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

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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
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