Institute of Molecular Biotechnology, Beutenbergstr. 11, 07745 Jena, Germany
Received 9 October 2003; received in revised form 4 May 2004; accepted 4 June 2004
Abstract
It is argued to study better non-binary (weighted) food webs where links are measured quantitatively, instead of binary(unweighted) food webs where links are just present or absent. Binary webs can simply be deduced from the more gen-eral non-binary ones: if a flux is larger than a critical flux value (below denoted by “cut”) a link is drawn, otherwise it isnot. However, the amount of a flux between two nodes critically depends on the biomass of these nodes. Unfortunately, aclear definition of what to subsume into one node is missing up to now. The simplest (and probably most important) sta-tistical description of non-binary food webs is given by the flux distribution function. However, this function may stronglybe changed by just merging two nodes into one (as we have shown). In order to solve this decisive problem we proposeto setup food webs with all nodes containing the same amount of biomass:equal-node-webs. A simple merging of just twonodes is forbidden here, because such a merged node would contain more biomass. The best basis for such a new foodweb theory would be provided byempirical equal-node-webs, established by field ecologists. However, such data has notbeen available up to now. Nevertheless, we demonstrate some advantages of studying equal-node-webs. For that purpose,fi hed non-b differents bs. There-f es of foodw
rst we show—by means of two example food webs—how equal-node-webs can be deduced from already publisinary webs. Second, we describe the deduced equal-node-webs by their flux distribution functions as well as bytatistical measures. It is shown that equal-node-webs are more scale invariant than common non-binary food weore, equal-node-webs provide a crucial, trustworthy basis for sound statistical descriptions and comparative analysebs.Moreover, the article contains a new detailed comparison between a highly productive lake and a low productive
hown that the food web of the latter has a higher complexity than that of the former.2004 Elsevier B.V. All rights reserved.
eywords:Food web theory; Standardization; Information theory; Connectance; Scale invariance
Most food web studies only deal with binaryformation: there is a link between two nodes or(Pimm et al., 1991; Solow and Beet, 1998; Williaand Martinez, 2000; Lassig et al., 2001; Winemille
216 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
et al., 2001). Of course, not all links have the sameimportance. Sometimes a very small flux may havebeen omitted, either intentionally or simply because itcould not have been observed. Thus,Paine (1988)wor-ried “that the manner in which field biologists such asmyself draw linkages is usually informal and idiosyn-cratic”.
The straightforward way out of this ambiguity is toestablish non-binary webs. Here, each observed flow isreported quantitatively and, therefore, in its right sig-nificance. In contrast to the binary food webs, for non-binary webs one has to define an appropriate time-spanover which all quantities should be averaged. The un-ambiguousness and the much additional information ofnon-binary webs justify the effort. Accordingly, differ-ent empirical examples can be found (Riemann et al.,1986; Ulanowicz, 1997). Recently, we have shown thatnon-binary food webs are crucially more informativethan binary ones(Wilhelm, 2003).
However, non-binary webs per se only remove am-biguity concerning links, but not concerning nodes!Thus, it still remains the important question of whatto subsume into one node. In 1993, 24 “students offood webs” called for a “need for standardization” andcomplained “the lack of methodological standards fordefining, observing, and reporting webs” (Cohen etal., 1993). In a tentative tone of voice the authors of-fered “recommendations. . . on how food webs can beimproved”. One point was “the real goal of measur-ing links quantitatively”, that means to establish non-b ntot orsp des,o ey’sd ves,s rtedaP osea
laya s, itw ona her-m ciesd is as lit-t all)(
However, the amount of a flux between two nodesstrongly depends on the amount of biomass sub-sumed into these nodes. If the interaction is of Lotka–Volterra type, the flux is directly proportional to thetwo biomasses. Therefore, in order to compare differ-ent non-binary food webs, it should be clear how todefine the nodes. As long as there is no ultimate inter-national standard of assigning biomass into nodes infood webs, each non-binary food web is still a quitearbitrary one. This cannot be the basis for sound sta-tistical analyses. Consider, for example, the distribu-tion of the amount of fluxes: if one merges two nodesinto one node, the absolute number of fluxes proba-bly decreases, but the number of large fluxes increases(probably some fluxes of the two original nodes sum tolarger fluxes). The important characteristic trait of thefood web, the flux distribution function, is changed.Clearly, such merging falsifies any flux statistics andany measures based on the amount of the fluxes. With-out a standard, two independently working ecologistswould very likely propose different food webs for ex-actly the same ecosystem. On the other hand, with aclear standard two independent ecologists would es-tablish exactly the same food web for the same ecosys-tem. Such a standard is, therefore, the presuppositionto find meaningful differences for different ecosystems,for instance, for near-natural and man-made disturbedones. The straightforward way to remove the arbitrari-ness concerning nodes is to establishequal-node-webs,where all nodes contain the same biomass. We are nota datah ane
oft assd ebs)f ec-t oodw r al onf anso criben oths e. Inp s tot ebm es-t r? It
inary webs. Furthermore, “lumping organisms irophic species. . . is not desirable”, instead the authroposed to use individual biological species as nor “when different consumers specialize on a prifferent parts, such as a plant’s sap, phloem, leatems, or roots, the different parts should be repos distinct units”. The papers ofCousins (1990)andahl-Wostl (1993)represent other attempts to propstandardisation.However, at least for microorganisms, which pcentral role in material recycling in ecosystemould be nearly impossible to quantify all fluxesspecies level. Here, lumping is inevitable. Furtore, a problem already arises at the level of speefinition: “we can no longer comfortably say whatpecies anymore” (D. Drell), and must confess “howle we understand about species’ definitions” (J. Wcitations fromPennisi, 2001).
ware of any empirical example where food webave been collected with the aim of establishingqual-node-web.
In Section 2, we demonstrate the deductionwo different non-binary webs with reduced biomependency (normalized webs and equal-node-w
rom already published non-binary food webs. In Sion 3, these webs are compared with the original febs (one for a highly productive lake and one fo
ow-productive lake) on the basis of flux distributiunctions. In Section 4, we compare all webs by mef different statistical measures developed to deson-binary webs. One main point of discussion in bections concerns the problem of scale invariancarticular, we consider the problem: what happen
he flux distributions and to the statistical food weasures if the original food web would have been
ablished with some of the nodes lumped togethe
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 217
is shown that equal-node-webs are more scale invari-ant than the original non-binary webs. This is a furtherargument for equal-node-webs as a basis for sound sta-tistical analyses and statistical comparisons betweendifferent food webs.
2. Deduction of equal-node-webs
In order to deduce non-binary webs with reducedbiomass-dependence of the fluxes, one has to know thebiomass of all nodes in the original food web.Riemannet al. (1986)published thoroughly established non-binary food webs which fulfill this condition. The webscontain quantitative information of all trophic levels,but are nevertheless simple enough for our demonstra-tion purpose.Fig. 1a shows the original food webTfor a highly productive lake, andFig. 2a that for a low-productive one (dimensions in the legend ofFigs. 1 and2; in each case averaged values for one lake and 1 year).In the original publication (Riemann et al., 1986) thesum of the influxes equals the sum of the effluxes foreach node. For the sake of simplicity and because of theirrelevance for our demonstration purpose, the “outer”fluxes from and to the surrounding have been omittedhere.
Qualitatively, both food webs contain the samefluxes. The only exception is the missing flux fromphytoplankton to fish in the low-productive lake. Rie-mann et al. were aware that, also in the low-productivel nk-t notq s int ller,w x inF uset hant fluxfm mps sq
2
unto
Fig. 1. Food webs of a highly productive lake. (a) The original foodweb (T ) (Riemann et al., 1986). The numbers in the nodes quantifythe standing stocks (g C m−2), averaged over 1 year. The numberson the arrows indicate carbon flows in g C m−2 per year. (b) The cor-responding normalized food web (Tn). (c) The corresponding equal-node-web (Teq). Nodes 1–50 correspond to the phytoplankton node,etc.
ake, there should be a small flux from phytoplaon to fish, but because of its smallness they haveuantified it. Because of the lower standing stock
he low-productive lake, here all fluxes are smahether not about the same factor. The largest fluig. 1a is from phytoplankton to bacteria, but beca
he biomass of microzooplankton is much smaller that of phytoplankton, one can assume that, e.g. therom bacteria (b) to microzooplankton (mi)tbmi is muchore intense. In Fig. 2a, the largest flux points frohytoplankton to macrozooplankton, but also heretbmihould be more intense. The normalized food webTn
uantify the intensity.
.1. Normalized food webs (Tn)
In food webs, in a first approximation the amof a flux tij from nodei to nodej is proportional to
218 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
Fig. 2. Food webs of a low-productive lake. (a) The original foodweb (T ) (Riemann et al., 1986). The numbers in the nodes quantifythe standing stocks (g C m−2), averaged over 1 year. The numberson the arrows indicate carbon flows in g C m−2 per year. (b) The cor-responding normalized food web (Tn). (c) The corresponding equal-node-web (Teq).
the biomasses of both the nodes:tij ∼ mimj (mi de-notes biomass of nodei). If the interaction betweentwo nodes is of the Lotka–Volterra type, the propor-tionality is exactly valid, in other cases only approxi-mately. Note that for Holling Type II-kinetics linearityholds for low prey concentrations, and for Beddington–DeAngelis- and Crowley–Martin-kinetics only for lowprey and low predator concentrations(Skalski andGilliam, 2001). However, with the simple normaliza-tion tnij = tij/(mi mj) for each fluxtij one obtains thenormalized food webTn, which is much more indepen-dent of the biomasses than the original food webT . Fig.1b showsTn for the highly productive lake, andFig. 2bTn for the low-productive lake. In both the cases,tnbmiis the most intense flux. Furthermore, the normalizedflux from phytoplankton (p) to fish (f)tnpf in Fig. 1b ismore than an order of magnitude smaller than the otherfluxes, so it could be omitted for the highly productivelake, too.
However, the procedure of normalization removesthe dependence of biomasses not exactly (kinetics canbe of Holling Type II, for instance). Another possibil-ity for a reduction of the biomass-dependence is thenon-dimensionalizationtnonij = tij M/(TST
√mimj),
whereM denotes the sum of all biomasses and TSTthe total system throughflow, i.e. the sum of all fluxes.However, in both the cases,Tn and Tnon, the origi-nal information about biomasses is lost. Two differ-ent food webs could give the same normalized ornon-dimensionalized web, information is lost in botht h allb om-p ameT , ared x-a witht don
2
ebsf idet des,e nallys ot al-w der.I sses
he cases. For instance, a second food web witiomasses doubled and all fluxes quadrupled (in carison to a hypothetical first food web) gives the sn than the first one. If both, biomasses and fluxesoubled one obtaines the sameTnon. These are just emples, there are other more heterogenous webs
he sameTn or Tnon. Equal-node-webs, in contrast,ot loose any information.
.2. Equal-node-webs (Teq)
Here, it is shown the deduction of equal-node-wrom usual non-binary webs. The idea is to subdivhe nodes containing much biomass into various noach one containing as much biomass as the origimallest node. Of course, the smallest biomass nays divides the larger biomasses without a remain
n order to ensure this, the according excess bioma
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 219
of the larger nodes must at first be removed and thecorresponding fluxes must be reduced accordingly.
We demonstrate the simple deduction procedureof Teq with the help of the example: flux fromphytoplankton to bacteria (tpb) in the low-productivelake. In Fig. 2a, we seetpb = 20. The smallest nodeis microzooplankton with a biomass of 0.15. Thisnumber divides neither 0.2 nor 1 (biomasses of bacteriaand phytoplankton, respectively) without remainder.We divide the phytoplankton node into six nodes(6 × 0.15 = 0.9) and neglect the surplus biomass of0.1 phytoplankton. That means we consider a flux of20× 0.9 = 18 from phytoplankton to bacteria. Ac-cordingly we neglect a biomass of 0.05 bacteria. Thus,only the amount of 18× 0.15/0.2 = 13.5 reaches theremaining bacteria node with the biomass 0.15. Aswe have six equal phytoplankton nodes, from eachnode flows 13.5/6 = 2.25 to the one bacteria node.All other fluxes are handled with the same procedure.
Figs. 1c and 2c show the equal-node-websTeq forthe highly productive lake and the low-productive one.In the case of the highly productive lake (Fig. 1c), phy-toplankton is divided into 50, and fish into 33 equalnodes. The fluxes between these nodes are by far thesmallest ones, in accordance withtnpf .
The splitting up of an original fluxtij into equalfluxesteqij is the simplest possibility, but one could alsouse other procedures, such as dividing with the result ofa power law distribution or Gaussian distribution withmeanteq . Furthermore, it is the simplest possibilityt atedf uitec Thes nda bef
justa bs.I re noe ebsw eent
3
ts be
defined as:
C = L
n2,
L denotes the actual number of links in the binary webandn the number of nodes (seeWarren (1994)for a listof other connectance definitions). Besides this simplepossibility of calculation, there are two further reasonsfor its popularity.
First,C enters directly the famous May–Wigner for-mulaα2nC < 1 (May, 1972), which states that a steadystate of a non-linear ordinary differential equation islocally stable with high probability if this condition isfulfilled (α denotes standard deviation of a Gaussiandistribution with zero mean from which the elementsof the Jacobian has been taken, the main diagonal el-ements furthermore have been reduced by one). Forn → ∞ the probability goes to one. This result hasoften been discussed in the context of the diversity–stability debate (cf.Wilhelm and Bruggemann, 2001).
Secondly, apparent regularities ofC has been dis-covered in many food webs. Some authors found thatnC should be constant (i.e.L ∼ n, linear link scal-ing) (Cohen and Briand, 1984; Sugihara et al., 1989;Solow and Beet, 1998). In contrast,Martinez (1992)proposed theconstant connectancehypothesis, i.e.C =const. (quadratic scaling). However, his linear regres-sion analysis resulted in a scaling with an exponent1.54. Recently, we have proposed an elementary dy-n ebsy
ijo assume no flux between the equal nodes originrom a subdividing of one node. This surely is not qorrect for all nodes, for example pike eats roach.tatistically most realistic variant of splitting nodes attributing fluxes between the equal nodes could
ound in a further study.However, the deduction of equal-node-webs is
n insufficient first step for studying equal-node-wet is necessary and needed as long as there ampirical equal-node-webs. Such empirical would automatically establish the right fluxes betw
he equal nodes.
. Connectance and connectance functions
Without doubt, the connectanceC is the mostudied measure of binary food webs. It can
amic model where the self organization of food wields an exponent 1.5(Wilhelm, 2003).
However, the connectanceC is still a matter of deate.Bersier et al. (1999)summarized the opinionsifferent authors: “elucidating the mechanisms resible for determining the number of trophic linksommunities remains a primary aim of food web egy”. Nevertheless, by dealing with just binary wne always faces the difficulty of arbitrariness. S
lar to Riemann et al. (1986)who not addedtpf inhe food web of the low-productive lake (Fig. 2a), its always a question of what to do with very smuxes. An important step forward wentGoldwassend Roughgarden (1997)who clearly demonstrate
hat link density is influenced by sampling effort: wore effort one also recognizes more small fluxes
he connectance increases. Thus, it has been proo establish yield–effort curves (Cohen et al., 1993).
220 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
Fig. 3. The connectance ofT, Tn, andTeq as function of “cut” for the highly productive lake (indicated by index 1, small filled symbols), andthe low-productive lake (index 2, large empty symbols).
However, we think it is better to setup non-binary foodwebs, because here all (observed) fluxes are countedin their right weighting just from the beginning. Forthese non-binary webs one can analyze the cumulativeflux distribution function (CDF) of the fluxesp(tij >
cut). Note thatp(tij > cut) ≡ p(cut) = L(cut)/L(0),and C(cut) = L(cut)/n2 (L(cut) denotes the numberof links in the binary web for the given value of cut,andL(0) the number of links at cut= 0). It followsC(cut) = p(cut)L(0)/n2, thus the CDFp(cut) and theconnectance functionC(cut) are equivalent up to thefactorL(0)/n2.
Fig. 3 shows the connectance functionC(cut)for T, Tn, Teq for the highly and low-productivelakes.
It can be seen that for the original food websT theconnectance of the highly productive lakeT1 is always(for all values of “cut”) higher than the connectance ofthe low-productive lakeT2. In contrast, for the normal-ized websTn, as well as for the equal-node-websTeq,the connectance is always higher in the low-productivelake. This supports the idea that the connectance in
lower charged lakes is generally higher than in highlyproductive lakes.
A straightforward way to study the scale invarianceof measures describing a network is to merge somenodes and to look what happens to the measures. Here,we have merged the nodes for each pair of vertices in theoriginal food webs (T ). For each of then(n − 1)/2 =10 different four-node-websT4ij (nodesiandj merged)the corresponding normalized webT4nij and the equal-node-webT4eqij has been calculated.
Fig. 4 shows the connectance functions for all net-works (T, Tn, Teq) of the highly productive lake, andFig. 5 that of the low-productive lake. In contrast toFig. 3, here the “cut”-range has not been linearly di-vided, but the connectance has been calculated only for“cut”-values whereC changes. This results in a muchmore concise picture and is a better basis for quantita-tive statistical analyses (for instance, fitting of the rightdistribution).
Figs. 4a and 5a show the connectance functions ofT andT4ij. It can be seen that in the case of the highlyproductive lake the function roughly follows a power
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 221
Fig. 4. The connectance functions of the highly productive lake. (a) Original food web; (b) normalized food web; (c) equal-node-web. Thenumbersi, j indicate the merged nodesi andj.
222 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
Fig. 4. (Continued).
law, whereas in the low-productive lake it seems to bean exponential law. For both lakes, the connectance ofeach merged network (T4ij) is always higher than in thecorresponding unmerged five-node-web.
Figs. 4b and 5b show the connectance functionsof the normalized networks, and Figs. 4c and 5c thatof the equal-node-webs. First, in contrast to the orig-inal webs, here the graphs for the five-node-webs arequalitatively the same in the highly and low-productivelake: all functions seem to follow an exponential decay.Secondly, in all these four cases (Figs. 4b and c, and5b and c) the four-node-networks havenot an alwayshigher connectance than their corresponding five-node-networks (which was the case for the original networksT ). Thus, with respect to connectance functions, it fol-lows that both, normalized networks as well as equal-node-webs are more scale invariant than the originalnetworks.
Nevertheless, in all the cases, a merging of twonodes, of course, changes the connectance function.Therefore, the best would be to have a standard of whatto subsume into one node. Then it would be excludedthat one ecologist would (unconsciously) merge somenodes, in contrast to an independently working ecolo-
gist who would consider these nodes explicitely. Ourequal-node proposal may be an important step towardssuch a standardisation.
4. Information theoretic measures for adescription of non-binary food webs
In addition to direct analyses of flux distributions,information theoretic measures are another possibil-ity to characterize non-binary networks.MacArthur(1955)was the first who has described food webs inthis manner. He proposed that entropy, defined in termsof the fluxestij, should grow during the successionof ecosystems (cf.Wilhelm and Bruggemann (2000)for a detailed discussion of ecosystem succession andmaturity). Rutledge et al. (1976)were the first whodistinguished two distributions in directed non-binaryfood webs: the distribution of the in- and that of the ef-fluxes. Accordingly, one can define measures dealingwith two distributions which are more appropriate de-scriptors of networks than MacArthur’s entropy. Here,we consider the joint entropyH (average uncertaintyabout the origin and the destination of a single flux),
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 223
Fig. 5. The connectance functions of the low-productive lake. (a) Original food web; (b) normalized food web; (c) equal-node-web. The numbersi, j indicate the merged nodesi andj.
224 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
Fig. 5. (Continued).
the redundancyR, and the mutual informationI (aver-age information about an influx if an efflux is known,and vice versa). The precise definitions are as follows(Pahl-Wostl, 1995):
H = −∑
i
∑
j
tij log tij,
R = −∑
i
∑
j
tij logt2ij∑
k tkj∑
l til,
I = H − R =∑
i
∑
j
tij logtij∑
k tkj∑
l til.
Here, tij denotes the normalized flux from nodei tonodej: tij = tij/TST, with TST= ∑
i
∑j tij (tij is the
non-normalized flux).It is interesting to note that for many years there was
as controversial debate about which measures are maxi-mized in natural food webs. For instance,Perez-Espanaand Arreguin-Sanchez (1999)proposed a scaled ver-sion of the joint entropy. These authors follow the ar-gumentation for the diversity of flows ofOdum (1953),Rutledge et al. (1976), etc. Others, inspired byMay(1972), proposed the opposite. It has been argued
(Hirata and Ulanowicz, 1984; Ulanowicz, 1997) thatduring succession of ecosystems autocatalytic cyclesarise which lead to veryarticulated food webs. Ac-cordingly, these authors proposed a scaled version ofmutual information, called ascendency, as the measurewhich is maximized by nature.
Recently, we proposed a resolution of this contra-diction (Wilhelm and Bruggemann, 2001). We arguedthat the truth lies in between and defined as appropriatemeasure themedium articulation:
MA = IR = (H − R)R.
This quantity has the typical feature of a complexitymeasure which is zero at the extremes (I = Imax, R =Rmax) and maximal in between (cf.Shiner et al.,1999). We have given evidence that natural foodwebs have indeed maximized their medium articu-lation (Wilhelm and Bruggemann, 2001). From thedefinitions of the measures one can easily deducetheir corresponding maximal values:Hmax = Rmax =2 log n; Imax = log n; MA max = (log n)2/2(Wilhelmand Bruggemann, 2001). With these maximal valuesone can normalize these measures.Table 1shows the
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 225
Table 1Information theoretic measures forT , Tn andTeq of the highly andlow-productive lake
MAT 0.627 0.535 0.351 (35) 0.745 (12)Tn 0.539 0.531 0.277 (45) 0.724 (25)Teq 0.488 0.356 0.261 (35) 0.527 (14)a In parenthesis are given the numbers of the merged nodes of the
corresponding four-node-webs.
four normalized measuresH, R, I and MA for T , Tn
and Teq for the highly productive lake and the low-productive one.
The first column shows the values for the five-node-webs (T5). It can be seen that the medium articulation(MA) is higher for the food webs of the low-productivelake. This result holds forT , Tn andTeq, but forTn andTeq this difference is even more pronounced. This isconsistent with the idea that near-natural ecosystems
are more complex than man-made disturbed ones (MAis a complexity measure).
The second column shows the average values of allfour-node-webs (T4ij), and the third and fourth columnsshow the minimal and maximal values of the corre-sponding four-node-webs. For instance, for the orig-inal food web (T ) of the highly productive lake thefour-node-web, which results from a merging of theNodes 3 and 5, has the lowest joint entropyH , namelyH = 0.554.
Generally, the deduced normalized websTn andequal-node-websTeq are more scale invariant than theoriginal food websT . Except for the joint entropyH ,in Tn andTeq the differenceT5 − 〈T4〉 is smaller thanin T . Furthermore, only forT theT5-value may be out-side the range ofT4min andT4max: this happens for theredundancyR of both lakes.
These questions need further studies. In a first stepthis can be done with equal-node-webs deduced frommore detailed non-binary food webs (such as the onesgiven in Ulanowicz, 1997). More importantly, futurefield studies could createempiricalequal-node-webs.These webs would be the best basis for appropriatestudies of scale-invariance.
5. Discussion
Food webs are mostly described by binary measures,such as diameter of the network, clustering coefficient,c rym i-n foodw rt”(F on-s ardf as af
arem s, ort tly,w uchmIe en;Ms s, e.g.
ycling index, etc. (for definitions and further binaeasures seeWilhelm, 2003). Unfortunately, these bary measures are not unambiguously defined forebs, but depend on “cut”, or the “sampling effo
which are closely connected, cf.Bersier et al., 1999).or the connectance this “cut”-dependence is demtrated inFig. 3. As long as there is no definite standor “cut”, all binary measures should be discussedunction of “cut”.
We have argued that non-binary food websore appropriate to compare different ecosystem
o find general regularities for food webs. Recene have shown that non-binary measures are more meaningful than binary ones(Wilhelm, 2003).
nstead of the often discussed functionsC = f (n) (lin-ar or quadratic link scaling, or something in betweartinez, 1992; Bersier et al., 1999; Wilhelm, 2003) it
eems better to discuss some non-binary measure
226 T. Wilhelm / Ecological Modelling 181 (2005) 215–227
MA = f (n). In the same spirit of preferring non-binarymeasures,Ulanowicz (2002)introduced an effectiveconnectance, defined with the help of the redundancyR: CE = exp(R/2). Recently,Zorach and Ulanowicz(2003)furthermore introduced an effective number offlows LE, as well as an effective number of nodesnE(for binary networks the effective numbers of flowsand nodes are identical to the usual numbers of flowsand nodes). The quotientnE/CE is termed “numberof roles” and is proposed to be an appropriate com-plexity measure for non-binary food webs(Zorach andUlanowicz, 2003).
However, also non-binary food webs need a stan-dard for their establishment. Here, we proposed tostudy equal-node-webs. We have shown how normal-ized food webs and equal-node-webs can be deducedfrom already published non-binary food webs (Section2), and how all these non-binary webs can be charac-terized by their connectance functions (equivalent toflux distribution functions, Section 3) and by differentinformation theoretic measures (Section 4).
The obvious problem of scale invariance of foodwebs has often been discussed(Sugihara et al., 1989;Solow and Beet, 1998; Bersier et al., 1999). We haveshown that both, normalized food webs and equal-node-webs are more scale invariant than the originalnon-binary webs.
McCann et al. (1998)described non-binary fluxes infood webs qualitatively: “Data on interaction strengthin natural food webs indicate that food web interactions inter-a sedo oodw tureo nal-y forc sys-t lity( uldd utd ribu-t hus,fl fore ayte
ebs( ;
Williams et al., 2002; Montoya and Sole, 2003;Wilhelm, 2003) seems better to be done in standard-ized webs. A standardization for food webs is perfectif two different ecologists would setup independentlyfrom each other exactly the same web for the sameecosystem (if they consider the same time and space).For that purpose for equal-node-webs one has to definethe parameteramount of biomass. Furthermore, one hasto definewhichbiomasses should be subsumed into theequal nodes. We are not aware of any other suggestionfor a standardization of food webs with as few ambi-guity as ourequal-nodeproposal.
We argued that for quantitative comparative anal-yses equal-node-webs are the right basis and thinkthat one should createempirical equal-node food websand analyse them. Of course, ecologists are usedto deal with non-equal nodes, if the nodes repre-sent species, for instance, than these nodes have notthe same biomass. However, for such usual descrip-tions the equal-node-webs can subsequently simplyand unambiguously be merged into the common morewell-known form. The advantages of equal-node-websshould justify the effort to establish them.
However, as long as there are no empirical equal-node-webs available, one can nevertheless study typi-cal features and advantages of such webs by using ourdeduction procedure given in this article. Our studiescould be expanded by using more detailed non-binaryfood webs (e.g.de Ruiter et al., 1995; Lyche et al.,1996; Ulanowicz, 1997). In order to evaluate the ro-b anal-y idedn andw orig-i rentfl
trengths are indeed characterized by many weakctions and a few strong interactions.” However, ban a clear standard of establishing non-binary febs (our proposal: equal-node-webs), in the fune can achieve a more quantitative description. Ases of flux distributions should be a good methodomparisons between food webs of different ecoems. The distributions could be of different quae.g. power-law or exponential distribution) or coiffer only quantitatively (same distribution type, bifferent parameters). Perhaps, differences in dist
ions could indicate disturbances of ecosystems. Tux distributions could be used as an instrumentvaluating the integrity of ecosystems in a similar whan biomass-distributions are already used(Steinbergt al., 1998).
However, the search for regularities in food wLassig et al., 2001; Wilhelm and Bruggemann, 2001
ustness of the deduced equal-node-webs one canse what happens if the original fluxes are subdivot equally, but according to other distributions,hat happens if the equal nodes arising from one
nal node are connected among each other by diffeuxes.
For both, empirical and/or deduced equal-noebs it is interesting to study whether some measu
eally maximized, for instance MA. In order to stucale invariance one should also merge more thanwo nodes. For empirical equal-node-webs it is impant to study the relevance ofhow muchandwhichiomass is subsumed into the equal nodes. A poseneral result could be: the larger the resolutionmaller the dependence onwhich. Last but not the leasne should investigate the problem: is it for a comp
ive analysis of ecosystems necessary to considerebs of the whole ecosystem, or does already a
T. Wilhelm / Ecological Modelling 181 (2005) 215–227 227
system (e.g. a food web of microorganisms) indicateall relevant characteristics?
If someday a clear standard for food webs exists andthe regularities (flux distribution functions, appropriatemeasures for non-binary webs) are known, then one cantry to understand the evolutionary mechanisms gener-ating these regularities (Drossel et al., 2001; Wilhelm,2003).
References
Bersier, L.-F., Dixon, P., Sugihara, G., 1999. Scale invariant or scale-dependent behavior of the link density property in food webs: amatter of sampling effort? Am. Nat. 153, 676–682.
Cohen, J.E., Briand, F., 1984. Trophic links of community food web.Proc. Natl. Acad. Sci. U.S.A. 81, 4105–4109.
Cousins, S.H., 1990. Countable ecosystems deriving from a new foodweb entity. Oikos 57, 270–275.
de Ruiter, P.C., Neutel, A.-M., Moore, J.C., 1995. Energetics, patternsof interaction strengths, and stability in real ecosystems. Science269, 1257–1260.
Drossel, B., Higgs, P.G., McKane, A.J., 2001. The influence ofpredator–prey population dynamics on the long-term evolutionof food web structure. J. Theor. Biol. 208, 91–107.
Goldwasser, L., Roughgarden, J., 1997. Sampling effects and the.
H ysis
L ape
L rger(Part
M ea-
M food
M 238,
M rac-
M od102
O lphia.P cross
Pahl-Wostl, C., 1995. The Dynamic Nature of Ecosystems. Wiley,New York.
Paine, R.T., 1988. Food webs: road maps of interactions or grist fortheoretical development? Ecology 69, 1648–1654.
Pennisi, E., 2001. Sequences reveal borrowed genes. Science 294,1634–1635.
Perez-Espana, H., Arreguin-Sanchez, F., 1999. A measure of ecosys-tem maturity. Ecol. Model. 119, 79–85.
Riemann, B., Søndergaard, M., Persson, L., Johansson, L., 1986.Carbon metabolism and community regulation in eutrophic, tem-perate lakes. In: Riemann, B., Søndergaard, M. (Eds.), CarbonDynamics in Eutrophic, Temperate Lakes. Elsevier, New York,Chapter 7.
Rutledge, R.W., Basore, B.L., Mulholland, R.J., 1976. Ecologicalstability: an information theory viewpoint. J. Theor. Biol. 57,355–371.
Shiner, J.S., Davison, M., Landsberg, P.T., 1999. Simple measure forcomplexity. Phys. Rev. E 59, 1459–1464.
Skalski, G.T., Gilliam, J.F., 2001. Functional responses with predatorinterference: viable alternatives to the Holling Type II model.Ecology 82, 3083–3092.
Solow, A.R., Beet, A.R., 1998. On lumping species in food webs.Ecology 79, 2013–2018.
Steinberg, C., Schafer, H., Beisker, W., Bruggemann, R., 1998. De-riving restoration goals for acidified lakes from ataxonomic phy-toplankton studies. Restor. Ecol. 6, 327–335.
Sugihara, G., Schoenly, K., Trombla, A., 1989. Scale invariance infood web properties. Science 245, 48–52.
Ulanowicz, R.E., 1997. Ecology, the Ascendent Perspective.Columbia University Press.
Ulanowicz, R.E., 2002. The balance between adaptability and adap-tation. Biosystems 64, 13–22.
Warren, P.H., 1994. Making connections in food webs. Trends Ecol.
W ary
W lop-
W uresH.,
l andn, pp.
W lex
W ,ebs.
W webNat.
Z xity(3),
estimation of food web properties. Ecology 74, 1216–1233irata, H., Ulanowicz, R.E., 1984. Information theoretical anal
of ecological networks. Int. J. Syst. Sci. 15, 261–270.assig, M., Bastolla, U., Manrubia, S.C., Valleriani, A., 2001. Sh
of ecological networks. Phys. Rev. Lett. 86, 4418–4421.yche, A., Andersen, T., Christoffersen, K., Hessen, D.O., Be
Hansen, P.H., Klysner, A., 1996. Mesocosm tracer studies1 and 2). Limnol. Ocean 41, 460–487.
acArthur, R., 1955. Fluctuations of animal populations and a msure of community stability. Ecology 36, 533–536.
ay, R., 1972. Will a large complex system be stable? Nature413–414.
cCann, K., Hastings, A., Huxel, G.R., 1998. Weak trophic intetions and the balance of nature. Nature 395, 794–798.
ontoya, J.M., Sole, R.V., 2003. Topological properties of fowebs: from real data to community assembly models. Oikos(3), 614–622.
dum, E.P., 1953. Fundamentals of Ecology. Saunders, Philadeahl-Wostl, C., 1993. Food webs and ecological networks a
temporal and spatial scales. Oikos 67, 415–432.
Evol. 9, 136–141.ilhelm, T., 2003. An elementary dynamic model for non-bin
food webs. Ecol. Model. 168, 145–152.ilhelm, T., Bruggemann, R., 2000. Goal functions for the deve
ment of natural systems. Ecol. Model. 132, 231–246.ilhelm, T., Bruggemann, R., 2001. Information theoretic meas
for the maturity of ecosystems. In: Matthies, M., Malchow,Kriz, J. (Eds.), Integrative Systems Approaches to NaturaSocial Sciences—Systems Science, 2000. Springer, Berli263–273.
