A. (1985). Geographical variation in food webstructure in Nepenthes pitcher plants. Ecol. Entomol. 10,241–248.Borgatti, S.P.&Everett, M. G. (1989). The class of allregular equivalences: algebraic structure and computation.Soc. Networks 11, 65–88.Borgatti,S.P.&Everett, M. G. (1993). Two algorithmsfor computing regular equivalence. Soc. Networks 16,43–55.Borgatti, S. P., Everett, M.G.&Shirey, P. (1990). LSsets, lambda sets, and other cohesive subsets. Soc.Networks 12, 337–357.Borgatti, S. P., Everett, M.G.&Freeman, L. (1999).UCINET V. Software for Social Network Analysis.Harvard, MA: Analytical Technologies.Boyd, J. P. (1969). The algebra of group kinship. J. Math.Psychol. 6, 139–167.Christian, R.R.&Luczkovich, J. J. (1999). Organizingand understanding a winter’s seagrass foodweb networkthrough effective trophic levels. Ecol. Model. 117, 99–124.Cohen, J. E. & Newman, C. M. (1985). A stochastictheory of community food webs. I. Models andaggregated data. Proc. R. Soc. London Biol. Sci. 224,421–448.Cousins, S. H. (1985). Ecologists build pyramids again.New Sci. 106, 50–54.Cousins, S. H. (1987). The decline of the trophic levelconcept. Trends Ecol. Evol. 2, 312–316.Elton, C. S. (1927). Animal Ecology. London: Sidgwickand Jackson.

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DEFINITION OF TROPHIC ROLE SIMILARITY 319(13) micro-fauna, (14) meiofauna, (15) depositfeedingamphipods, (16) detritus-feeding crustaceans,(17) hermit crab, (18) spider crab, (19)omnivorous crabs, (20) blue crab, (21) isopod,(22) brittle stars, (23) herbivorous shrimp, (24)predatory shrimp, (25) deposit-feeding gastropod,(26) deposit-feeding polychaetes, (27) predatorypolychaetes, (28) predatory gastropod,(29) epiphyte-grazing gastropod, (30) othergastropods, (31) catfish and stingrays, (32)tonguefish , (33) gulf flounder and needle fish,(34) southern hake and sea robins, (35)atlantic silverside and bay anchovies, (36)gobies and blennies, (37) pinfish, (38) spot, (39)pipefish and seahorses, (40) sheepsheadminnow, (41) red drum, (42) killifish, (43)herbivorous ducks, (44) benthos-eating birds,(45) fish-eating birds, (46) fish and crustaceaneating birds, (47) gulls, (48) raptors, (49)dissolved organic carbon (DOC), (50) suspendedparticulate organic carbon, (51) sedimentPOC.