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

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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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