OF SPECIES IN FOOD WEBS: A SIGNED DIGRAPH APPROACH3
INTRODUCTION
A food web is in essence a network model of an ecosystem that describes the trophic interactions between species. Species in a food web are often connected, either directly or indirectly, thus changes in the abundance of one species could potentially affect the abundance of others (Wootton 1994a). Some human activities (e.g. overfishing) clearly change the abundance of exploited species (Jennings and Kaiser 1998), thus there are concerns over how those effects might ripple through and affect the entire ecosystem (Frank et al. 2005, Baum and Worm 2009). With the fear of species extinction and the collapse of the entire ecosystem in mind, there have been a growing number of food web studies in the past decade (Ings et al. 2009). These include studies on the assemblage and structural organisation of food webs (Sole and Montoya 2001, Dunne et al. 2002b, Sole et al. 2006, Dunne and Williams 2009) as it is imperative to
understand how species are embedded in a food web such that the potential effect of disturbance in species abundance can be easily understood (Wootton 1994a, Dambacher et al. 2003b, Ebenman et al.
2004, Hosack et al. 2009, Montoya et al. 2009). On a more practical side, especially as a guide for conservation efforts, many food web studies provide informative indices measuring species importance and
3 This chapter has published and cited as Liu, W.-C., H.-W. Chen, F. Jordan, W.-H. Lin, and C. W.-J.
Liu. 2010. Quantifying the interaction structure and the topological importance of species in food webs:
A signed digraph approach. Journal of Theoretical Biology 267:355-362 (see Appendix 2). Author‟s contribution: HWC, WCL, FJ, CWJL conceived the study. WHL wrote the programs. HWC, WCL
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subsequently identify the so-called topological keystone species (Ebenman and Jonsson 2005, Jordan et al. 2006, Jordan 2009).
Since a food web is a network, the topological importance of a species can be defined as the positional importance of a node in the network (Jordan et al. 2006, Jordan 2009). There exist several indices for measuring this from the graph theoretical perspective (Wasserman and Faust 1994). Starting from the most local measure we have the degree centrality, or simply the number of direct neighbours of a node. Then there are indices which consider a node‟s indirect neighbours as well as direct ones. One of such indices is the closeness centrality that measures how close a node is to others in the same network; and another non-local measure widely used is the betweenness centrality which quantifies how often a node is incident to all shortest paths between node pairs.
Furthermore, since networks can be represented as adjacency matrices, indices such as eigenvector and information centralities employ
matrix-related properties and operations to quantify nodal importance (Allesina and Pascual 2009).
Species importance can also be measured by examining the interaction between species and how a focal species can directly and indirectly affect others in the same food web (Menge 1995, Yodzis 2000).
Indices for species importance viewed from this perspective differ from those based on the graph theoretical approach in that they consider inter-specific interactions that are of ecological relevance. For instance species importance in host-parasitoid communities can be quantified by measuring the strength of apparent competition between species
(Memmott et al. 1994, Muller et al. 1999, Rott and Godfray 2000), and the long indirect effects of trophic cascade and indirect-food supply have
been used as proxies to species importance in multi-trophic level food webs (Jordan and Scheuring 2002, Quince et al. 2005). In a more general manner, Jordán et al. (2003) developed a centrality index basing on the direct and indirect effects of a focal species on all species, and such a methodology has been used to study the interaction structure and species importance of host-parasitioid communities (Jordan et al. 2003) and marine wasp-waist systems (Jordan et al. 2005); and more recently it has also been used to investigate the trophic field overlap of species in an ecosystem (Jordan 2009). However, a trophic interaction implies in fact two directed links of opposite signs (i.e. positive food supply from resource to consumer and negative feeding effect from consumer to resource), and above studies on indirect effects generally do not consider this. Although there have been a lot of researches on signed, direct and indirect effects in networks (Levine 1976, Wootton 1994b, Dambacher et al. 2003a, Dambacher et al. 2003b, Ebenman et al. 2004, Hosack et al.
2009, Montoya et al. 2009), but they are rarely considered in the same framework with interaction structure and topological importance (but see (Vasas and Jordan 2006)). An important exception is the “mixed trophic impact” (MTI for short) analysis (Ulanowicz and Puccia 1990,
Christensen and Walters 2004) where the sign and direction of trophic interactions are considered in determining the effects of one species on others. However, this approach relies on a mass-balance model where parameters related to a species‟s trophic input and output are often required for computation, and many large food webs are not
parameterized so richly. Furthermore, this approach does not explicitly separate different types of indirect effects such as indirect food supply, trophic cascade, exploitative and apparent competitions that might be of interest to ecologists in general.
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In the light of those points above, we present an approach to quantify the interaction structure of a food web (i.e. a signed digraph) by
extending the methodology of Jordán et al. (2003). Topological
importance of a node is then defined by structurally how effects from it can affect all nodes in the same network. The paper is organised as follows. First, we describe how we convert a food web to a signed
digraph, and we describe briefly the food web we use to demonstrate our methodology. Second, we present the methodology of Jordán et al. (2003) and show how it can be modified for signed digraphs. Third, we present our result and compare our ranking of species importance with those obtained from other indices in network literature. Next, we then compare the interaction structure obtained from our methodology and that from MTI as means to test our new methodology. Lastly, we discuss the implication of our results and future applications of our methodology.
MATERIAL AND METHODS
Converting a binary food web into a signed digraph
A food web is a network where nodes and links represent species and trophic interactions respectively. A trophic interaction between a prey and a predator implies that the prey has direct positive influence on the
predator (food supply) whereas the predator has direct negative influence on the prey (feeding effect). Thus a trophic interaction can be
decomposed into two directed and signed links. Here we define that a species influences another positively if the former increases the
abundance of the latter; and negatively if the former decreases the abundance of the latter.
Data
The food web we used describes the Kuosheng Bay ecosystem in North-East Taiwan (Lin et al. 2004). We use the version modified by Jordán et al. (2009) where there are 15 trophic groups and 36 trophic links (Fig. 4-1). Following the decomposition method above, the food web can be converted into a signed digraph with 15 nodes and 72 links (36 positive and 36 negative).
Topological importance basing on direct and indirect effects for undirected and unsigned networks
We first describe the methodology of Jordán et al. (2003). Let N be the number of nodes in a graph. Assuming node i is a direct neighbor of node j, we define aij as the direct effect (i.e. one-step effect) of i on j
j
ij D
a 1
, (1)
where Dj is the number of node j‟s direct neighbors. Next, for a pathway consisted of several steps (a step means travelling along a link), we assume an indirect effect is multiplicative in the sense that it is the product of one-step effects. Consider a simple case where node i can reach node j in 2 steps and there is only one such a pathway via node k (i.e. i-k-j). In this case, the effect of i on j via k is simply:
kj
ik a
a . (2)
Furthermore, we assume that effects are additive in the sense that if there exist more than one pathway of length n from node i to 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 and there are exactly two
pathways for doing so (e.g. i-k-j and i-h-j), then the total effect from i on j in 2 steps is:
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For step lengths from 1 to h, we can organise those various effects 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 and columns of a matrix refer to effect originators and receivers respectively, and construct matrix TEn whose elements are of the form TEij,n. Jordán et al. (2003) suggested that the sum of the ith row in TEn quantifies the total effect of node i on all nodes for n steps, and this can be regarded as the total effect node i on the network for n steps (γi,n):
Since a node can affect another via pathways of different step lengths, one needs to calculate the cumulative total effect by summing up effects of all lengths if he or she wishes to quantify the overall effect of one node on another. For effects up to n steps, we determine TE1, TE2, TE3… TEn,
where CTEn is a matrix whose (i,j)th element represents the cumulative total effect of node i on node j up to n steps. Jordán et al. (2003)
From now we refer the topological importance of node i based on the approach of Jordán et al. (2003), or the cumulative total effect of node i
up to n steps, simply as ηi,n. Since ηi,n increases without bound as n
approaches infinity, Jordán et al. (2003) suggested one should predefine a maximum n when he or she quantifies the topological importance of nodes, and they further define topological importance of node i as the averaged effect of i on all nodes up to n steps.
Similar indices can also be defined for networks with weighted links.
All effects are defined in the same way as above with the exception of direct effects, which is now defined as:
j strength of the link connecting i and j. We denote all indices derived from weighted links with a superscript “W” to distinguish them from their un-weighted counterparts.
Topological importance basing on direct and indirect effects for signed diagraphs
Essentially all things are the same except for how we define direct effects. First, we define i→jS as the directed link from node i to node j (i.e.
i is influencing j). The superscript “S“ is the sign of the interaction: it is positive (+1) if node i influences node j positively and negative (–1) otherwise. Let node j has Dj neighbours, and the magnitude of the unsigned direct effect from a neighbour i on j is
j
ij D
a 1
. (8)
However, if the sign of interaction is taken into account, then the signed direct effect from i on j is:
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Here, direct effects from j‟s 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 way as their unsigned and undirected counterparts. Since an indirect effect is the product of several direct effects and some of those might take positive or negative values, thus an indirect effect following a particular pathway is either positive or negative. In general, if there are m pathways from node i to node j in n steps then there will be m individual indirect effects, and each of which is either positive or negative. We define Eij,n,+ as the sum of positive effects of i on j in n steps and similarly define Eij,n,- for negative effects. The net effect of i on j for n steps can then be defined as
which can be positive, negative or zero. Note that the sum of the magnitude of Eij,n,+ and Eij,n,- is the total effect of i on j for n steps
which is always positive. Note that when n=2, Eij,2,+ represents the sum of all 2-step positive effects from i on j, and this includes the 2-step indirect effects of indirect food supply (＋×＋=＋) and trophic cascade (－×－=
＋). Moreover, when n=2, Eij,2,- is the sum of all 2-step negative effects from i on j, and this includes the 2-step indirect effects of exploitive competition (－×＋=－) and apparent competition (＋×－=－).
Again, for step length n, we can construct matrices NEn, En,+ and E n,-whose elements are NEij,n,Eij,n,+ andEij,n,- respectively. The sum of the ith row in NEn is the net effect of node i for n steps (xi,n):
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 define the sum of the ith row in CNEn as the topological importance of node i in
From now on we refer the topological importance of node i based on our approach, or the cumulative net effect of node i up to n steps, simply as yi,n.
The same approach is also applicable to signed digraphs with weighted links with the exception of how we define direct effects. A direct effect of node i on node j is now defined as:
W strength of the link connecting i to j. Again, we denote indices derived from weighted links with a superscript “W” to distinguish them from their un-weighted counterparts.
RESULTS
We analysed the food web representing the Kuosheng Bay ecosystem.
We computed CTEn, CTEWn, CNEn and CNEWn for n=1 to 20. From CTE20, CTEW20, CNE20 and CNEW20 we then determined ηi,20, ηWi,20, yi,20
and yWi,20. For the weighted indices, we used the consumption rates of
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consumers on resources (Lin et al. 2004) as link strengths. Although we have carried out the same analysis on the weighted version of the
Kuosheng Bay food web, however, for simplicity, we will restrict our presentation and discussion to the un-weighted version of topological importance.
Topological Importance
We first present the result derived from the methodology of Jordán et al. (2003). Fig. 4-2 is a plot of ηi,n for n=1 to 20 for individual nodes. For small n, there are crossovers between lines implying changes in the topological importance ranking among nodes; and when n increases, the importance ranking becomes fixed. Note that ηi,n increases indefinitely as the n increases, and Jordán et al. (2003) suggested the topological
importance of a node should be divided by the number of step length considered, n. Fig. 4-3 is a plot of yi,n for n=1 to 20 for individual nodes.
Again as above, there are changes in the ranking of topological
importance among nodes when n is small, but the ranking becomes fixed when n becomes large. Interestingly, in contrast to the plot in Fig. 4-2, yi,n
do not increase without bound to infinity. Instead, they stabilize to a fixed and non-zero value: some are positive and some are negative. This shows that some nodes are predominately positive interactors while the
remaining ones are negative interactors. The reason why topological importance here stabilizes as n increases is 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 step numbers, the net effects are almost zero and contribute a little to topological importance.
Another difference between our methodology and that of Jordán et al.
(2003) is that our approach is able to quantify topological importance not only basing on the size of the effect a node exerts on the whole network, but also in terms of the manner in which it influences the whole network.
For instance, if the food web is unsigned and undirected, then nodes 4 and 3 are the most influential nodes according to ηi,20; however, if the food web is treated as signed digraph, then nodes 4 and 3 are only weak positive interactors according to yi,20 (Fig. 4-4). Furthermore, according to yi,20, our methodology reveals node 2 is the most important positive
interactor despite the observation that it is one of the least influential nodes according to ηi,20 (Fig. 4-4). As far as the Kuosheng Bay ecosystem is concerned, both methodologies do suggest node 14 being the most topologically important, and our approach suggests it is important as a negative interactor (Fig. 4-4). Both methodologies also agree on node 8 being the least important node in the system (Fig. 4-4).
Interaction structure
More interesting information is embedded in the matrix CNEn as it in a sense records the interaction structure in a signed digraph (Table 4-1).
The (i,j)th element of this matrix denotes the cumulative 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 on individual nodes in the same network. First from this matrix, we observe that the diagonal entries representing self-looping back effects are all relatively strong and negative. Second, we examine the interaction structure of three selected nodes (i.e. given a focal node i, we wish to know how it affects all nodes in the same network): node 1, one of the basal trophic groups of the ecosystem; node 15, the only top trophic group of the system; and then node 10, one of the trophic groups occupying intermediate trophic levels of the food web.
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Fig. 4-5 depicts the cumulative net effects of node 1 on individual nodes: here, a filled circle indicates the effect a node receives from node 1 is positive, and an open circle means the effect from node 1 is negative;
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 trophic level, and the effects it exerts on those at higher trophic levels are mainly positive except for nodes 2, 4, 7, 11 and 14. Nodes 1 and 2 are both consumed by node 12, therefore node 2 is expected to be influenced by node 1 via apparent competition resulting in a negative effect. Node 1 can be regarded as indirect food supply for node 4 and the corresponding effect should therefore be positive, however node 4 is also consumed by many of node 1‟s predators (i.e. nodes 5, 6 and 12) resulting in node 4 also being
involved in apparent competition with node 1; since apparent competition dominates the interaction from node 1 on node 4, and the resulting net effect should also be negative. Although nodes 7 and 11 are of higher trophic level than node 1, but they are not influenced by node 1 via indirect food supply (as they consume directly from node 2); therefore they are not expected to be influenced by node 1 positively. We can also observe from Fig. 5 that nodes that are distant from node 1 tend to be influenced weakly by it, but node 3 is one exception here. Node 3 is a direct neighbour of node 1 and therefore it should be influenced by node 1 strongly and positively, however, such direct effect should be
compensated by many negative effects from node 1 via node 3‟s
consumers (i.e. apparent competition via nodes 5, 6 and 12) resulting in a small positive net effect.
Fig. 4-6 shows the cumulative net effects of node 15 on individual nodes. We can observe that the effects node 15 has on its direct
neighbours are all negative whereas the effects on the remaining nodes
are mostly positive. By being the top predator, node 15 influences its direct neighbours negatively due to their predator-prey relationships. The effects of node 15 on other nodes at lower trophic levels are mostly positive due to the process of trophic cascade. And lastly Fig. 4-7 shows the cumulative net effect of node 10 on individual nodes in the network.
Node 10 occupies a position located in the middle of several food chains.
The effects of node 10 on those at higher trophic level are predominantly positive simply via direct or indirect food supply while those on its prey are understandably negative via predator-prey interaction. One interesting observation here is that through trophic cascade node 10 can indirectly influence node 1 in a positive manner, and the size of such an indirect effect is greater than the positive effects node 10 has on its direct predators (i.e. nodes 13 and 14).
Relationship with other indices of topological importance Using the signed digraph approach, we calculated the size the cumulative net effects (signs are ignored here) for both the un-weighted
Relationship with other indices of topological importance Using the signed digraph approach, we calculated the size the cumulative net effects (signs are ignored here) for both the un-weighted