Glossary - tuvalu.santafe.edutuvalu.santafe.edu/files/Teachers 2010/Network... · Food Webs F 3661 FoodWebs JENNIFER A. DUNNE1,2 1 Santa FeInstitute, SantaFe,USA 2 Pacific Ecoinformatics

Food Webs F 3661

FoodWebsJENNIFER A DUNNE121 Santa Fe Institute Santa Fe USA2 Pacific Ecoinformatics and Computational Ecology LabBerkeley USA

Article Outline

GlossaryDefinition of the SubjectIntroduction Food Web Concepts and DataEarly FoodWeb Structure ResearchFood Web PropertiesFood Webs Compared to Other NetworksModels of Food Web StructureStructural Robustness of FoodWebsFood Web DynamicsEcological Networks

Future DirectionsBibliography

Glossary

Connectance(C) The proportion of possible links ina food web that actually occur There are many algo-rithms for calculating connectance The simplest andmost widely used algorithm sometimes referred to asldquodirected connectancerdquo is links per species2 (LS2)where S2 represents all possible directed feeding inter-actions among S species and L is the total number ofactual feeding links Connectance ranges from 003to 03 in food webs with a mean of 010 to 015

Consumer-resource interactions A generic way of refer-ring to a wide variety of feeding interactions suchas predator-prey herbivore-plant or parasite-host in-teractions Similarly ldquoconsumerrdquo refers generically toanything that consumes or preys on something elseand ldquoresourcerdquo refers to anything that is consumed orpreyed upon Many taxa are both consumers and re-sources within a particular food web

Food web The network of feeding interactions among di-verse co-occurring species in a particular habitat

Trophic species (S) Defined within the context of a par-ticular food web a trophic species is comprised of a setof taxa that share the same set of consumers and re-sources A particular trophic species is represented bya single node in the network and that node is topologi-cally distinct from all other nodes ldquoTrophic speciesrdquo isa convention introduced to minimize bias due to un-even resolution in food web data and to focus analysisand modeling on functionally distinct network com-ponents S is used to denote the number of trophicspecies in a food web The terms ldquotrophic speciesrdquoldquospeciesrdquo and ldquotaxardquo will be used somewhat inter-changeably throughout this article to refer to nodes ina food web ldquoOriginal speciesrdquo will be used specificallyto denote the taxa found in the original dataset priorto trophic species aggregation

Definition of the Subject

Food webs refer to the networks of feeding (ldquotrophicrdquo)interactions among species that co-occur within partic-ular habitats Research on food webs is one of the fewsubdisciplines within ecology that seeks to quantify andanalyze direct and indirect interactions among diversespecies rather than focusing on particular types of taxaFood webs ideally represent whole communities includingplants bacteria fungi invertebrates and vertebrates Feed-ing links represent transfers of biomass and encompass

3662 F Food Webs

a variety of trophic strategies including detritivory her-bivory predation cannibalism and parasitism At the baseof every food web are one or more types of autotrophsorganisms such as plants or chemoautotrophic bacteriawhich produce complex organic compounds from an ex-ternal energy source (e g light) and simple inorganic car-bon molecules (e g CO2) Food webs also have a detritalcomponentmdashnon-living particulate organic material thatcomes from the body tissues of organisms Feeding-medi-ated transfers of organic material which ultimately traceback to autotrophs or detritus via food chains of varyinglengths provide the energy organic carbon and nutrientsnecessary to fuel metabolism in all other organisms re-ferred to as heterotrophs

While food webs have been a topic of interest in ecol-ogy for many decades some aspects of contemporary foodweb research fall within the scope of the broader cross-disciplinary research agenda focused on complex ldquoreal-worldrdquo networks both biotic and abiotic [283101] Usingthe language of graph theory and the framework of net-work analysis species are represented by vertices (nodes)and feeding links are represented by edges (links) betweenvertices As with any other network the structure and dy-namics of food webs can be quantified analyzed andmod-eled Links in food webs are generally considered directedsince biomass flows from a resource species to a consumerspecies (A B) However trophic links are sometimestreated as undirected since any given trophic interactionalters the population and biomass dynamics of both theconsumer and resource species ( A $ B) The types ofquestions explored in food web research range from ldquoDofood webs from dierent habitats display universal topo-logical characteristics and how does their structure com-pare to that of other types of networksrdquo to ldquoWhat fac-tors promote dierent aspects of stability of complex foodwebs and their components given internal dynamics andexternal perturbationsrdquo Two fundamental measures usedto characterize food webs are S the number of species ornodes in a web and C connectancemdashthe proportion ofpossible feeding links that are actually realized in a web(C D LS2 where L is the number of observed directedfeeding links and S2 is the number of possible directedfeeding interactions among S taxa)

This article focuses on research that falls at the inter-section of food webs and complex networks with an em-phasis on network structure augmented by a brief discus-sion of dynamics This is a subset of a wide variety of eco-logical research that has been conducted on feeding inter-actions and food webs Refer to the ldquoBooks and Reviewsrdquoin the bibliography for more information about a broaderrange of research related to food webs

Introduction FoodWeb Concepts and Data

The concept of food chains (e g grass is eaten bygrasshoppers which are eaten by mice which are eaten byowls A B C D) goes back at least several hun-dred years as evidenced by two terrestrial and aquatic foodchains briefly described by Carl Linnaeus in 1749 [42] Theearliest description of a food web may be the mostly detri-tal-based feeding interactions observed by Charles Darwinin 1832 on the island of St Paul which had only two birdspecies (Darwin 1939 as reported by Egerton [42])

By the side of many of these [tern] nests a smallflying-fish was placed which I suppose had beenbrought by the male bird for its partner quicklya large and active crab (Craspus) which inhabits thecrevices of the rock stole the fish from the side ofthe nest as soon as we had disturbed the birds Nota single plant not even a lichen grows on this is-land yet it is inhabited by several insects and spi-ders The following list completes I believe the ter-restrial fauna a species of Feronia and an acaruswhich must have come here as parasites on thebirds a small brown moth belonging to a genusthat feeds on feathers a staphylinus (Quedius) anda woodlouse from beneath the dung and lastly nu-merous spiders which I suppose prey on these smallattendants on and scavengers of the waterfowl

The earliest known diagrams of generalized food chainsand food webs appeared in the late 1800s and diagrams ofspecific food webs began appearing in the early 1900s forexample the network of insect predators and parasites oncotton-feeding weevils (ldquothe boll weevil complexrdquo [87])By the late 1920s diagrams and descriptions of terres-trial and marine food webs were becoming more common(e g Fig 1 from [103] see also [48104]) Charles Eltonintroduced the terms ldquofood chainrdquo and ldquofood cyclerdquo in hisclassic early textbook Animal Ecology [43] By the timeEugene Odum published a later classic textbook Funda-mentals of Ecology [84] the term ldquofood webrdquo was startingto replace ldquofood cyclerdquo

From the 1920s to the 1980s dozens of system-specificfood web diagrams and descriptions were published aswell as some webs that were more stylized (e g [60]) andthat quantified link flows or species biomasses In 1977Joel Cohen published the first comparative studies of em-pirical food web network structure using up to 30 foodwebs collected from the literature [2324] To standardizethe data he transformed the diagrams and descriptions ofwebs in the literature into binarymatrices withm rows andn columns [24] Each column is headed by the number of

Food Webs F 3663

Food Webs Figure 1A diagram of a terrestrial Arctic food web with a focus on nitrogen cycling for Bear Island published in 1923 [103]

one of the consumer taxa in a particular web and each rowis headed by the number of one of the resource taxa forthat web If wij represents the entry in the ith row and thejth column it equals 1 if consumer j eats resource i or 0 if jdoes not eat i This matrix-based representation of data isstill often used particularly in a full S by S format (whereS is the number of taxa in the web) but for larger datasetsa compressed two- or three-column notation for observedlinks is more ecient (Fig 2)

By the mid-1980s those 30 initial webs had expandedinto a 113-web catalog [30] which included webs mostlyculled from the literature dating back to the 1923 BearIsland food web ([103] Fig 1) However it was appar-ent that there were many problems with the data Mostof the 113 food webs had very low diversity compared tothe biodiversity known to be present in ecosystems witha range of only 5 to 48 species in the original datasets and3 to 48 trophic species This low diversity was largely dueto very uneven resolution and inclusion of taxa in mostof these webs The webs were put together in many dif-ferent ways and for various purposes that did not includecomparative quantitative research Many types of organ-isms were aggregated underrepresented or missing alto-gether and in a few cases animal taxa had no food chainsconnecting them to basal species In addition cannibalis-tic links were purged when the webs were compiled intothe 113-web catalog To many ecologists these food webslooked like little more than idiosyncratic cartoons of muchricher and more complex species interactions found in

natural systems and they appeared to be an extremely un-sound foundation on which to build understanding andtheory [8692]

Another catalog of ldquosmallrdquo webs emerged in the late1980s a set of 60 insect-dominated webs with 2 to 87 orig-inal species (mean = 22) and 2 to 54 trophic species (mean= 12) [102] Unlike the 113-web catalog these webs arehighly taxonomically resolved mostly to the species levelHowever they are still small due to their focus in mostcases on insect interactions in ephemeral microhabitatssuch as phytotelmata (i e plant-held aquatic systems suchas water in tree holes or pitcher plants) and singular detri-tal sources (e g dung paddies rotting logs animal car-casses) Thus while the 113-web catalog presented foodwebs for communities at fairly broad temporal and spa-tial scales but with low and uneven resolution the 60-webcatalog presented highly resolved but very small spatialand temporal slices of broader communities These twovery dierent catalogs were compiled into ECOWeB theldquoEcologists Co-Operative Web Bankrdquo a machine readabledatabase of food webs that was made available by Joel Co-hen in 1989 [26] The two catalogs both separately andtogether as ECOWeB were used in many studies of regu-larities in food web network structure as discussed in thenext Sect ldquoEarly Food Web Structure Researchrdquo

A new level of detail resolution and comprehensive-ness in whole-community food web characterization waspresented in two seminal papers in 1991 Gary Polis [92]published an enormous array of data for taxa found in the

3664 F Food Webs

Food Webs Figure 2Examples of formats for standardized notation of binary food webdata A hypothetical webwith 6 taxa and 12 links is usedNumbers1ndash6 correspond to the different taxa a Partial matrix format the 1s or 0s inside the matrix denote the presence or absence of a feed-ing link between a consumer (whose numbers 3ndash6 head columns) and a resource (whose numbers 1ndash6 head rows) b Full matrixformat similar to a but all 6 taxa are listed at the heads of columns and rows c Two-column format a consumerrsquos number appearsin the first column and one of its resourcersquos numbers appears in the second column d Three-column format similar to c but wherethere is a third number the second and third numbers refer to a range of resource taxa In this hypothetical web taxa numbers 1 and2 are basal taxa (i e taxa that do not feed on other taxamdashautotrophs or detritus) and taxa numbers 3 5 and 6 have cannibalisticlinks to themselves

Coachella Valley desert (California) Over two decades hecollected taxonomic and trophic information on at least174 vascular plant species 138 vertebrate species 55 spiderspecies thousands of insect species including parasitoidsand unknown numbers of microorganisms acari and ne-matodes He did not compile a complete food web includ-ing all of that information but instead reported a numberof detailed subwebs (e g a soil web a scorpion-focusedweb a carnivore web etc) each of which was more di-verse than most of the ECOWeB webs On the basis ofthe subwebs and a simplified aggregated 30-taxa web ofthe whole community he concluded that ldquo most cata-logued webs are oversimplified caricatures of actual com-munities [they are]grossly incomplete representations ofcommunities in terms of both diversity and trophic connec-tionsrdquo

At about the same time Neo Martinez [63] publisheda detailed food web for Little Rock Lake (Wisconsin) thathe compiled explicitly to test food web theory and patterns(see Sect ldquoEarly Food Web Structure Researchrdquo) By piec-ing together diversity and trophic information from mul-tiple investigators actively studying various types of taxa inthe lake he was able to put together a relatively completeand highly resolved food web of 182 taxa most identifiedto the genus species or ontogentic life-stage level includ-ing fishes copepods cladocera rotifers diptera and otherinsects mollusks worms porifera algae and cyanobac-teria In later publications Martinez modified the origi-

nal dataset slightly into one with 181 taxa The 181 taxaweb aggregates into a 92 trophic-species web with nearly1000 links among the taxa (Fig 3) This dataset and theaccompanying analysis set a new standard for food webempiricism and analysis It still stands as the best whole-community food web compiled in terms of even detailedcomprehensive resolution

Since 2000 the use of the ECOWeB database for com-parative analysis and modeling has mostly given way toa focus on a smaller set of more recently published foodwebs [10373999110] These webs available throughwwwfoodwebsorg or from individual researchers arecompiled for particular broad-scale habitats such as StMarkrsquos Estuary [22] Little Rock Lake [63] the island ofSt Martin [46] and the Northeast US Marine Shelf [61]Most of the food webs used in contemporary compara-tive research are still problematicmdashwhile they generallyare more diverse andor evenly resolved than the earlierwebs most could still be resolved more highly and evenlyAmong several issues organisms such as parasites are usu-ally left out (but see [51596774]) microorganisms are ei-ther missing or highly aggregated and there is still a ten-dency to resolve vertebrates more highly than lower levelorganisms An important part of future food web researchis the compilation of more inclusive evenly resolved andwell-defined datasetsMeanwhile the careful selection andjustification of datasets to analyze is an important part ofcurrent research that all too often is ignored

Food Webs F 3665

Food Webs Figure 3Food web of Little Rock Lake Wisconsin [63] 997 feeding links among 92 trophic species are shown Image produced with Food-Web3D written by RJ Williams available at the Pacific Ecoinformatics and Computational Ecology Lab (wwwfoodwebsorg)

How exactly are food web data collected In generalthe approach is to compile as complete a species list aspossible for a site and then to determine the diets ofeach species present at that site However researchers havetaken a number of dierent approaches to compiling foodwebs In some cases researchers base their food webs onobservations they make themselves in the field For ex-ample ecologists in New Zealand have characterized thestructure of stream food webs by taking samples fromparticular patches in the streams identifying the speciespresent in those samples taking several individuals of eachspecies present and identifying their diets through gut-content analysis [106] In other cases researchers compilefood web data by consulting with experts and conductingliterature searches For example Martinez [63] compiledthe Little Rock Lake (WI) food web by drawing on the ex-pertise of more than a dozen biologists who were special-ists on various types of taxa and who had been working atLittle Rock Lake for many years Combinations of thesetwo approaches can also come into playmdashfor examplea researcher might compile a relatively complete specieslist through field-based observations and sampling andthen assign trophic habits to those taxa through a combi-nation of observation consulting with experts and search-ing the literature and online databases

It is important to note that most of the webs usedfor comparative research can be considered ldquocumulativerdquowebs Contemporary food web data range from time- andspace-averaged or ldquocumulativerdquo (e g [63]) to more finely

resolved in time (e g seasonal websmdash[6]) andor space(e g patch-scale websmdash[106] microhabitat websmdash[94])The generally implicit assumption underlying cumulativefood web data is that the set of species in question co-existwithin a habitat and individuals of those species have theopportunity over some span of time and space to inter-act directly To the degree possible such webs documentwho eats whom among all species within a macrohabitatsuch as a lake or meadow over multiple seasons or yearsincluding interactions that are low frequency or repre-sent a small proportion of consumption Such cumulativewebs are used widely for comparative research to look atwhether there are regularities in food web structure acrosshabitat (see Sect ldquoFood Webs Compared to Other Net-worksrdquo and Sect ldquoModels of Food Web Structurerdquo) Morenarrowly defined webs at finer scales of time or space orthat utilize strict evidence standards (e g recording linksonly through gut content sampling) have been useful forcharacterizing how such constraints influence perceivedstructure within habitats [105106] but are not used asmuch to look for cross-system regularities in trophic net-work structure

Early FoodWeb Structure Research

The earliest comparative studies of food web structurewere published by Joel Cohen in 1977 Using data fromthe first 30-web catalog one study focused on the ratioof predators to prey in food webs [23] and the other in-

3666 F Food Webs

vestigated whether food webs could be represented by sin-gle dimension interval graphs [24] a topic which contin-ues to be of interest today (see Sect ldquoFood Webs Com-pared to Other Networksrdquo) In both cases he found regu-laritiesmdash(1) a ratio of prey to predators of 34 regard-less of the size of the web and (2) most of the webs areinterval such that all species in a food web can be placedin a fixed order on a line such that each predatorrsquos set ofprey forms a single contiguous segment of that line Theprey-predator ratio paper proved to be the first salvo ina quickly growing set of papers that suggested that a va-riety of food web properties were ldquoscale-invariantrdquo In itsstrong sense scale invariance means that certain prop-erties have constant values as the size (S) of food webschange In its weak sense scale-invariance refers to prop-erties not changing systematically with changing S Otherscale-invariant patterns identified include constant pro-portions of top species (Top species with no predators)intermediate species (Int species with both predators andprey) and basal species (Bas species with no prey) col-lectively called ldquospecies scaling lawsrdquo [12] and constantproportions of T-I I-B T-B and I-I links between T Iand B species collectively called ldquolink scaling lawsrdquo [27]Other general properties of food webs were thought to in-clude food chains are short [31435089] cyclingloopingis rare (e g A B C A [28]) compartments orsubwebs with many internal links that have few links toother subwebs are rare [91] omnivory or feeding at morethan one trophic level is uncommon [90] and webs tendto be interval with instances of intervality decreasing as Sincreases [2429116]Most of these patterns were reportedfor the 113-web catalog [31] and some of the regularitieswere also documented in a subset the 60 insect-dominatedwebs [102]

Another related prominent line of early comparativefood web research was inspired by Bob Mayrsquos work fromthe early 1970s showing that simple abstract communitiesof interacting species will tend to transition sharply fromlocal stability to instability as the complexity of the systemincreasesmdashin particular as the number of species (S) theconnectance (C) or the average interaction strength (i) in-crease beyond critical values [6970] He formalized this asa criterion that ecological communities near equilibriumwill tend to be stable if i(SC)12 lt 1 This mathematicalanalysis flew in the face of the intuition of many ecolo-gists (e g [44506284]) who felt that increased complex-ity (in terms of greater numbers of species and links be-tween them) in ecosystems gives rise to stability

Mayrsquos criterion and the general question of how di-versity is maintained in communities provided a frame-work within which to analyze some readily accessible em-

pirical data namely the numbers of links and species infood webs Assuming that average interaction strength (i)is constant Mayrsquos criterion suggests that communities canbe stable given increasing diversity (S) as long as con-nectance (C) decreases This can be empirically demon-strated using food web data in three similar ways by show-ing that 1) C hyperbolically declines as S increases so thatthe product SC remains constant 2) the ratio of links tospecies (LS) also referred to as link or linkage densityremains constant as S increases or 3) L plotted as a func-tion of S on a log-log graph producing a power-law rela-tion of the form L D ˛Sˇ displays an exponent of ˇ D 1(the slope of the regression) indicating a linear relation-ship between L and S These relationships were demon-strated across food webs in a variety of studies (see detailedreview in [36]) culminating with support from the 113-web catalog and the 60 insect-dominated web catalog Co-hen and colleagues identified the ldquolink-species scaling lawrdquoof LS 2 using the 113 web catalog (i e there are twolinks per species on average in any given food web regard-less of its size) [2830] and SCwas reported as ldquoroughly in-dependent of species numberrdquo in a subset of the 60 insect-dominated webs [102]

However these early conclusions about patterns offood web structure began to crumble with the advent ofimproved data and new analysis methods that focused onthe issues of species aggregation sampling eort and sam-pling consistency [36] Even before there was access toimproved data Tom Schoener [93] set the stage for cri-tiques of the conventional paradigm in his Ecological So-ciety of America MacArthur Award lecture in which heexplored the ramifications of a simple conceptual modelbased on notions of ldquogeneralityrdquo (what Schoener referredto as ldquogeneralizationrdquo) and ldquovulnerabilityrdquo He adopted thebasic notion underlying the ldquolink-species scaling lawrdquo thathow many dierent taxa something can eat is constrainedwhich results in the number of resource taxa per consumertaxon (generality) holding relatively steadywith increasingS However he further hypothesized that the ability of re-source taxa to defend again consumers is also constrainedsuch that the number of consumer taxa per resource taxon(vulnerability) should increase with increasing S A majorconsequence of this conceptual model is that total links perspecies (LS which includes links to resources and con-sumers) and most other food web properties should dis-play scale-dependence not scale-invariance A statisticalreanalysis of a subset of the 113-web catalog supported thiscontention as well as the basic assumptions of his concep-tual model about generality and vulnerability

Shortly thereafter more comprehensive detaileddatasets like the ones for Coachella Valley [92] and Little

Food Webs F 3667

Rock Lake [63] began to appear in the literature Theseand other new datasets provided direct empirical coun-terpoints to many of the prevailing notions about foodwebs their connectance and links per species were muchhigher than expected from the ldquolink-species scaling lawrdquofood chains could be quite long omnivory and cannibal-ism and looping could be quite frequent etc In additionanalyzes such as the one byMartinez [63] in which he sys-tematically aggregated the Little Rock Lake food web taxaand links to generate small webs that looked like the earlierdata demonstrated that ldquomost published food web patternsappear to be artifacts of poorly resolved datardquo Compara-tive studies incorporating newly available data further un-dermined the whole notion of ldquoscale invariancerdquo of mostproperties particularly LS(e g [6566])

For many researchers the array of issues brought tolight by the improved data and more sophisticated ana-lyzes was enough for them to turn their back on struc-tural food web research A few hardy researchers soughtto build new theory on top of the improved data ldquoCon-stant connectancerdquo was suggested as an alternative hy-pothesis to constant LS (the ldquolink-species scaling lawrdquo)based on a comparative analysis of the relationship ofL to S across a subset of available food webs includingLittle Rock Lake [64] The mathematical dierence be-tween constant C and constant LS can be simply statedusing a log-log graph of links as a function of species(Fig 4) If a power law exists of the form L D ˛Sˇ in thecase of the link-species scaling law ˇ D 1 which meansthat L D ˛S LS D ˛ indicating constant LS In thecase of constant connectance ˇ D 2 and thus L D ˛S2LS2 D ˛ indicating constant C (LS2) Constant con-nectance means that LS increases as a fixed proportion ofS One ecological interpretation of constant connectanceis that consumers are likely to exploit an approximatelyconstant fraction of available prey species so as diversityincreases links per species increases [108]

Given the L D ˛Sˇ framework ˇ D 2 was reportedfor a set of 15 webs derived from an English pond [108]and ˇ D 19 for a set of 50 Adirondack lakes [65] sug-gesting connectance may be constant across webs withina habitat or type of habitat Across habitats the picture isless clear While ˇ D 2 was reported for a small subset ofthe 5 most ldquocrediblerdquo food webs then available from dier-ent habitats [64] several analyzes of both the old ECOWeBdata and the more reliable newer data suggest that the ex-ponent lies somewhere between 1 and 2 suggesting that Cdeclines non-linearly with S (Fig 4 [273036647993])For example Schoenerrsquos reanalysis of the 113-web catalogsuggested thatˇ D 15 indicating that L23 is proportionalto S A recent analysis of 19 recent trophic-species food

Food Webs Figure 4The relationship of links to species for 19 trophic-species foodwebs froma variety of habitats (black circles) The solid line showsthe log-log regression for the empirical data the dashed lineshows the prediction for constant connectance and the dottedline shows the prediction for the link-species scaling law (repro-duced from [36] Fig 1)

webs with S of 25 to 172 also reported ˇ D 15 with muchscatter in the data (Fig 4)

A recent analysis has provided a possible mechanisticbasis for the observed constrained variation in C ( 003to 03 in cumulative community webs) as well as the scal-ing of C with S implied by ˇ intermediate between 1 and2 [10] A simple diet breadth model based on optimal for-aging theory predicts both of these patterns across foodwebs as an emergent consequence of individual foragingbehavior of consumers In particular a contingency modelof optimal foraging is used to predict meandiet breadth forS animal species in a food web based on three parametersfor an individual of species j (1) net energy gain from con-sumption of an individual of species i (2) the encounterrate of individuals of species i and (3) the handling timespent attacking an individual of species i This allows es-timation of C for the animal portion of food webs oncedata aggregation and cumulative sampling well-knownfeatures of empirical datasets are taken into account Themodel does a good job of predicting values of C observedin empirical food webs and associated patterns of C acrossfood webs

FoodWeb Properties

Food webs have been characterized by a variety of prop-erties or metrics several of which have been mentionedpreviously (Sect ldquoEarly Food Web Structure Researchrdquo)Many of these properties are quantifiable just using the ba-sic network structure (ldquotopologyrdquo) of feeding interactionsThese types of topological properties have been used to

3668 F Food Webs

evaluate simple models of food web structure (Sect ldquoFoodWeb Propertiesrdquo) Any number of properties can be cal-culated on a given networkmdashecologists tend to focus onproperties that are meaningful within the context of eco-logical research although other properties such as pathlength (Path) and clustering coecient (Cl) have beenborrowed from network research [109] Examples of sev-eral types of food web network structure properties withcommon abbreviations and definitions follow

Fundamental Properties These properties characterizevery simple overall attributes of food web network struc-ture

S number of nodes in a food webL number of links in a food webLS links per speciesC or LS2 connectance or the proportion of possible

links that are realized

Types of Taxa These properties characterize what pro-portion or percentage of taxa within a food web fall intoparticular topologically defined roles

Bas percentage of basal taxa (taxa without resources)Int percentage of intermediate taxa (taxa with both con-

sumers and resources)Top percentage of top taxa (taxa with no consumers)Herb percentage of herbivores plus detritivores (taxa that

feed on autotrophs or detritus)Can percentage of cannibals (taxa that feed on their own

taxa)Omn percentage of omnivores (taxa that feed that feed

on taxa at dierent trophic levelsLoop percentage of taxa that are in loops food chains in

which a taxon occur twice (e g A B C A)

Network Structure These properties characterize otherattributes of network structure based on how links are dis-tributed among taxa

TL trophic level averaged across taxa Trophic level rep-resents how many steps energy must take to get froman energy source to a taxon Basal taxa haveTL = 1 andobligate herbivores are TL = 2 TL can be calculatedusing many dierent algorithms that take into accountmultiple food chains that can connect higher level or-ganisms to basal taxa (Williams and Martinez 2004)

ChLen mean food chain length averaged over all speciesChSD standard deviation of ChLenChNum log number of food chains

LinkSD normalized standard deviation of links ( linksper taxon)

GenSD normalized standard deviation of generality (resources per taxon)

VulSD normalized standard deviation of vulnerability (consumers per taxon)

MaxSim mean across taxa of the maximum trophic sim-ilarity of each taxon to other taxa

Ddiet the level of diet discontinuitymdashthe proportion oftriplets of taxa with an irreducible gap in feeding linksover the number of possible triplets [19]mdasha local esti-mate of intervality

Cl clustering coecient (probability that two taxa linkedto the same taxon are linked)

Path characteristic path length the mean shortest set oflinks (where links are treated as undirected) betweenspecies pairs

The previous properties (most of which are describedin [110] and [39]) each provide a single metric that charac-terizes some aspect of food web structure There are otherproperties such as Degree Distribution which are notsingle-number properties ldquoDegreerdquo refers to the numberof links that connect to a particular node and the de-gree distribution of a network describes (in the formatof a function or a graph) the total number of nodes ina network that have a given degree for each level of de-gree (Subsect ldquoDegree Distributionrdquo) In food web analy-sis LinkSD GenSD and VulSD characterize the variabil-ity of dierent aspects of degree distribution Many foodweb structure properties are correlated with each otherand vary in predictable ways with S andor C This pro-vides opportunities for topological modeling that are dis-cussed below (Sect ldquoModels of Food Web Structurerdquo)

In addition to these types of metrics based on networkswith unweighted links and nodes it is possible to calculatea variety of metrics for food webs with nodes andor linksthat are weighted by measures such as biomass numer-ical abundance frequency interaction strength or bodysize [11336781] However few food web datasets areldquoenrichedrdquo with such quantitative data and it remains tobe seen whether such approaches are primarily a toolfor richer description of particular ecosystems or whetherthey can give rise to novel generalities models and predic-tions One potential generality was suggested by a studyof interaction strengths in seven soil food webs where in-teraction strength reflects the size of the eects of specieson each otherrsquos dynamics near equilibrium Interactionstrengths appear to be organized such that long loops con-tain many weak links a pattern which enhances stabilityof complex food webs [81]

Food Webs F 3669

FoodWebs Compared to Other Networks

Small-World Properties

How does the structure of food webs compare to thatof other kinds of networks One common way that vari-ous networks have been compared is in terms of whetherthey are ldquosmall-worldrdquo networks Small-world networksare characterized by two of the properties described previ-ously characteristic path length (Path) and clustering co-ecient (Cl) [109] Most real-world networks appear tohave high clustering like what is seen on some types ofregular spatial lattices (such as a planar triangular latticewhere many of a nodersquos neighbors are neighbors of oneanother) but have short path lengths like what is seenon ldquorandom graphsrdquo (i e networks in which links aredistributed randomly) Food webs do display short pathlengths that are similar to what is seen in random webs(Table 1 [163778113]) On average taxa are about twolinks from other taxa in a food web (ldquotwo degrees of sep-arationrdquo) and path length decreases with increasing con-nectance [113]

However clustering tends to be quite low in manyfood webs closer to the clustering expected on a randomnetwork (Table 1) This relatively low clustering in foodwebs appears consistent with their small size comparedto most other kinds of networks studied since the ra-tio of clustering in empirical versus comparable random

Food Webs Table 1Topological properties of empirical and random food webs listed in order of increasing connectance Path refers to characteristicpath length and Cl refers to the clustering coefficient Pathr and Clr refer to the mean D and Cl for 100 random webs with the sameS and C Modified from [37] Table 1

Food Web S C(LS2) LS Path Pathr Cl Clr ClClrGrassland 61 0026 159 374 363 011 003 37

Scotch Broom 85 0031 262 311 282 012 004 30

Ythan Estuary 1 124 0038 467 234 239 015 004 38

Ythan Estuary 2 83 0057 476 220 219 016 006 27

El Verde Rainforest 155 0063 974 220 195 012 007 14

Canton Creek 102 0067 683 227 201 002 007 03

Stony Stream 109 0070 761 231 196 003 007 04

Chesapeake Bay 31 0071 219 265 240 009 009 10

St Marks Seagrass 48 0096 460 204 194 014 011 13

St Martin Island 42 0116 488 188 185 014 013 11

Little Rock Lake 92 0118 1084 189 177 025 012 21

Lake Tahoe 172 0131 2259 181 174 014 013 11

Mirror Lake 172 0146 2513 176 172 014 015 09

Bridge Brook Lake 25 0171 428 185 168 016 019 08

Coachella Valley 29 0312 903 142 143 043 032 13

Skipwith Pond 25 0315 788 133 141 033 033 10

networks increases linearly with the size of the network(Fig 5)

Degree Distribution

In addition to small-world properties many real-worldnetworks appear to display power-law degree distribu-tions [2] Whereas regular graphs have the same numberof links per node and random graphs display a Poissondegree distribution many empirical networks both bioticand abiotic display a highly skewed power-law (ldquoscale-freerdquo) degree distribution where most nodes have fewlinks and a few nodes have many links However someempirical networks display less-skewed distributions suchas exponential distributions [4] Most empirical food websdisplay exponential or uniform degree distributions notpower-law distributions [1637] and it has been suggestedthat normalized degree distributions in food webs followuniversal functional forms [16] although there is a quitea bit of scatter when a wide range of data are consid-ered (Fig 6 [37]) Variable degree distributions like whatis seen in individual food webs could result from sim-ple mechanisms For example exponential and uniformfood web degree distributions are generated by a modelthat combines (1) random immigration to local webs froma randomly linked regional set of taxa and (2) random ex-tinctions in the local webs [5] The general lack of power-

3670 F Food Webs

Food Webs Figure 5Trends in clustering coefficient across networks The ratio ofclustering in empirical networks (Clempirical) to clustering inrandom networks with the same number of nodes and links(Clrandom) is shown as a function of the size of the network (num-ber of nodes) Reproduced from [37] Fig 1

law degree distributions in food webs may result partlyfrom the small size and large connectance of such net-works which limits the potential for highly skewed dis-tributions Many of the networks displaying power-lawdegree distributions are much larger and much moresparsely connected than food webs

Other Properties

Assortative mixing or the tendency of nodes with similardegree to be linked to each other appears to be a pervasivephenomenon in a variety of social networks [82] How-ever other kinds of networks including technological andbiological networks tend to show disassortative mixingwhere nodes with high degree tend to link to nodes withlow degree Biological networks and particularly two foodwebs examined show strong disassortativity [82] Some ofthis may relate to a finite-size eect in systems like foodwebs that have limits on how many links are recorded be-tween pairs of nodes However in food webs it may alsoresult from the stabilizing eects of having feeding special-ists linked to feeding generalists as has been suggested forplant-animal pollination and frugivory (fruit-eating) net-works ([7] Sect ldquoEcological Networksrdquo)

Another aspect of structure that has been directly com-pared across several types of networks including food websare ldquomotifsrdquo defined as ldquorecurring significant patterns of

Food Webs Figure 6Log-log overlay plot of the cumulative distributions of links perspecies in 16 food webs The link data are normalized by the av-erage number of linksspecies in each web If the distributionsfollowed a power law the data would tend to follow a straightline Instead they follow a roughly exponential shape Repro-duced from [37] Fig 3

interconnectionsrdquo [77] A variety of networks (transcrip-tional gene regulation neuron connectivity food webstwo types of electronic circuits the World Wide Web)were scanned for all possible subgraphs that could beconstructed out of 3 or 4 nodes (13 and 199 possiblesubgraphs respectively) Subgraphs that appeared signifi-cantly more often in empirical webs than in their random-ized counterparts (i e networks with the same number ofnodes and links and the same degree for each node butwith links otherwise randomly distributed) were identi-fied For the seven food webs examined there were twoldquoconsensus motifsrdquo shared by most of the websmdasha three-node food chain and a four-species diamond wherea predator has two prey which in turn prey on the samespecies (Fig 7) The four-node motif was shared by twoother types of networks (neuron connectivity one typeof electronic circuit) and nothing shared the three-nodechain The WWW and food web networks appear mostdissimilar to other types of networks (and to each other)in terms of significant motifs

Complex networks can be decomposed into minimumspanning trees (MST) A MST is a simplified version ofa network created by removing links to minimize the dis-tance between nodes and some destination For examplea food web can be turned into MST by adding an ldquoen-vironmentrdquo node that all basal taxa link to tracing theshortest food chain from each species to the environmentnode and removing links that do not appear in the short-est chains Given this algorithm a MST removes links that

Food Webs F 3671

Food Webs Figure 7The two 3 or 4-node network motifs found to occur significantlymore often than expected in most of seven food webs exam-ined There is one significant 3-node motif (out of 13 possiblemotifs) a food chain of the form A eats B eats C There is onesignificant 4-node motif (out of 199 possible motifs) a trophicdiamond (ldquobi-parallelrdquo) of the form A eats B and C which botheat D

occur in loops and retains a basic backbone that has a tree-like structure In a MST the quantity Ai is defined as thenumber of nodes in a subtree rooted at node i and can beregarded as the transportation rate through that nodeCi isdefined as the integral ofAi (i e the sumofAi for all nodesrooted at node i) and can be regarded as the transporta-tion cost at node i These properties can be used to plot Civersus Ai for each node in a networks or to plot whole-system Co versus Ao across multiple networks to identifywhether scaling relationships of the form C(A) An arepresent indicating self-similarity in the structure of theMST (see [18] for review) In a food web MST the mostecient configuration is a star where every species linksdirectly to the environment node resulting in an expo-nent of 1 and the least ecient configuration is a singlechain where resources have to pass through each speciesin a line resulting in an exponent of 2 It has been sug-gested that food webs display a universal exponent of113 [1845] reflecting an invariant functional food webproperty relating to very ecient resource transportationwithin an ecosystem However analyzes based on a largerset of webs (17 webs versus 7) suggest that exponents forCi as a function of Ai range from 109 to 126 and arethus not universal that the exponents are quite sensitive tosmall changes in food web structure and that the observedrange of exponent values would be similarly constrainedin any network with only 3 levels as is seen in food webMSTs [15]

Models of FoodWeb Structure

An important area of research on food webs has beenwhether their observed structure which often appearsquite complex can emerge from simple rules or modelsAs with other kinds of ldquoreal-worldrdquo networks models that

assign links among nodes randomly according to fixedprobabilities fail to reproduce the network structure ofempirically observed food webs [2428110] Instead sev-eral models that combine stochastic elements with simplelink assignment rules have been proposed to generate andpredict the network structure of empirical food webs

The models share a basic formulation [110] Thereare two empirically quantifiable parameters (1) S thenumber of trophic species in a food web and (2) C theconnectance of a food web defined as links per speciessquared LS2 Thus S specifies the number of nodes ina network and C specifies the number of links in a net-work with S nodes Each species is assigned a ldquoniche valuerdquoni drawn randomly and uniformly from the interval [01]The models dier in the rules used to distribute linksamong species The link distribution rules follow in the or-der the models were introduced in the literature

Cascade Model (Cohen and Newman [28]) Eachspecies has the fixed probability P D 2CS(S 1) of con-suming species with niche values less than its own Thiscreates a food web with hierarchical feeding since it doesnot allow feeding on taxa with the same niche value(cannibalism) or taxa with higher niche values (loop-ingcycling) This formulation [110] is a modified versionof the original cascade model that allows LS equivalentto the CS term in the probability statement above to varyas a tunable parameter rather than be fixed as a con-stant [28]

Niche Model (Williams and Martinez [110] Fig 8)Each species consumes all species within a segment of the[01] interval whose size ri is calculated using the feedingrange width algorithm described below The rirsquos center ciis set at a random value drawn uniformly from the interval[ri 2 ni ] or [ri 2 1 ri 2] if ni gt 1 ri 2 which placesci equal to or lower than the niche value ni and keeps theri segment within [01] The ci rule relaxes the strict feed-ing hierarchy of the cascade model and allows for the pos-sibility of cannibalism and looping Also the ri rule en-sures that species feed on a contiguous range of speciesnecessarily creating interval graphs (i e species can belined up along a single interval such that all of their re-source species are located in contiguous segments alongthe interval)

Feeding range width algorithm The value ofri D xni where 0 lt x lt 1 is randomly drawn fromthe probability density function p(x) D ˇ(1x)b1

(the beta distribution) whereˇ D (12C)1 to ob-tain a C close to the desired C

3672 F Food Webs

Food Webs Figure 8Graphical representation of the niche model Species i feeds on4 taxa including itself and one with a higher niche value

Nested-Hierarchy Model (Cattin et al [19]) Like theniche model the number of prey items for each species isdrawn randomly from a beta distribution that constrainsCclose to a target value Once the number of prey items foreach species is set those links are assigned in a multistepprocess First a link is randomly assigned from species i toa species j with a lower ni If j is fed on by other speciesthe next feeding links for i are selected randomly from thepool of prey species fed on by a set of consumer speciesdefined as follows they share at least one prey species andat least one of them feeds on j If more links need to be dis-tributed they are then randomly assigned to species with-out predators and with niche values lt ni and finally tothose with niche value $ ni These rules were chosen torelax the contiguity rule of the niche model and to allowfor trophic habit overlap among taxa in a manner whichthe authors suggest evokes phylogenetic constraints

Generalized Cascade Model (Stouer et al [99])Species i feeds on species j if nj ni with a probabilitydrawn from the interval [01] using the beta or an expo-nential distribution This model combines the beta distri-bution introduced in the niche model with the hierarchi-cal non-contiguous feeding of the cascade model

Thesemodels have been evaluatedwith respect to theirrelative fit to empirical data in a variety of ways In a se-ries of six papers published from 1985 to 1990 with thecommon title ldquoA stochastic theory of community foodwebsrdquo the cascade model was proposed as a means of ex-plaining ldquothe phenomenology of observed food web struc-ture using a minimum of hypothesesrdquo [31] This was notthe first simple model proposed for generating food webstructure [258889116] but it was the most well-devel-oped model Cohen and colleagues also examined sev-eral model variations most of which performed poorlyWhile the cascade model appeared to generate structuresthat qualitatively fit general patterns in the data from the113-web catalog subsequent statistical analyzes suggestedthat the fit between the model and that early data was

poor [939697] Once improved data began to emerge itbecame clear that some of the basic assumptions built in tothe cascade model such as no looping and minimal over-lap and no clustering of feeding habits across taxa are vi-olated by common features of multi-species interactions

The niche model was introduced in 2000 along witha new approach to analysis numerical simulations to com-pare statistically the ability of the niche model the cascademodel and one type of random network model to fit em-pirical food web data [110] Because of stochastic varia-tion in how species and links are distributed in any par-ticular model web analysis begins with the generation ofhundreds to thousands of model webs with the same S andsimilar C as an empirical food web of interest Model websthat fall within 3 of the targetC are retained Model-gen-erated webs occasionally contain species with no links toother species or species that are trophically identical Ei-ther those webs are thrown out or those species are elim-inated and replaced until every model web has no dis-connected or identical species Also each model web mustcontain at least one basal species These requirements en-sure that model webs can be sensibly comparable to em-pirical trophic-species webs

Once a set of model webs are generated with the sameS and C as an empirical web model means and standarddeviations are calculated for each food web property ofinterest which can then be compared to empirical val-ues Raw error the dierence between the value of anempirical property and a model mean for that propertyis normalized by dividing it by the standard deviation ofthe propertyrsquos simulated distribution This approach al-lows assessment not only of whether a model over- or un-der-estimates empirical properties as indicated by the rawerror but also to what degree a modelrsquos mean deviatesfrom the empirical value Normalized errors within ˙2are considered to indicate a good fit between the modelprediction and the empirical value This approach has alsobeen used to analyze network motifs [77] (Subsect ldquoOtherPropertiesrdquo)

The initial niche model analyzes examined sevenmorerecent diverse food webs (S D 24 to 92) and up to 12 net-work structure properties for each web [110] The randommodel (links are distributed randomly among nodes) per-forms poorly with an average normalized error (ANE) of271 (SD D 202) The cascade model performs better withANE of -30 (SD D 141) The niche model performs anorder of magnitude better than the cascade model withANE of 022 (SD D 18) Only the niche model falls within˙2 ANE considered to show a good fit to the data Notsurprisingly there is variability in how all three models fitdierent food webs and properties For example the niche

Food Webs F 3673

model generally overestimates food-chain length Specificmismatches are generally attributable either to limitationsof the models or biases in the data [110] A separate testof the niche and cascade models with three marine foodwebs a type of habitat not included in the original analysisobtained similar results [39] These analyzes demonstratethat the structure of food webs is far from random and thatsimple link distribution rules can yield apparently com-plex network structure similar to that observed in empir-ical data In addition the analyzes suggest that food websfrom a variety of habitats share a fundamentally similarnetwork structure and that the structure is scale-depen-dent in predictable ways with S and C

The nested-hierarchy model [19] and generalized cas-cade model [99] variants of the niche model do not ap-pear to improve on the niche model and in fact may beworse at representing several aspects of empirical networkstructure Although the nested-hierarchymodel breaks theintervality of the niche model and uses a complicated-sounding set of link distribution rules to try to mimicphylogenetic constraints on trophic structure it ldquogener-ates webs characterized by the same universal distribu-tions of numbers of prey predators and linksrdquo as the nichemodel [99] Both the niche and nested-hierarchy modelshave a beta distribution at their core The beta distributionis reasonably approximated by an exponential distributionfor C lt 012 [99] and thus reproduces the exponential de-gree distributions observed in many empirical webs par-ticularly those with average or less-than-average C [37]The generalized cascade model was proposed as a simpli-fiedmodel that would return the same distributions of taxaand links as the niche and nested-hierarchy models It isdefined using only two criteria (1) taxa form a totally or-dered setmdashthis is fulfilled by the arrangement of taxa alonga single ldquonicherdquo interval or dimension and (2) each specieshas an exponentially decaying probability of preying ona given fraction of species with lower niche values [99]

Although the generalized cascade model does capturea central tendency of successful food web models onlysome food web properties are derivable from link distribu-tions (e g Top Bas Can VulSD GenSD Clus) There area variety of food web structure properties of interest thatare not derivable from degree distributions (e g LoopOmnHerb TL food-chain statistics intervality statistics)The accurate representation of these types of propertiesmay depend on additional factors for example the con-tiguous feeding ranges specified by the niche model butabsent from the cascade nested-hierarchy and general-ized cascademodelsWhile it is known that empirical foodwebs are not interval until recently it was not clear hownon-interval they are Intervality is a brittle property that

is broken by a single gap in a single feeding range (i ea single missing link in a food web) and trying to ar-range species in a food web into their most interval or-dering is a computationally challenging problem A morerobust measure of intervality in food webs has been devel-oped in conjunction with the use of simulated annealingto estimate the most interval ordering of empirical foodwebs [100] This analysis suggests that complex food websldquodo exhibit a strong bias toward contiguity of prey that istoward intervalityrdquo when compared to several alternativeldquonullrdquo models including the generalized cascade modelThus the intervality assumption of the niche model ini-tially critiqued as a flaw of themodel [19] helps to producea better fit to empirical data than the non-interval alternatemodels

Structural Robustness of FoodWebs

A series of papers have examined the response of a va-riety of networks including the Internet and WWW webpages [1] and metabolic and protein networks [5253] tothe simulated loss of nodes In each case the networks allof which display highly skewed power-law degree distribu-tions appear very sensitive to the targeted loss of highly-connected nodes but relatively robust to random loss ofnodes When highly-connected nodes are removed fromscale-free networks the average path length tends to in-crease rapidly and the networks quickly fragment intoisolated clusters In essence paths of information flow inhighly skewed networks are easily disrupted by loss ofnodes that are directly connected to an unusually largenumber of other nodes In contrast random networkswith much less skewed Poisson degree distributions dis-play similar responses to targeted loss of highly-connectednodes versus random node loss [101]

Within ecology species deletions on small (S lt 14)hypothetical food web networks as well as a subset of the113-web catalog have been used to examine the reliabilityof network flow or the probability that sources (produc-ers) are connected to sinks (consumers) in food webs [54]The structure of the empirical webs appears to conformto reliable flow patterns identified using the hypotheticalwebs but that result is based on low diversity poorly re-solved data The use of more highly resolved data withnode knock-out algorithms to simulate the loss of speciesallows assessment of potential secondary extinctions incomplex empirical food webs Secondary extinctions resultwhen the removal of taxa results in one ormore consumerslosing all of their resource taxa Even though most foodwebs do not have power-law degree distributions theyshow similar patterns of robustness to other networks re-

3674 F Food Webs

moval of highly-connected species results in much higherrates of secondary extinctions than random loss of species([383995] Fig 9) In addition loss of high-degree speciesresults in more rapid fragmentation of the webs [95] Pro-tecting basal taxa from primary removal increases the ro-bustness of the web (i e fewer secondary extinctions oc-cur) ([38] Fig 9) While removing species with few linksgenerally results in few secondary extinctions in a quarterof the food webs examined removing low-degree speciesresults in secondary extinctions comparable to or greaterthan what is seen with removal of high-degree species [38]This tends to occur in webs with relatively high C

Beyond dierential impacts of various sequences ofspecies loss in food webs food web lsquostructural robustnessrsquocan be defined as the fraction of primary species loss thatinduces some level of species loss (primary + secondary ex-tinctions) for a particular trophic-species web Analysis ofR50 (i e what proportion of species have to be removedto achieve $ 50 total species loss) across multiple foodwebs shows that robustness increases approximately log-arithmically with increasing connectance ([3839] Fig 910) In essence from a topological perspective food webswith more densely interconnected taxa are better pro-tected from species loss since it takes greater species lossfor consumers to lose all of their resources

It is also potentially important from a conservationperspective to identify particular species likely to resultin the greatest number of secondary extinctions throughtheir loss The loss of a particular highly-connected speciesmay or may not result in secondary extinctions One

Food Webs Figure 9Secondary extinctions resulting from primary species loss in 4 food webs ordered by increasing connectance (C) The y-axis showsthe cumulative secondary extinctions as a fraction of initial S and the x-axis shows the primary removals of species as a fractionof initial S 95 error bars for the random removals fall within the size of the symbols and are not shown For the most connected(circles) least connected (triangles) and random removal (plus symbols) sequences the data series end at the black diagonal dashedline where primary removals plus secondary extinctions equal S and the web disappears For the most connected species removalswith basal species preserved (black dots) the data points end when only basal species remain The shorter red diagonal lines showthe points at which 50of species are lost through combined primary removals and secondary extinctions (ldquorobustnessrdquo or R50)

Food Webs Figure 10The proportion of primary species removals required to inducea total loss (primary removals plus secondary extinctions) of 50of the species in each of 16 food webs (ldquorobustnessrdquo see theshorter red line of Fig 9 for visual representation) as a functionof the connectance of each web Logarithmic fits to the threedata sets are shown with a solid line for the most connecteddeletion order a long dashed line for the most connected withbasal species preserved deletion order and a short dashed linefor random deletion order The maximum possible y value is050 The equations for the fits are y D 0162 ln(x) C 0651 formost connected species removals y D 0148 ln(x) C 0691 formost connected species removals with basal species preservedand y D 0067 ln(x) C 0571 for random species removals Re-produced from [38] Fig 2

way to identify critical taxa is to reduce the topologi-cal structure of empirical food webs into linear pathwaysthat define the essential chains of energy delivery in eachweb A particular species can be said to ldquodominaterdquo other

Food Webs F 3675

species if it passes energy to them along a chain in thedominator tree The higher the number of species thata particular species dominates the greater the secondaryextinctions that may result from its removal [3] This ap-proach has the advantage of going beyond assessment ofdirect interactions to include indirect interactions

As in food webs the order of pollinator loss has aneect on potential plant extinction patterns in plant-pol-linator networks [75] (see Sect ldquoEcological Networksrdquo)Loss of plant diversity associated with targeted removal ofhighly-connected pollinators is not as extreme as compa-rable secondary extinctions in food webs which may bedue to pollinator redundancy and the nested topology ofthose networks

While the order in which species go locally extinctclearly aects the potential for secondary extinctions inecosystems the focus on high-degree random or evendominator species does not provide insight on ecologi-cally plausible species loss scenarios whether the focus ison human perturbations or natural dynamics The issue ofwhat realistic natural extinction sequences might look likehas been explored using a set of pelagic-focused food websfor 50 Adirondack lakes [49] with up to 75 species [98]The geographic nestedness of species composition acrossthe lakes is used to derive an ecologically plausible extinc-tion sequence scenario with the most restricted taxa themost likely to go extinct This sequence is corroborated bythe pH tolerances of the species Species removal simula-tions show that the food webs are highly robust in termsof secondary extinctions to the ldquorealisticrdquo extinction orderand highly sensitive to the reverse order This suggests thatnested geographical distribution patterns coupled with lo-cal food web interaction patterns appear to buer eectsof likely species losses This highlights important aspectsof community organization that may help to minimizebiodiversity loss in the face of a naturally changing envi-ronment However anthropogenic disturbances may dis-rupt the inherent buering of how taxa are organized ge-ographically and trophically reducing the robustness ofecosystems

FoodWeb Dynamics

Analysis of the topology of food webs has proven very use-ful for exploring basic patterns and generalities of ldquowhoeats whomrdquo in ecosystems This approach seeks to iden-tify ldquothe most universal high-level persistent elements oforganizationrdquo [35] in trophic networks and to leverageunderstanding of such organization for thinking aboutecosystem robustness However food webs are inher-ently dynamical systems since feeding interactions in-

volve biomass flows among species whose ldquostocksrdquo can becharacterized by numbers of individuals andor aggregatepopulation biomass All of these stocks and flows changethrough time in response to direct and indirect trophicand other types of interactions Determining the interplayamong network structure network dynamics and variousaspects of stability such as persistence robustness and re-silience in complex ldquoreal-worldrdquo networks is one of thegreat current challenges in network research [101] It isparticularly important in the study of ecosystems sincethey face a variety of anthropogenic perturbations such asclimate change habitat loss and invasions and since hu-mans depend on them for a variety of ldquoecosystem servicesrdquosuch as supply of clean water and pollination of crops [34]

Because it is nearly impossible to compile detailedlong-term empirical data for dynamics of more than twointeracting species most research on species interactiondynamics relies on analytical or simulation modelingMost modeling studies of trophic dynamics have focusednarrowly on predator-prey or parasite-host interactionsHowever as the previous sections should make clear innatural ecosystems such interaction dyads are embeddedin diverse complex networks where many additional taxaand their direct and indirect interactions can play impor-tant roles for the stability of focal species as well as thestability or persistence of the broader community Mov-ing beyond the two-species population dynamics model-ing paradigm there is a strong tradition of research thatlooks at interactions among 3ndash8 species exploring dynam-ics and simple variations in structure in slightlymore com-plex systems (see reviews in [4055]) However these in-teraction modules still present a drastic simplification ofthe diversity and structure of natural ecosystems Otherdynamical approaches have focused on higher diversitymodel systems [69] but ignore network structure in orderto conduct analytically tractable analyzes

Researchers are increasingly integrating dynamicswith complex food web structure in modeling studies thatmove beyond small modules The LotkandashVolterra cascademodel [202132] was an early incarnation of this type ofintegration As its name suggests the LotkandashVolterra cas-cade model runs classic L-V dynamics including a non-saturating linear functional response on sets of species in-teractions structured according to the cascade model [28]The cascademodel was also used to generate the structuralframework for a dynamical food web model with a lin-ear functional response [58] used to study the eects ofprey-switching on ecosystem stability Improving on as-pects of biological realism of both dynamics and struc-ture a bioenergetic dynamical model with nonlinear func-tional responses [119] was used in conjunction with em-

3676 F Food Webs

pirically-defined food web structure among 29 species tosimulate the biomass dynamics of a marine fisheries foodweb [117118] This type of nonlinear bioenergetic dynam-ical modeling approach has been integrated with nichemodel network structure and used to study more com-plex networks [131468112] A variety of types of dy-namics are observed in these non-linear models includ-ing equilibrium limit cycle and chaotic dynamics whichmay ormay not be persistent over short or long time scales(Fig 11) Other approaches model ecological and evolu-tionary dynamics to assemble species into networks ratherthan imposing a particular structure on them These mod-els which typically employ an enormous amount of pa-rameters are evaluated as to whether they generate plau-sible persistence diversity and network structure (see re-view by [72]) All of these approaches are generally used toexamine stability characterized in a diversity of ways inecosystems with complex structure and dynamics [7185]

While it is basically impossible to empirically validatemodels of integrated structure and dynamics for com-plex ecological networks in some situations it is possi-ble to draw interesting connections between models anddata at more aggregated levels This provides opportuni-ties to move beyond the merely heuristic role that suchmodels generally play For example nonlinear bioener-getic models of population dynamics parametrized by bi-ological rates allometrically scaled to populationsrsquo averagebody masses have been run on various types of model foodweb structures [14] This approach has allowed the com-parison of trends in two dierent measures of food webstability and how they relate to consumer-resource body-size ratios and to initial network structure One measureof stability is the fraction of original species that displaypersistent dynamics i e what fraction of species do notgo extinct in the model when it is run over many timesteps (ldquospecies persistencerdquo) Another measure of stabilityis how variable the densities of all of the persistent speciesare (ldquopopulation stabilityrdquo)mdashgreater variability across allthe species indicates decreased stability in terms of popu-lation dynamics

Brose and colleagues [14] ran the model using dier-ent hypothetical consumer-resource body-size ratios thatrange from 102 (consumers are 100 times smaller thantheir resources) to 105 (consumers are 100000 times largerthan their resources) (Fig 12) Species persistence in-creases dramatically with increasing body-size ratios un-til inflection points are reached at which persistence shiftsto high levels ( 080) of persistence (Fig 12a) How-ever population stability decreases with increasing body-size ratios until inflection points are reached that showthe lowest stability and then increases again beyond those

Food Webs Figure 115 different types of population dynamics shown as time series ofpopulation density (from [40] Fig 101) The types of dynamicsshown include a stable equilibrium damped oscillations limitcycles amplified oscillations and chaotic oscillations

points (Fig 12b) In both cases the inflection points corre-spond to empirically observed consumer-resource body-size ratios both for webs parametrized to represent in-vertebrate dominated webs and for webs parametrizedto represent ectotherm vertebrate dominated webs Thusacross thousands of observations invertebrate consumer-resource body size ratios are 101 and ectotherm verte-brate consumer-resource body size ratios are 102 whichcorrespond to themodelrsquos inflection points for species per-sistence and population stability (Fig 12) It is interesting

Food Webs F 3677

Food Webs Figure 12a shows the fraction of species that display persistent dynamics as a function of consumer-resource body-size ratios for modelfood webs parametrized for invertebrates (gray line) and ectotherm vertebrates (black line) The inflection points for shifts to high-persistence dynamics are indicated by red arrows for both curves and those inflection points correspond to empirically observedconsumer-resource body size ratios for invertebrate dominated webs (101mdashconsumers are on average 10 times larger than theirresources) and ectotherm vertebrate dominated webs (102mdashconsumers are on average 100 times larger than their resources) bshows results for population stability the mean of how variable species population biomasses are in persistent webs In this casethe inflection points for shifts to low population stability are indicated by red arrows and those inflection points also correspond tothe empirically observed body-size ratios for consumers and resources Figure adapted from [14]

to note that high species persistence is coupled to low pop-ulation stabilitymdashi e an aspect of increased stability of thewhole system (species persistence) is linked to an aspect ofdecreased stability of components of that system (popula-tion stability) It is also interesting to note that in this for-mulation using initial cascade versus niche model struc-ture had little impact on species persistence or populationstability [14] although other formulations show increasedpersistence when dynamics are initiated with niche modelversus other structures [68] How structure influences dy-namics and vice-versa is an open question

Ecological Networks

This article has focused on food webs which gener-ally concern classic predator-herbivore-primary producerfeeding interactions However the basic concept of foodwebs can be extended to a broader framework of ldquoecolog-ical networksrdquo that is more inclusive of dierent compo-nents of ecosystem biomass flow and that takes into con-sideration dierent kinds of species interactions that arenot classic ldquopredator-preyrdquo interactions Three examplesare mentioned here First parasites have typically beengiven short shrift in traditional food webs although ex-ceptions exist (e g [516774]) This is changing as it be-comes clear that parasites are ubiquitous often have sig-nificant impacts on predator-prey dynamics and may bethe dominant trophic habitat in most food webs poten-tially altering our understanding of structure and dynam-

ics [59] The dynamical models described previously havebeen parametrized with more conventional non-parasiteconsumers in mind An interesting open question is howaltering dynamical model parameters such as metabolicrate functional response and consumer-resource bodysize ratios to reflect parasite characteristics will aect ourunderstanding of food web stability

Second the role of detritus or dead organic matterin food webs has yet to be adequately resolved in eitherstructural or dynamical approaches Detritus has typicallybeen included as one or several separate nodes in many bi-nary-link and flow-weighted food webs In some cases it istreated as an additional ldquoprimary producerrdquo while in othercases both primary producers and detritivores connect toit Researchersmust think much more carefully about howto include detritus in all kinds of ecological studies [80]given that it plays a fundamental role in most ecosystemsand has particular characteristics that dier from otherfood web nodes it is non-living organic matter all speciescontribute to detrital pools it is a major resource for manyspecies and the forms it takes are extremely heterogeneous(e g suspended organic matter in water columns fecalmaterial rotting trees dead animal bodies small bits ofplants and molted cuticle skin and hair mixed in soiletc)

Third there are many interactions that species par-ticipate in that go beyond strictly trophic interactionsPlant-animal mutualistic networks particularly pollina-tion and seed dispersal or ldquofrugivoryrdquo networks have re-

3678 F Food Webs

ceived the most attention thus far They are character-ized as ldquobipartiterdquo (two-level) graphs with links from an-imals to plants but no links among plants or among ani-mals [79565773107] While both pollination and seeddispersal do involve a trophic interaction with animalsgaining nutrition from plants during the interactions un-like in classic predator-prey relationships a positive bene-fit is conferred upon both partners in the interaction Theevolutionary and ecological dynamics of such mutualis-tic relationships place unique constraints on the networkstructure of these interactions and the dynamical stabil-ity of such networks For example plant-animal mutualis-tic networks are highly nested and thus asymmetric suchthat generalist plants and generalist animals tend to in-teract among themselves but specialist species (whetherplants or animals) also tend to interact with the most gen-eralist species [7107] When simple dynamics are run onthese types of ldquocoevolutionaryrdquo bipartite networks it ap-pears that the asymmetric structure enhances long-termspecies coexistence and thus biodiversity maintenance [9]

Future Directions

Food web research of all kinds has expanded greatly overthe last several years and there are many opportunitiesfor exciting new work at the intersection of ecology andnetwork structure and dynamics In terms of empiricismthere is still a paucity of detailed evenly resolved com-munity food webs in every habitat type Current theorymodels and applications need to be tested against morediverse more complete and more highly quantified dataIn addition there are many types of datasets that could becompiled which would support novel research For exam-ple certain kinds of fossil assemblagesmay allow the com-pilation of detailed paleo food webs which in turn couldallow examination of questions about how and why foodweb structure does or does not change over deep time orin response tomajor extinction events Another example isdata illustrating the assembly of food [41] webs in partic-ular habitats over ecological time In particular areas un-dergoing rapid successional dynamics would be excellentcandidates such as an area covered by volcanic lava flowsa field exposed by a retreating glacier a hillside denuded byan earth slide or a forest burned in a large fire This type ofdata would allow empirically-based research on the topo-logical dynamics of food webs Another empirical frontieris the integration of multiple kinds of ecological interac-tion data into networks with multiple kinds of linksmdashforexample networks that combine mutualistic interactionssuch as pollination and antagonistic interactions such aspredator-prey relationships In addition more spatially

explicit food web data can be compiled across microhab-itats or linked macrohabitats [8] Most current food webdata is eectively aspatial even though trophic interactionsoccur within a spatial context More could also be done tocollect food web data based on specific instances of trophicinteractions This was done for the insects that live insidethe stems of grasses in British fields The web includesmultiple grass species grass herbivores their parasitoidshyper-parasitoids and hyper-hyper parasitoids [67] Dis-section of over 160000 grass stems allowed detailed quan-tification of the frequency with which the species (S D 77insect plus 10 grass species) and dierent trophic interac-tions (L D 126) were observed

Better empiricism will support improved and novelanalysis modeling and theory development and testingFor example while food webs appear fundamentally dif-ferent in some ways from other kinds of ldquoreal-worldrdquo net-works (e g they donrsquot display power-law degree distribu-tions) they also appear to share a common core networkstructure that is scale-dependent with species richness andconnectance in predicable ways as suggested by the suc-cess of the niche and related models Some of the dispar-ity with other kinds of networks and the shared structureacross food webs may be explicable through finite-size ef-fects or other methodological or empirical constraints orartifacts However aspects of these patterns may reflect at-tributes of ecosystems that relate to particular ecologicalevolutionary or thermodynamic mechanisms underlyinghow species are organized in complex bioenergetic net-works of feeding interactions Untangling artifacts fromattributes [63] and determining potential mechanisms un-derlying robust phenomenological patterns (e g [10]) isan important area of ongoing and future research Asa part of this there is much work to be done to continueto integrate structure and dynamics of complex ecologi-cal networks This is critical for gaining a more compre-hensive understanding of the conditions that underlie andpromote dierent aspects of stability at dierent levels oforganization in response to external perturbations and toendogenous short- and long-term dynamics

As the empiricism analysis and modeling of food webstructure and dynamics improves food web network re-search can play a more central and critical role in con-servation and management [76] It is increasingly appar-ent that an ecological network perspective which encom-passes direct and indirect eects among interacting taxais critical for understanding predicting andmanaging theimpacts of species loss and invasion habitat conversionand climate change Far too often critical issues of ecosys-tem management have been decided on extremely limitedknowledge of one or a very few taxa For example this

Food Webs F 3679

has been an ongoing problem in fisheries science The nar-row focus of most research driving fisheries managementdecisions has resulted in overly optimistic assessments ofsustainable fishing levels Coupled with climate stressorsover-fishing appears to be driving steep rapid declines indiversity of common predator target species and proba-bly many other kinds of associated taxa [114] Until weacknowledge that species of interest to humans are em-bedded within complex networks of interactions that canproduce unexpected eects through the interplay of directand indirect eects we will continue to experience nega-tive outcomes from our management decisions [118] Animportant part of minimizing and mitigating human im-pacts on ecosystems also involves research that explicitlyintegrates human and natural dynamics Network researchprovides a natural framework for analyzing and modelingthe complex ways in which humans interact with and im-pact the worldrsquos ecosystems whether through local forag-ing or large-scale commercial harvesting driven by globaleconomic markets

These and other related research directions will de-pend on ecient management of increasingly dispersedand diversely formatted ecological and environmentaldata Ecoinformatic toolsmdashthe technologies and practicesfor gathering synthesizing analyzing visualizing stor-ing retrieving and otherwise managing ecological knowl-edge and informationmdashare playing an increasingly impor-tant role in the study of complex ecosystems includingfood web research [47] Indeed ecology provides an ex-cellent testbed for developing implementing and testingnew information technologies in a biocomplexity researchcontext (e g Semantic Prototypes in Research Ecoinfor-maticsSPiRE spireumbceduus Science Environmentfor Ecological KnowledgeSEEK seekecoinformaticsorgWebs on the WebWoW wwwfoodwebsorg) Synergis-tic ties between ecology physics computer science andother disciplines will dramatically increase the ecacy ofresearch that takes advantage of such interdisciplinary ap-proaches as is currently happening in food web and re-lated research

Bibliography

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ecosystem Energy flow bottlenecks and cascading extinc-tions J Theor Biol 230351ndash358

4 Amaral LAN Scala A BerthelemyM Stanley HE (2000) Classesof small-world networks Proc Nat Acad Sci USA 9711149ndash11152

5 Arii K Parrott L (2004) Emergence of non-random structurein local food webs generated from randomly structured re-gional webs J Theor Biol 227327ndash333

6 Baird D Ulanowicz RE (1989) The seasonal dynamics of theChesapeake Bay ecosystem Ecol Monogr 59329ndash364

7 Bascompte J Jordano P Meliaacuten CJ Olesen JM (2003) Thenested assembly of plant-animal mutualistic networks ProcNatl Acad Sci USA 1009383ndash9387

8 Bascompte J Meliaacuten CJ Sala E (2005) Interaction strengthcombinations and the overfishing of amarine food web ProcNatl Acad Sci 1025443ndash5447

9 Bascompte J Jordano P Olesen JM (2006) Asymmetric coevo-lutionary networks facilitate biodiversity maintenance Sci-ence 312431ndash433

10 Beckerman AP Petchey OL Warren PH (2006) Foraging biol-ogy predicts food web complexity Proc Natl Acad Sci USA10313745ndash13749

11 Bersier L-F Banaek-Richter C Cattin M-F (2002) Quantitativedescriptors of food web matrices Ecology 832394ndash2407

12 Briand F Cohen JE (1984) Community food webs have scale-invariant structure Nature 398330ndash334

13 BroseUWilliams RJMartinezND (2003) Comment on ldquoForag-ing adaptation and the relationship between food-web com-plexity and stabilityrdquo Science 301918b

14 Brose U Williams RJ Martinez ND (2006) Allometric scalingenhances stability in complex food webs Ecol Lett 91228ndash1236

15 Camacho J Arenas A (2005) Universal scaling in food-webstructure Nature 435E3-E4

16 Camacho J Guimeragrave R Amaral LAN (2002) Robust patterns infood web structure Phys Rev Lett 88228102

17 Camacho J Guimeragrave R Amaral LAN (2002) Analytical solutionof a model for complex food webs Phys Rev Lett E 65030901

18 Cartoza CC Garlaschelli D Caldarelli G (2006) Graph theoryand food webs In Pascual M Dunne JA (eds) Ecological Net-works Linking Structure to Dynamics in Food Webs OxfordUniversity Press New York pp 93ndash117

19 Cattin M-F Bersier L-F Banaek-Richter C Baltensperger MGabriel J-P (2004) Phylogenetic constraints and adaptationexplain food-web structure Nature 427835ndash839

20 Chen X Cohen JE (2001) Global stability local stability andpermanence in model food webs J Th Biol 212223ndash235

21 Chen X Cohen JE (2001) Transient dynamics and food webcomplexity in the LotkandashVolterra cascade model Proc RoySoc Lond B 268869ndash877

22 Christian RR Luczkovich JJ (1999) Organizing and under-standing awinterrsquos seagrass foodweb network through effec-tive trophic levels Ecol Model 11799ndash124

23 Cohen JE (1977) Ratio of prey to predators in community foodwebs Nature 270165ndash167

24 Cohen JE (1977) Food webs and the dimensionality of trophicniche space Proc Natl Acad Sci USA 744533ndash4563

25 Cohen JE (1978) Food Webs and Niche Space Princeton Uni-versity Press NJ

26 Cohen JE (1989) Ecologists Co-operative Web Bank(ECOWeBtrade) Version 10 Machine Readable Data Base ofFood Webs Rockefeller University NY

3680 F Food Webs

27 Cohen JE Briand F (1984) Trophic links of community foodwebs Proc Natl Acad Sci USA 814105ndash4109

28 Cohen JE Newman CM (1985) A stochastic theory of commu-nity food webs I Models and aggregated data Proc R SocLond B 224421ndash448

29 Cohen JE Palka ZJ (1990) A stochastic theory of commu-nity food webs V Intervality and triangulation in the trophicniche overlap graph Am Nat 135435ndash463

30 Cohen JE Briand F Newman CM (1986) A stochastic theory ofcommunity food webs III Predicted and observed length offood chains Proc R Soc Lond B 228317ndash353

31 Cohen JE Briand F Newman CM (1990) Community FoodWebs Data and Theory Springer Berlin

32 Cohen JE Luczak T Newman CM Zhou Z-M (1990) Stochasticstructure and non-linear dynamics of food webs qualitativestability in a LotkandashVolterra cascade model Proc R Soc LondB 240607ndash627

33 Cohen JE Jonsson T Carpenter SR (2003) Ecological commu-nity description using the food web species abundance andbody size Proc Natl Acad Sci USA 1001781ndash1786

34 Daily GC (ed) (1997) Naturersquos services Societal dependenceon natural ecosystems Island Press Washington DC

35 Doyle J Csete M (2007) Rules of engagement Nature 44686036 Dunne JA (2006) The network structure of food webs In Pas-

cual M Dunne JA (eds) Ecological Networks Linking Struc-ture to Dynamics in Food Webs Oxford University Press NewYork pp 27ndash86

37 Dunne JA Williams RJ Martinez ND (2002) Food web struc-ture and network theory the role of connectance and sizeProc Natl Acad Sci USA 9912917ndash12922

38 Dunne JA Williams RJ Martinez ND (2002) Network structureand biodiversity loss in food webs robustness increases withconnectance Ecol Lett 5558ndash567

39 Dunne JA Williams RJ Martinez ND (2004) Network struc-ture and robustness of marine food webs Mar Ecol Prog Ser273291ndash302

40 Dunne JA Brose U Williams RJ Martinez ND (2005) Model-ing food-web dynamics complexity-stability implications InBelgrano A Scharler U Dunne JA Ulanowicz RE (eds) AquaticFoodWebs An Ecosystem Approach Oxford University PressNew York pp 117ndash129

41 Dunne JA Williams RJ Martinez ND Wood RA ErwingDE (2008) Compilation and network analyses of Cambrianfood webs PLoS Biology 5e102 doi101371journalpbio0060102

42 Egerton FN (2007) Understanding food chains and foodwebs1700ndash1970 Bull Ecol Soc Am 88(1)50ndash69

43 Elton CS (1927) Animal Ecology Sidgwick and Jackson Lon-don

44 Elton CS (1958) Ecology of Invasions by Animals and PlantsChapman amp Hall London

45 Garlaschelli D Caldarelli G Pietronero L (2003) Universal scal-ing relations in food webs Nature 423165ndash168

46 Goldwasser L Roughgarden JA (1993) Construction of a largeCaribbean food web Ecology 741216ndash1233

47 Green JL Hastings A Arzberger P Ayala F Cottingham KLCuddington K Davis F Dunne JA Fortin M-J Gerber LNeubert M (2005) Complexity in ecology and conservationmathematical statistical and computational challenges Bio-science 55501ndash510

48 Hardy AC (1924) The herring in relation to its animate envi-

ronment Part 1 The food and feeding habits of the herringwith special reference to the East Coast of England Fish In-vestig Ser II 71ndash53

49 Havens K (1992) Scale and structure in natural food webs Sci-ence 2571107ndash1109

50 Hutchinson GE (1959) Homage to Santa Rosalia or why arethere so many kinds of animals Am Nat 93145ndash159

51 HuxhamM Beany S Raffaelli D (1996) Doparasites reduce thechances of triangulation in a real foodwebOikos 76284ndash300

52 Jeong H Tombor B Albert R Oltvia ZN Barabaacutesi A-L (2000)The large-scale organization of metabolic networks Nature407651ndash654

53 Jeong H Mason SP Barabaacutesi A-L Oltavia ZN (2001) Lethalityand centrality in protein networks Nature 41141

54 Jordaacuten F Molnaacuter I (1999) Reliable flows and preferred pat-terns in food webs Ecol Ecol Res 1591ndash609

55 Jordaacuten F Scheuring I (2004) Network ecology topologicalconstraints on ecosystem dynamics Phys Life Rev 1139ndash229

56 Jordano P (1987) Patterns of mutualistic interactions in polli-nation and seed dispersal connectance dependence asym-metries and coevolution Am Nat 129657ndash677

57 Jordano P Bascompte J Olesen JM (2003) Invariant proper-ties in co-evolutionary networks of plant-animal interactionsEcol Lett 669ndash81

58 Kondoh M (2003) Foraging adaptation and the relation-ship between food-web complexity and stability Science2991388ndash1391

59 Lafferty KD Dobson AP Kurls AM (2006) Parasites dominatefood web links Proc Nat Acad Sci USA 10311211ndash11216

60 Lindeman RL (1942) The trophic-dynamic aspect of ecologyEcology 23399ndash418

61 Link J (2002) Does food web theory work for marine ecosys-tems Mar Ecol Prog Ser 2301ndash9

62 MacArthur RH (1955) Fluctuation of animal populations anda measure of community stability Ecology 36533ndash536

63 Martinez ND (1991) Artifacts or attributes Effects of resolu-tion on the Little Rock Lake food web Ecol Monogr 61367ndash392

64 Martinez ND (1992) Constant connectance in communityfood webs Am Nat 1391208ndash1218

65 Martinez ND (1993) Effect of scale on food web structure Sci-ence 260242ndash243

66 Martinez ND (1994) Scale-dependent constraints on food-web structure Am Nat 144935ndash953

67 Martinez ND Hawkins BA Dawah HA Feifarek BP (1999)Effects of sampling effort on characterization of food-webstructure Ecology 801044ndash1055

68 Martinez ND Williams RJ Dunne JA (2006) Diversity com-plexity and persistence in large model ecosystems In Pas-cual M Dunne JA (eds) Ecological Networks Linking Struc-ture to Dynamics in Food Webs Oxford University Press NewYork pp 163ndash185

69 May RM (1972) Will a large complex system be stable Nature238413ndash414

70 May RM (1973) Stability andComplexity inModel EcosystemsPrinceton University Press Princeton Reprinted in 2001 asa ldquoPrinceton Landmarks in Biologyrdquo edition

71 McCann KS (2000) The diversity-stability debate Nature405228ndash233

72 McKane AJ Drossel B (2006) Models of food-web evolutionIn Pascual M Dunne JA (eds) Ecological Networks Linking

Food Webs F 3681

Structure to Dynamics in FoodWebs Oxford University PressNew York pp 223ndash243

73 Memmott J (1999) The structure of a plant-pollinator net-work Ecol Lett 2276ndash280

74 Memmott J Martinez ND Cohen JE (2000) Predators para-sitoids and pathogens species richness trophic generalityand body sizes in a natural food web J Anim Ecol 691ndash15

75 Memmott J Waser NM Price MV (2004) Tolerance of pollina-tion networks to species extinctions Proc Royal Soc Lond Se-ries B 2712605ndash2611

76 Memmott J AlonsoD Berlow EL Dobson A Dunne JA Sole RWeitz J (2006) Biodiversity loss and ecological network struc-ture In Pascual M Dunne JA (eds) Ecological Networks Link-ing Structure to Dynamics in Food Webs Oxford UniversityPress New York pp 325ndash347

77 Milo R Shen-Orr S Itzkovitz S Kashtan N Chklovskii D AlonU (2002) Network motifs simple building blocks of complexnetworks Science 298763ndash764

78 Montoya JM Soleacute RV (2002) Small world patterns in foodwebs J Theor Biol 214405ndash412

79 Montoya JM Soleacute RV (2003) Topological properties of foodwebs from real data to community assembly models Oikos102614ndash622

80 Moore JC Berlow EL ColemanDC de Ruiter PCDongQ Hast-ings A Collin Johnson N McCann KS Melville K Morin PJNadelhoffer K Rosemond AD Post DM Sabo JL Scow KMVanni MJ Wall DH (2004) Detritus trophic dynamics and bio-diversity Ecol Lett 7584ndash600

81 Neutel AM Heesterbeek JAP de Ruiter PC (2002) Stability inreal food webs weak links in long loops Science 2961120ndash1123

82 NewmanMEJ (2002) Assortativemixing in networks Phys RevLett 89208701

83 Newman M Barabasi A-L Watts DJ (eds) (2006) The Struc-ture and Dynamics of Networks Princeton University PressPrinceton

84 Odum E (1953) Fundamentals of Ecology Saunders Philadel-phia

85 Pascual M Dunne JA (eds) (2006) Ecological Networks Link-ing Structure to Dynamics in Food Webs Oxford UniversityPress New York

86 Paine RT (1988) Food webs road maps of interactions or gristfor theoretical development Ecology 691648ndash1654

87 Pierce WD Cushamn RA Hood CE (1912) The insect enemiesof the cotton boll weevil US Dept Agric Bull 1009ndash99

88 Pimm SL (1982) Food Webs Chapman and Hall LondonReprinted in 2002 as a 2nd edition by University of ChicagoPress

89 Pimm SL (1984) The complexity and stability of ecosystemsNature 307321ndash326

90 Pimm SL Lawton JH (1978) On feeding on more than onetrophic level Nature 275542ndash544

91 Pimm SL Lawton JH (1980) Are food webs divided into com-partments J Anim Ecol 49879ndash898

92 Polis GA (1991) Complex desert food webs an empirical cri-tique of food web theory Am Nat 138123ndash155

93 Schoener TW (1989) Food webs from the small to the largeEcology 701559ndash1589

94 Schoenly K Beaver R Heumier T (1991) On the trophic rela-tions of insects a food web approach Am Nat 137597ndash638

95 Soleacute RV Montoya JM (2001) Complexity and fragility in eco-logical networks Proc R Soc Lond B 2682039ndash2045

96 Solow AR (1996) On the goodness of fit of the cascademodelEcology 771294ndash1297

97 Solow AR Beet AR (1998) On lumping species in food websEcology 792013ndash2018

98 Srinivasan UT Dunne JA Harte H Martinez ND (2007) Re-sponse of complex food webs to realistic extinction se-quences Ecology 88671ndash682

99 Stouffer DB Camacho J Guimera R Ng CA Amaral LAN (2005)Quantitative patterns in the structure of model and empiricalfood webs Ecology 861301ndash1311

100 Stouffer DB Camacho J Amaral LAN (2006) A robustmeasureof food web intervality Proc Nat Acad Sci 10319015ndash19020

101 Strogatz SH (2001) Exploring complex networks Nature410268ndash275

102 Sugihara G Schoenly K Trombla A (1989) Scale invariance infood web properties Science 24548ndash52

103 Summerhayes VS Elton CS (1923) Contributions to the ecol-ogy of Spitzbergen and Bear Island J Ecol 11214ndash286

104 Summerhayes VS Elton CS (1928) Further contributions tothe ecology of Spitzbergen and Bear Island J Ecol 16193ndash268

105 Thompson RM Townsend CR (1999) The effect of seasonalvariation on the community structure and food-web at-tributes of two streams implications for food-web scienceOikos 8775ndash88

106 Thompson RM Townsend CR (2005) Food web topologyvaries with spatial scale in a patchy environment Ecology861916ndash1925

107 Vaacutesquez DP Aizen MA (2004) Asymmetric specializationa pervasive feature of plant-pollinator interactions Ecology851251ndash1257

108 Warren PH (1989) Spatial and temporal variation in the struc-ture of a freshwater food web Oikos 55299ndash311

109 Watts DJ Strogatz SH (1998) Collective dynamics of lsquosmall-worldrsquo networks Nature 393440ndash442

110 Williams RJ Martinez ND (2000) Simple rules yield complexfood webs Nature 404180ndash183

111 Williams RJ Martinez ND (2004) Trophic levels in complexfood webs theory and data Am Nat 163458ndash468

112 Williams RJ Martinez ND (2004) Diversity complexity andpersistence in large model ecosystems Santa Fe InstituteWorking Paper 04-07-022

113 Williams RJ Berlow EL Dunne JA Barabaacutesi AL Martinez ND(2002) Two degrees of separation in complex food webs ProcNatl Acad Sci USA 9912913ndash12916

114 Worm B Sandow M Oschlies A Lotze HK Myers RA (2005)Global patterns of predator diversity in the open oceans Sci-ence 3091365ndash1369

115 Yodzis P (1980) The connectance of real ecosystems Nature284544ndash545

116 Yodzis P (1984) The structure of assembled communities IIJ Theor Biol 107115ndash126

117 Yodzis P (1998) Local trophodynamics and the interaction ofmarine mammals and fisheries in the Benguela ecosystemJ Anim Ecol 67635ndash658

118 Yodzis P (2000) Diffuse effects in food webs Ecology 81261ndash266

119 Yodzis P Innes S (1992) Body-size and consumer-resource dy-namics Am Nat 1391151ndash1173

3682 F Foraging Robots

Books and ReviewsBelgrano A Scharler U Dunne JA Ulanowicz RE (eds) (2005)

Aquatic Food Webs An Ecosystem Approach Oxford Univer-sity Press Oxford

Berlow EL Neutel A-M Cohen JE De Ruiter P Ebenman B Emmer-son M Fox JW Jansen VAA Jones JI Kokkoris GD LogofetDO McKane AJ Montoya J Petchey OL (2004) Interactionstrengths in food webs issues and opportunities J AnimalEcol 73585ndash598

Borer ET Anderson K Blanchette CA Broitman B Cooper SDHalpern BS (2002) Topological approaches to food web anal-yses a few modifications may improve our insights Oikos99397ndash401

Christensen V Pauly D (1993) Trophic Models of Aquatic Ecosys-tems ICLARM Manila

Cohen JE Beaver RA Cousins SH De Angelis DL et al (1993) Im-proving food webs Ecology 74252ndash258

Cohen JE Briand F Newman CM (1990) Community Food WebsData and Theory Springer Berlin

DeAngelis DL Post WM Sugihara G (eds) (1983) Current Trends inFood Web Theory ORNL-5983 Oak Ridge Natl Laboratory

Drossel B McKane AJ (2003) Modelling food webs In Bornholt SSchuster HG (eds) Handbook of Graphs and Networks Fromthe Genome to the Internet Wiley-VCH Berlin

Hall SJ Raffaelli DG (1993) Food webs theory and reality Advancesin Ecological Research 24187ndash239

Lawton JH (1989) Food webs In Cherett JM (ed) Ecological Con-cepts Blackwell Scientific Oxford

Lawton JH Warren PH (1988) Static and dynamic explanationsfor patterns in food webs Trends in Ecology and Evolution3242ndash245

Martinez ND (1995) Unifying ecological subdisciplines with ecosys-tem food webs In Jones CG Lawton JH (eds) Linking Speciesand Ecosystems Chapman and Hall New York

Martinez ND Dunne JA (1998) Time space and beyond scale is-sues in food-web research In Peterson D Parker VT (eds) Eco-logical Scale Theory and Applications Columbia UniversityPress New York

May RM (1983) The structure of food webs Nature 301566ndash568May RM (2006) Network structure and the biology of populations

Trends Ecol Evol 21394ndash399Montoya JM PimmSL Sole RV (2006) Ecological networks and their

fragility Nature 442259ndash264Moore J de Ruiter P Wolters V (eds) (2005) Dynamic Food Webs

Multispecies Assemblages Ecosystem Development and En-vironmental Change Academic Press Elsevier Amsterdam

Pimm SL Lawton JH Cohen JE (1991) Food web patterns and theirconsequences Nature 350669ndash674

Polis GAWinemiller KO (eds) (1996) FoodWebs Integration of Pat-terns amp Dynamics Chapman and Hall

Polis GA Power ME Huxel GR (eds) (2003) Food Webs at the Land-scape Level University of Chicago Press

Post DM (2002) The long and short of food-chain length TrendsEcol Evol 17269ndash277

Strong DR (ed) (1988) Food web theory a ladder for picking straw-berries Special Feature Ecology 691647ndash1676

Warren PH (1994) Making connections in food webs Trends EcolEvol 9136ndash141

Woodward G Ebenman B Emmerson M Montoya JM Olesen JMValido A Warren PH (2005) Body size in ecological networksTrends Ecol Evol 20402ndash409