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Quantifying the interaction structure and the topological importance of species in food webs: A signed digraph approachWei-chung Liu a , Hsuan-Wien Chen b ,c ,n , Ferenc Jorda n d , Wen-Hsieh Lin a , Chester Wai-Jen Liu ea Institute of Statistical Science, Academia Sinica, Taiwanb Department of Life Science, National Taiwan Normal University, Taiwanc Biodiversity Research Center, Academia Sinica, Taiwand The Microsoft Research, University of Trento, Centre for Computational and Systems Biology, Italye Department of International Business, Chang Jung Christian University, Taiwan

a r t i c l e i n f o

Article history:Received 12 February 2010Received in revised form22 July 2010Accepted 31 August 2010Available online 8 September 2010

Keywords:NetworkCentrality indicesKeystone speciesMixed trophic impactTrophic cascade

a b s t r a c t

Due to the structural complexity of nature, it is not always easy to identify topologically importancespecies in an ecosystem. In the past decade, several studies in ecology have developed methods formeasuring species importance basing on direct and indirect inter-specic interactions. Here, byextending a previously developed methodology, we present an approach that can quantify theinteraction structure of a food web and consequently the topological importance of species whenthe food web is viewed as a signed digraph. The basic principle behind our approach is to determine thesign and strength of direct and indirect interactions for all pathways up to a predened number of steps.Our approach mainly differs from the previous methodology in that we are able to quantify the strengthof inter-specic interaction as well as in what way species interact with each other, as it can explicitlyquantify a wide range of ecological interactions such as cascading effect, indirect food supply effect,apparent and exploitive competitions in the same framework. This then allows us to quantify the

topological importance of a species and examine whether it is a predominately positive or negativeinteractor in a food web. Furthermore, our analysis reveals that positive and negative effects from onespecies on others eventually cancel each other out for longer pathways resulting in stable interactionstructure. Applications of our methodology include providing a more informative index forconservation biologists, and the potential use of interaction structure derived from our approach infood web robustness studies is also discussed.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A food web is in essence a network model of an ecosystem thatdescribes the trophic interactions between species. Species in afood web are often connected, either directly or indirectly, thuschanges in the abundance of one species could potentially affectthe abundance of others ( Wootton, 2002 ). Some human activities(e.g. overshing) clearly change the abundance of exploitedspecies ( Jennings and Kaiser, 1998; Daskalov, 2002 ), thus thereare concerns over how those effects might ripple through andaffect the entire ecosystem ( Frank et al., 2005; Baum and Worm,2009 ). With the fear of species extinction and the collapse of theentire ecosystem in mind, there have been a growing number of food web studies in the past decade ( Ings et al., 2009 ). Theseinclude studies on the assemblage and structural robustness of

food webs ( Sole and Montoya, 2001; Dunne et al., 2002;Dunne and Williams, 2009 ) as it is imperative to understandhow species are embedded in a food web such that the potentialeffect of disturbance in species abundance can be easilyunderstood ( Wootton, 1994a, 1998; Dambacher et al., 2003;

Ebenman et al., 2004; Montoya et al., 2009 ). On a more practicalside, especially as a guide for conservation efforts, many food webstudies provide informative indices measuring species impor-tance and subsequently identify the so-called topologicalkeystone species ( Ebenman and Jonsson, 2005; Jorda n et al.,2006; Jorda n, 2009 ).

Since a food web is a network, the topological importance of aspecies can be dened as the positional importance of a node inthe network ( Jorda n et al., 2006; Jorda n, 2009 ). There exist severalindices for measuring this from the graph theoretical perspective(Wasserman and Faust, 1994 ). Starting from the most localmeasure we have the degree centrality, or simply the number of direct neighbours of a node. Then there are indices which considera nodes indirect neighbours as well as direct ones. One of

such indices is the closeness centrality that measures how close a

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Journal of Theoretical Biology

0022-5193/$- see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2010.08.035

n Corresponding author at: Department of Life Science, National Taiwan NormalUniversity, Taiwan, Tel: +886 2 27899545x214.

E-mail address: [email protected] (H.-W. Chen).

Journal of Theoretical Biology 267 (2010) 355362
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node is to others in the same network; and another non-localmeasure widely used is the betweenness centrality whichquanties how often a node is incident to all shortest pathsbetween node pairs. Furthermore, since networks can berepresented as adjacency matrices, indices such as eigenvectorand information centralities employ matrix-related properties

and operations to quantify nodal importance ( Bonacich, 1987;Stephenson and Zelen, 1989 ).

Species importance can also be measured by examining theinteraction between species and how a focal species can directlyand indirectly affect others in the same food web ( Menge, 1995;Yodzis, 2000 ). Indices for species importance viewed from thisperspective differ from those based on the graph theoreticalapproach in that they consider inter-specic interactions thatare of ecological relevance. For instance species importance inhostparasitoid communities can be quantied by measuring thestrength of apparent competition between species ( Memmottet al., 1994; M uller et al., 1999; Rott and Godfray, 2000 ), and thelong indirect effects of trophic cascade and indirect-foodsupply have been used as proxies to species importance in

multi-trophic level food webs ( Jorda n and Scheuring, 2002;Quince et al., 2005 ). In a more general manner, Jorda n et al.(2003) developed a centrality index basing on the direct andindirect effects of a focal species on all species, and such amethodology has been used to study the interaction structure andspecies importance of hostparasitioid communities ( Jorda n et al.,2003 ) and marine waspwaist systems ( Jorda n et al., 2005 ); andmore recently it has also been used to investigate the trophic eldoverlap of species in an ecosystem ( Jorda n et al., 2009 ). However,a trophic interaction implies in fact two directed links of oppositesigns (i.e. positive food supply from resource to consumer andnegative feeding effect from consumer to resource), and abovestudies on indirect effects generally do not consider this. Althoughthere have been a lot of researches on signed, direct and indirect

effects in networks ( Levine, 1976; Wootton, 1994a; Dambacheret al., 2003; Ebenman et al., 2004; Hosack et al., 2009; Montoyaet al., 2009 ), but they are rarely considered in the sameframework with interaction structure and topological importance(but see Vasas and Jorda n, 2006 ). An important exception is themixed trophic impact (MTI for short) analysis ( Ulanowicz andPuccia, 1990; Christensen and Walters, 2004 ) where the sign anddirection of trophic interactions are considered in determining theeffects of one species on others. However, this approach relies ona mass-balance model where parameters related to a speciestrophic input and output are often required for computation, andmany large food webs are not parameterized so richly. Further-more, this approach does not explicitly separate different types of indirect effects such as indirect food supply, trophic cascade,exploitative and apparent competitions ( Holt, 1977 ) that might beof interest to ecologists in general.

In the light of those points above, we present an approach toquantify the interaction structure of a food web (i.e. a signeddigraph) by extending the methodology of Jorda n et al. (2003) .Topological importance of a node is then dened by structurallyhow effects from it can affect all nodes in the same network. Thepaper is organised as follows. First, we describe how we convert afood web to a signed digraph, and we describe briey the foodweb we use to demonstrate our methodology. Second, we presentthe methodology of Jorda n et al. (2003) and show how it can bemodied for signed digraphs. Third, we present our result andcompare our ranking of species importance with those obtainedfrom other indices in network literature. Next, we then comparethe interaction structure obtained from our methodology and thatfrom MTI as means to test our new methodology. Lastly, wediscuss the implication of our results and future applications of our methodology.

2. Methods

2.1. Converting a binary food web into a signed digraph

A food web is a network where nodes and links representspecies and trophic interactions, respectively. A trophic interac-tion between a prey and a predator implies that the prey hasdirect positive inuence on the predator (food supply) whereasthe predator has direct negative inuence on the prey (feedingeffect). Thus a trophic interaction can be decomposed into twodirected and signed links. Here we dene that a species inuencesanother positively if the former increases the abundance of thelatter; and negatively if the former decreases the abundance of the latter.

2.2. Data

The food web we used describes the Kuosheng Bay ecosystemin North-East Taiwan ( Lin et al., 2004 ). We use the versionmodied by Jorda n et al. (2009) where there are 15 trophic groupsand 36 trophic links ( Fig. 1). Following the decomposition methodabove, the food web can be converted into a signed digraph with15 nodes and 72 links (36 positive and 36 negative).

2.3. Topological importance basing on direct and indirect effects for undirected and unsigned networks

We rst describe the methodology of Jorda n et al. (2003) . Let N be the number of nodes in a graph. Assuming node i is a directneighbour of node j, we dene a ij as the direct effect (i.e. one-stepeffect) of i on j

a ij 1D j

, 1

where D j is the number of node js direct neighbours. Next, for apathway consisted of several steps (a step means travelling alonga link), we assume an indirect effect is multiplicative in the sensethat it is the product of one-step effects. Consider a simple case

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Fig. 1. A modied food web of the Kuosheng Bay ecosystem in northern Taiwan(Lin et al., 2004; Jorda n et al., 2009 ). A circle represents a trophic group and a linkhere represents a trophic (or predatorprey) interaction linking one trophic groupto another. A lled circle indicates that the corresponding trophic group exerts apositive net effect on the network. An open circle indicates the correspondingtrophic group exerts a negative net effect on the network. The size of a circle isproportional to the magnitude of net effects exerted by the corresponding trophicgroup. Trophic groups are: (1) phytoplankton, (2) periphyton, (3) herbivorous

zooplankton, (4) carnivorous zooplankton, (5) infauna, (6) barnacles, (7) gastro-pods, (8) bivalves, (9) shrimp, (10) crabs, (11) detritivorous sh, (12) herbivoroussh, (13) zooplanktivorous sh, (14) benthic-feeding sh, and (15) piscivoroussh. The network is drawn by using UCINET ( Borgatti et al., 2002 ).

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where node i can reach node j in 2 steps and there is only one sucha pathway via node k (i.e. ik j). In this case, the effect of i on j viak is simply

a ik a kj : 2

Furthermore, we assume that effects are additive in the sensethat if there exist more than one pathway of length n from node ito node j, then the total effect will be the sum of indirect effects.For instance, in a case where node i can reach node j in 2 steps andthere are exactly two pathways for doing so (e.g. ik j and ih j),then the total effect from i on j in 2 steps is

a ik a kj a ih ahj : 3

In general we dene TE ij ,n as the total effect of i on j for n steps,and if there are m n -step pathways from i to j then TE ij ,n is the sumof m indirect effects.

For step lengths from 1 to h , we can organise those variouseffects of one node on another in matrix forms, and in the end

there will be h matrices of dimension N N . Here we let rows andcolumns of a matrix refer to effect originators and receivers,respectively, and construct matrix TEn whose elements are of theform TE ij ,n . Jorda n et al. (2003) suggested that the sum of the ithrow in TEn quanties the total effect of node i on all nodes for nsteps, and this can be regarded as the total effect node i on thenetwork for n steps ( g i,n )

g i , n XN

j 1TE ij , n : 4

Since a node can affect another via pathways of different steplengths, one needs to calculate the cumulative total effect bysumming up effects of all lengths if he or she wishes to quantifythe overall effect of one node on another. For effects up to n steps,we determine TE1 , TE2 , TE3 y TEn , and dene

CTEn TE1 TE2 TE3 ::: TEn , 5

where CTEn is a matrix whose ( i, j)th element represents thecumulative total effect of node i on node j up to n steps. Jorda net al. (2003) suggested that the sum of the ith row in CTEn is thenthe topological importance of species i in the network up to nsteps ( Z i,n )

Z i, n XN

j 1CTE ij , n : 6

From now we refer the topological importance of node i basedon the approach of Jorda n et al. (2003) , or the cumulative totaleffect of node i up to n steps, simply as Z i,n . Since Z i,n increaseswithout bound as n approaches innity, Jorda n et al. (2003)suggested one should predene a maximum n when he or shequanties the topological importance of nodes, and they furtherdene topological importance of node i as the averaged effect of ion all nodes up to n steps.

Similar indices can also be dened for networks with weightedlinks. All effects are dened in the same way as above with theexception of direct effects, which is now dened as:

a W ij e ij

m j, 7

where m j is the sum of strength of links connected to j while e ij isthe strength of the link connecting i and j. We denote all indicesderived from weighted links with a superscript W to distinguishthem from their un-weighted counterparts.

2.4. Topological importance basing on direct and indirect effects for signed diagraphs

Essentially all things are the same except for how we denedirect effects. First, we dene i- jS as the directed link from node ito node j (i.e. i is inuencing j). The superscript S is the sign of theinteraction: it is positive (+1) if node i inuences node j positivelyand negative ( 1) otherwise. Let node j has D j neighbours, and themagnitude of the unsigned direct effect from a neighbour i on j is

a ij 1D j

: 8

However, if the sign of interaction is taken into account, thenthe signed direct effect from i on j is

c i- jS S 1D j

S a ij: 9

Here, direct effects from js neighbours can be partitioned into

two sets: one for positive effects and the other for negative effects.Signed indirect effects from i on j is determined in the same wayas their unsigned and undirected counterparts. Since an indirecteffect is the product of several direct effects and some of thosemight take positive or negative values, thus an indirect effectfollowing a particular pathway is either positive or negative. Ingeneral, if there are m pathways from node i to node j in n stepsthen there will be m individual indirect effects, and each of which iseither positive or negative. We dene E ij ,n , + as the sum of positiveeffects of i on j in n steps and similarly dene E ij ,n ,- for negativeeffects. The net effect of i on j for n steps can then be dened as

NE ij, n E ij, n E ij , n , 10

which can be positive, negative or zero. Note that the sum of themagnitude of E ij ,n , + and E ij ,n , is the total effect of i on j for n steps

TE ij, n 9E ij , n 9 9E ij, n 9, 11

which is always positive. Note that when n 2, E ij,2,+ representsthe sum of all 2-step positive effects from i on j, and this includes the2-step indirect effects of indirect food supply (+ + +) and trophiccascade ( +). Moreover, when n 2, E ij, 2,- is the sum of all2-step negative effects from i on j, and this includes the 2-stepindirect effects of exploitive competition ( + ) and apparentcompetition (+ ).

Again, for step length n, we can construct matrices NEn , En ,+ andEn , whose elements are NE ij ,n , E ij ,n ,+ and E ij ,n , respectively. Thesum of the ith row in NEn is the net effect of node i for n steps ( xi,n ):

xi , n XN

j 1NE ij , n : 12

Furthermore we dene

CNEn NE1 NE2 NE3 ::: NEn , 13

where CNEn is a matrix whose ( i, j)th element is the cumulative net effect of node i on node j up to n steps in a signed digraph.Similarly, we dene the sum of the ith row in CNEn as thetopological importance of node i in the network up to n steps ( yi,n )

yi , n XN

j 1CNE ij , n : 14

From now on we refer the topological importance of node ibased on our approach, or the cumulative net effect of node i up ton steps, simply as yi,n .

Wei-chung Liu et al. / Journal of Theoretical Biology 267 (2010) 355362 357
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The same approach is also applicable to signed digraphs withweighted links with the exception of how we dene direct effects.A direct effect of node i on node j is now dened as

c W i- jS S e W ij

mW j, 15

where mW j is the sum of strength of links connected to j while e W ij isthe strength of the link connecting i to j. Again, we denote indicesderived from weighted links with a superscript W to distinguishthem from their un-weighted counterparts.

3. Results

We analysed the food web representing the Kuosheng Bayecosystem. We computed CTEn , CTE W n , CNEn and CNE W n for n 1 to20. From CTE20 , CTE W 20 , CNE20 and CNE W 20 we then determined Z i,20 ,Z W i, 20 , yi,20 and y

W i , 20 . For the weighted indices, we used the

consumption rates of consumers on resources ( Lin et al., 2004 ) aslink strengths. Although we have carried out the same analysis on

the weighted version of the Kuosheng Bay food web, however, forsimplicity, we will restrict our presentation and discussion to theun-weighted version of topological importance.

3.1. Topological importance

We rst present the result derived from the methodology of Jorda n et al. (2003) . Fig. 2 is a plot of Z i,n for n 120 for individualnodes. For small n , there are crossovers between lines implyingchanges in the topological importance ranking among nodes; andwhen n increases, the importance ranking becomes xed. Notethat Z i,n increases indenitely as the n increases, and Jorda n et al.(2003) suggested the topological importance of a node should bedivided by the number of step length considered, n . Fig. 3 is a plot

of yi,n for n 120 for individual nodes. Again as above, there arechanges in the ranking of topological importance among nodeswhen n is small, but the ranking becomes xed when n becomeslarge. Interestingly, in contrast to the plot in Fig. 2, yi,n do notincrease without bound to innity. Instead, they stabilize to axed and non-zero value: some are positive and some arenegative. This shows that some nodes are predominately positiveinteractors while the remaining ones are negative interactors. Thereason why topological importance here stabilizes as n increasesis that the magnitude of positive and negative effects for longer

pathways exerted by one node on the network are similar in size,therefore they cancel each other out. As a result, for large stepnumbers, the net effects are almost zero and contribute a little totopological importance.

Another difference between our methodology and that of Jorda n et al. (2003) is that our approach is able to quantifytopological importance not only basing on the size of the effect anode exerts on the whole network, but also in terms of themanner in which it inuences the whole network. For instance, if the food web is unsigned and undirected, then nodes 4 and 3 arethe most inuential nodes according to Z i,20 ; however, if the foodweb is treated as signed digraph, then nodes 4 and 3 are onlyweak positive interactors according to yi,20 (Fig. 4). Furthermore,according to yi,20 , our methodology reveals node 2 is the mostimportant positive interactor despite the observation that it is oneof the least inuential nodes according to Z i,20 (Fig. 4). As far asthe Kuosheng Bay ecosystem is concerned, both methodologies dosuggest node 14 being the most topologically important, and ourapproach suggests it is important as a negative interactor ( Fig. 4).Both methodologies also agree on node 8 being the leastimportant node in the system ( Fig. 4).

Step length, n

T o p o

l o g i c a

l i m p o r t a n c e

b a s e

d o n c u m u l a t

i v e

t o t a l e f f e c

t

35

30

25

20

15

10

5

00 5 10 15 20

Fig. 2. Topological importance of nodes as measured by their cumulative totaleffects ( Z i,n) for step length n 120. Each line represents a particular trophicgroup of the Kuosheng Bay ecosystem in Taiwan.

Step length, n

T o p o l o g

i c a l

i m p o r t a n c e

b a s e

d o n c u m u

l a t i v e

n e t e f

f e c t

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.50 5 10 15 20

Fig. 3. Topological importance of nodes as measured by their cumulative neteffects ( yi,n) for step length n 120. Each line represents a particular trophicgroup of the Kuosheng Bay ecosystem in Taiwan.

Topological importance based on cumulative total effects

T o p o l o g i c a

l i m p o r

t a n c e

b a s e d o n c u m u

l a t i v e

n e t e f

f e c t s

1

0.5

0

-0.5

-1

-1.5

0 5 10 15 20 25 30 35

Fig. 4. A scatter plot of trophic groups of the Kuosheng Bay ecosystem when thetopological importance as measured by the cumulative total effect ( Z i,n) is plottedagainst that measured by the cumulative net effect ( yi,n ). Each trophic group islabeled with the corresponding node number as it rst appears in Fig. 1.

Wei-chung Liu et al. / Journal of Theoretical Biology 267 (2010) 355362358
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3.2. Interaction structure

More interesting information is embedded in the matrix CNEnas it in a sense records the interaction structure in a signeddigraph ( Table 1 ). The ( i, j)th element of this matrix denotes thecumulative net effect of node i on node j up to n steps, therefore

the ith row of this matrix records the effects of node i onindividual nodes in the same network. First from this matrix, weobserve that the diagonal entries representing self-looping backeffects are all relatively strong and negative. Second, we examinethe interaction structure of three selected nodes (i.e. given a focalnode i, we wish to know how it affects all nodes in the samenetwork): node 1, one of the basal trophic groups of theecosystem; node 15, the only top trophic group of the system;and then node 10, one of the trophic groups occupyingintermediate trophic levels of the food web.

Fig. 5 depicts the cumulative net effects of node 1 on individualnodes: here, a lled circle indicates the effect a node receives fromnode 1 is positive, and an open circle means the effect from node 1 isnegative; and the size of a circle is proportional to the size of the

effect from node 1. Node 1 is a basal species at the lowest trophiclevel, and the effects it exerts on those at higher trophic levels aremainly positive except for nodes 2, 4, 7, 11 and 14. Nodes 1 and 2 areboth consumed by node 12, therefore node 2 is expected to beinuenced by node 1 via apparent competition resulting in anegative effect. Node 1 can be regarded as indirect food supply fornode 4 and the corresponding effect should therefore be positive,however node 4 is also consumed by many of node 1s predators(i.e. nodes 5, 6 and 12) resulting in node 4 also being involved inapparent competition with node 1; since apparent competitiondominates the interaction from node 1 on node 4, and the resultingnet effect should also be negative. Although nodes 7 and 11 areof higher trophic level than node 1, but they are not inuenced by

node 1 via indirect food supply (as they consume directly fromnode 2); therefore they are not expected to be inuenced by node 1positively. We can also observe from Fig. 5 that nodes that are distantfrom node 1 tend to be inuenced weakly by it, but node 3 is oneexception here. Node 3 is a direct neighbour of node 1 and therefore itshould be inuenced by node 1 strongly andpositively, however, such

direct effect should be compensated by many negative effects fromnode 1 via node 3s consumers (i.e. apparent competition via nodes 5,6 and 12) resulting in a small positive net effect.

Table 1A summary of cumulative total and net effects from a node on another up to 20 step lengths: an ( i, j)th entry of the table has two numbers: the rst and second numberscorrespond to the cumulative total effect (i.e. C TE ij,20 ) and cumulative net effect (i.e. C NE ij,20 ) from node i (identied by the left column) on node j (identied by the top row),respectively, up to 20 step lengths. The last column gives the sums of those two cumulative effects exerted by the corresponding node in the network (i.e. Z i,20 and yi,20 ).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Row sum

1 1.22 0.94 1.25 1.11 1.30 1.37 0.89 0.90 1.07 1.11 0.90 1.20 1.04 0.96 0.97 16.21 0.13 0.03 0.04 0.07 0.15 0.20 0.02 0.00 0.02 0.05 0.03 0.13 0.02 0.00 0.03 0.37

2 0.71 1.36 0.69 0.70 0.65 0.65 1.56 0.81 0.69 0.68 1.65 1.00 0.76 0.85 1.02 13.78 0.02 0.25 0.01 0.01 0.01 0.01 0.35 0.03 0.01 0.01 0.31 0.10 0.01 0.02 0.11 0.53

3 2.49 1.84 2.25 2.31 2.44 2.49 1.82 1.91 2.34 2.35 1.79 2.24 2.24 2.02 1.96 32.46 0.28 0.00 0.12 0.02 0.06 0.12 0.03 0.05 0.15 0.14 0.04 0.06 0.16 0.05 0.06 0.29

4 2.21 1.88 2.31 2.20 2.37 2.40 1.93 2.09 2.36 2.35 1.83 2.20 2.26 2.20 2.01 32.61 0.03 0.01 0.19 0.12 0.09 0.14 0.07 0.13 0.10 0.11 0.04 0.08 0.11 0.13 0.07 0.27

5 1.62 1.08 1.52 1.48 1.47 1.47 1.11 1.20 1.54 1.52 1.06 1.30 1.33 1.27 1.17 20.14 0.15 0.02 0.11 0.13 0.12 0.12 0.00 0.01 0.13 0.09 0.01 0.06 0.01 0.01 0.01 0.47

6 1.37 0.87 1.24 1.20 1.17 1.21 0.87 0.94 1.08 1.26 0.84 1.05 1.05 0.99 0.93 16.08 0.16 0.01 0.10 0.11 0.08 0.12 0.01 0.01 0.05 0.10 0.01 0.05 0.03 0.01 0.02 0.58

7 0.45 1.04 0.45 0.48 0.44 0.44 0.84 0.67 0.49 0.48 0.80 0.56 0.51 0.70 0.62 8.990.00 0.23 0.01 0.00 0.01 0.01 0.16 0.10 0.01 0.01 0.11 0.03 0.02 0.10 0.02 0.56

8 0.23 0.27 0.24 0.26 0.24 0.23 0.34 0.42 0.27 0.27 0.27 0.25 0.27 0.43 0.29 4.270.00 0.02 0.01 0.01 0.01 0.00 0.05 0.11 0.02 0.01 0.00 0.00 0.02 0.11 0.02 0.06

9 1.34 1.15 1.46 1.47 1.54 1.35 1.23 1.37 1.41 1.39 1.14 1.29 1.46 1.44 1.26 20.310.07 0.03 0.06 0.10 0.18 0.00 0.03 0.09 0.10 0.08 0.00 0.03 0.07 0.09 0.03 0.39

10 1.67 1.35 1.76 1.76 1.82 1.89 1.43 1.59 1.66 1.69 1.33 1.54 1.71 1.68 1.48 24.370.11 0.02 0.04 0.07 0.16 0.22 0.03 0.09 0.09 0.11 0.01 0.01 0.08 0.09 0.03 0.49

11 0.45 1.10 0.45 0.46 0.42 0.42 0.80 0.53 0.45 0.44 0.97 0.62 0.52 0.56 0.89 9.090.01 0.22 0.01 0.01 0.00 0.01 0.09 0.03 0.02 0.01 0.19 0.06 0.02 0.03 0.17 0.34

12 1.80 2.00 1.68 1.65 1.56 1.58 1.69 1.47 1.55 1.54 1.85 1.70 1.71 1.56 1.87 25.21 0.16 0.20 0.09 0.09 0.05 0.07 0.07 0.05 0.05 0.05 0.18 0.13 0.06 0.05 0.17 0.92

13 1.82 1.78 1.96 1.98 1.86 1.84 1.80 1.91 2.05 2.00 1.84 1.99 1.91 2.01 2.09 28.840.03 0.07 0.04 0.06 0.06 0.02 0.02 0.03 0.19 0.16 0.05 0.17 0.12 0.04 0.17 0.41

14 1.68 1.98 1.77 1.93 1.77 1.74 2.45 3.02 2.02 1.96 1.96 1.82 2.01 2.12 2.13 30.34

0.03 0.11 0.04 0.05 0.04 0.02 0.32 0.76 0.11 0.09 0.01 0.02 0.16 0.23 0.15 1.4015 0.97 1.36 0.98 1.01 0.94 0.93 1.23 1.16 1.01 0.99 1.78 1.25 1.19 1.22 1.30 17.32

0.02 0.11 0.01 0.03 0.01 0.01 0.11 0.09 0.05 0.04 0.36 0.09 0.10 0.09 0.16 0.35

1

1011

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5 6

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Fig. 5. A pictorial representation of the cumulative net effect of node 1 on allnodes in the Kuosheng Bay ecosystem in northern Taiwan. A circle represents atrophic group and a link here represents a trophic (or predatorprey) interactionlinking one trophic group to another. A lled circle indicates that the correspond-ing trophic group is inuenced by node 1 positively. An open circle indicates thecorresponding trophic group is affected by node 1 negatively. The size of a circle isproportional to the magnitude of net effects exerted by node 1 on thecorresponding trophic group.

Wei-chung Liu et al. / Journal of Theoretical Biology 267 (2010) 355362 359
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5 Liu and Chen Etal 2010 JTB