- Email: [email protected]
Structural food web regimes
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
Structural food web regimes Brian D. Fath a,b,∗ a b
Biology Department, Towson University, Towson, MD 21252, USA Dynamic Systems, International Institute for Applied Systems Analysis, Laxenburg, Austria
a r t i c l e
i n f o
a b s t r a c t
Article history:
Ecological network analysis allows for an investigation of the structural and functional inter-
Received 27 March 2006
connectedness in ecosystems. Typically, these interactions are seen to comprise a food web
Received in revised form 2 June 2007
of “who eats whom”, but more generally applies to the transfer of energy-matter within
Accepted 12 June 2007
the biotic and abiotic ecosphere. This web of transactions can be depicted as a digraph or
Published on line 24 July 2007
an adjacency matrix in which the presence of direct transactions are represented as a 1 and no transactions as 0. Each transaction between system components leads to an over-
Keywords:
all network structural pattern. These structures cluster into different categories or regimes
Network analysis
based on their cyclic nature. This paper demonstrates threshold effects of the placement
Regime changes
or removal of links, such that certain changes essentially keep the structure in the same
Food webs
regime whereas others shift it to another regime in a non-linear manner. © 2007 Elsevier B.V. All rights reserved.
Cycling
1.
Introduction
One of the most important ecosystem properties is the presence of cyclic pathways (Lindeman, 1942; Ulanowicz, 1983; Patten, 1985; Burns, 1989); therefore, it is important to identify network structures that have this feature. Food webs provide a convenient way to depict who eats whom in an ecological community (Pimm et al., 1991; Pollis and Strong, 1996) and more generally assemble ecological data regarding trophic and non-trophic energy flow in the ecosphere. Structural analysis of ecological food webs has received considerable attention of late (Dunne et al., 2002; Borrett et al., 2007; Fath and Halnes, 2007). Properties such as link density, connectance, and the ratio of top predator-to-basal species have revealed new patterns and insights (e.g., Solow and Beet, 1998; Williams and Martinez, 2000; Quince et al., 2005). An older approach to investigating network structure uses matrix properties of the associated adjacency matrix, including identification of indi-
∗
Tel.: +1 410 704 2535; fax: +1 410 704 2405. E-mail address: [email protected]. 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.06.013
rect pathways and presence of cyclic pathways (Patten, 1982; Ulanowicz, 1983; Higashi and Patten, 1989; Burns et al., 1991; Fath and Patten, 1999). In particular, recent attention has been given to a straightforward measure for the presence of cyclic pathways in irreducible networks using the maximum eigenvalue (max ) of the adjacency matrix (Fath, 1998; Jain and Krishna, 2003; Fath and Halnes, 2007; in particular see Borrett et al., 2007 for a detailed explanation of the application of max ). This eigenvalue analysis produces three categories of networks, those in which the maximum eigenvalue is: 1) equal to zero (max = 0), 2) equal to one (max = 1), or 3) greater than one (max > 1). The systems that fall into the first category have no structural cycles present. Energy-matter flowing in such networks passes once through never returning to an originating compartment after its initial release. Those in the second category have weak cycling, meaning that closed loop pathways are present such that energy-matter may return to an originating compartment, but the number of
392
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
Table 1 – Number of structural three-compartment configurations for given cyclicity as links increase max = 0
max = 1
max > 1
2 3 4 5 6 7 8 9
12 6 0 0 0 0 0 0
0 50 72 36 6 0 0 0
0 0 39 87 78 36 9 1
Total
18
164
250
No. of links
pathways between two nodes does not increase geometrically as the path length between them increases. Networks in the third category exhibit strong cycling such that the number of pathways between two nodes increases geometrically as the path length between them increases. Networks with strong cycling contain a complex path structure with a large number of pathways between compartments not evident from simple inspection. In fact, in the limit as the path length, m, increases, the rate of pathway rise is equal to the maximum eigenvalue (m+1) (m) aij /aij = max as m → ∞1 (Hill, personal communication). Therefore, networks with a higher value of max correspond to a greater cycling potential. Here, this eigenvalue metric is used to identify the presence of structural cycling in basic network configurations and to show that the three categories produce a threshold effect such that certain links are more effective than others at determining the cycling regime.
2.
Structural regimes
2.1.
Three-compartment networks
Cycling presence presents a natural classification for networks. Starting with a simple three-node digraph representing a three-compartment model, there are a total of 432 irreducible distinct configurations,2 18 of which have no cycling pathways (max = 0), 164 with weak cycling (max = 1), and 250 configurations with strong cycling (max > 1). More refined inspection shows the number of typologies falling into each regime for a given number of links (Table 1). Fig. 1 shows the relation between number of links and cyclicity, where several points are evident. First, and obviously, it is impossible to connect three compartments with less than two links. Second, weak cycling first occurs with three links and strong cycling occurs with four links. Third, there is a general trend toward increasing cyclicity as the number of links increases, but there are many four-, five-, and even six-link configurations that do not have strong cycling. Other standard measures, such as connectance and link density are not dependent on the interrelations between nodes, but cyclicity directly relates to
1 Where a(m) are elements of the matrix Am , resulting from matrix multiplication. 2 Although there are 232ˆ = 512 possible combinations several of those do not link all three compartments, such as the null matrix, or A = [0 0 0; 1 0 0; 0 0 0], etc., and are not considered here.
Fig. 1 – Cyclicity vs. number of links for three-compartment network. In general, cyclicity increases as number of links increases, but there are three regimes in which the network typologies fall.
Fig. 2 – Six different configurations of the three-node network with six links and weak cycling.
the network organization. For example, a three-compartment network with six links has a link density = 2.0, however, the cyclicity of networks with the same link density could vary greatly depending on the placement of those links. Fig. 2 shows all six, six-link configurations that have weak cycling.3 These particular configurations represent networks in which links have not enabled the network to cross the threshold from weak to strong cycling. Therefore, certain links are less effective in the network typology at providing cyclic feed-
3 The MATLAB Spy plots a dot where a link exists and a blank otherwise. The matrices are oriented from columns to rows because in ecological networks the flow transaction is in the opposite direction of the ecological relations. For example, a flow in a three-chain module from i–j–k, has a grazing relation from j to i, and a predation relation from k to j. Therefore, to maintain the natural orientation for the relations requires transposing the flows.
393
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
Table 2 – Number of structural four-compartment configurations for given cyclicity as links increase max = 0
max = 1
max > 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0 128 186 108 24 0 0 0 0 0 0 0 0 0 0
0 0 758 2292 3091 2306 1006 240 23 0 0 0 0 0 0
0 0 0 864 3979 8630 11681 11160 7980 4368 1820 560 120 16 1
Total
446
9717
51179
No. of links
Fig. 3 – Cyclicity vs. number of links for four-compartment network. In general, cyclicity increases as number of links increases, but there are three regimes in which the network typologies fall.
2
back. Further note that three-node, six-link structures without strong cycling are poised for large increase in cyclicity with a further additional link. For example, considering the first structure of Fig. 2, in which max = 1:
⎡
1 A = ⎣0 0
1 1 0
⎤
1 1⎦ 1
A seventh link in either the a23 or a32 positions takes max to 2.0 and the seventh link in the a31 position gives max = 2.32.
2.2.
Four-compartment networks
As with the three-compartment network, a similar regime pattern emerges in which cyclicity is dependent heavily on the link placement (Fig. 3). In a four-compartment model, with a possibility of 61,342 irreducible structures,4 51,179 have strong cycling, 9717 have weak cycling, and 446 have no cycling. We see that four-compartment networks with one, two, three, or four links can all have zero cycling. Networks with anywhere from three to six links can have simple cycling, while four links, specifically placed, are sufficient to produce strong cycling (Table 2). In fact, a four-compartment network with five or six links could fall into any of the three regimes depending on where the links are in the network. The majority of typologies with five links have weak cycling and the majority with six links have strong cycling, but all three regimes are possible for the same number of links. Therefore, two networks each with the same link density or connectivity vary greatly in the network cyclicity. Furthermore, cyclicity is directly related to the performance of the network as characterized by its ability to transfer matter-energy and to provide indirect control. We could continue this exercise for larger and larger networks, but computationally the number of structural possi-
There are a total of 216 = 65,536 structural configurations including reducible and irreducible. 4
bilities increases as 2n , so even with just five compartments there are over 33 million distinct configurations. That alone should is pause for thought: only five nodes yield over 33 million ways to interact with each other. Therefore, let us turn to empirical food webs to investigate how they are structured. Although most earlier empirical food webs were constructed with only a few compartments, recently there have been several on the order of n = 100. Many of these networks exclude key elements of the web notably the detrital elements, but there is a subset of ecologists that construct energy-matter flow networks including all trophic and nontrophic pathways. A study of 16 empirically derived networks (e.g., Heymans et al., 2002; Sandberg et al., 2000; Leguerrier et al., 2003; Patricio et al., 2004) that included detrital links showed that strong cycling was a feature in all of them (Fath and Halnes, 2007), thereby indicating the ubiquity of this regime in real ecosystems. In another study, Borrett et al. (2007) showed that in 17 well-studied food webs, nine had strong cycling, five have weak cycling, and two had no cycling. In fact, the only reason those did not have greater cycling is because they systematically excluded the detrital loop from the structural network. Inclusion of the loop in the ten cases that had available data, as demonstrated in Fath and Halnes (2007), resulted in all ten also falling in the strong cycling regime.
3.
Conclusions and discussion
Network structures can be classified into one of three cycling regimes based on the maximum eigenvalue of the adjacency matrix, those with no cycling (max = 0), weak cycling (max = 1), and strong cycling (max > 1). There are thresholds regarding the number of links necessary for a network to fall into a particular regime, but there is also overlap meaning that certain links are more efficient at producing cycles than other. A wellplaced link can mean the difference in the network from no cycling to strong cycling. These cycling regimes are more relevant than other structural properties such as connectance or
394
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
link density because they reveal organizational properties of the network. Cycling is an important mechanism for feedback and self-regulation because it means that compartments are both reaching to and reachable from other compartments in the network (Fath, 2004). Since it is impossible for large-scale networks to physically “test” each configuration, network patterns emerge non-ergodically from the myriad of interactions for a given resource flow. Evidence shows that this self-organization of ecospheric webs results in networks that contain strong cycling pathways signifying that there is some advantage for systems to be in this regime. One may find that as system size increases the likelihood of strong cycling also increases. While it is clear that real ecosystems fall in the strong cycling regime, it is not yet clear whether or not there is an optimal level of cycling within this regime.
Acknowledgements The author thanks Audrey Mayer for organizing a workshop at EPA on Complex Systems that motivated this research and Stuart Borrett for review comments on the manuscript.
references
Borrett, S.R., Fath, B.D., Patten, B.C., 2007. Pathway proliferation and modularity in ecological networks. J. Theor. Biol. 245, 98–111. Burns, T.P., 1989. Lindeman’s contradiction and the trophic structure of ecosystems. Ecology 70, 1355–1362. Burns, T.P., Higashi, M., Wainright, S.C., Patten, B.C., 1991. Trophic unfolding of a continental shelf energy-flow network. Ecol. Model. 55, 1–26. Dunne, J.A., Williams, R.J., Martinez, N.D., 2002. Food-web structure and network theory: the role of connectance and size. Proc. Natl. Acad. Sci. U.S.A. 99, 12917–12922. Fath, B.D., 1998. Network analysis: foundations, extensions, and applications of a systems theory of the environment. Ph.D. Dissertation. University of Georgia, Athens, Georgia. Fath, B.D., 2004. Distributed control in ecological networks. Ecol. Model. 179, 235–246.
Fath, B.D., Halnes, G., 2007. Cyclic energy pathways in ecological food webs. Ecol. Model. 208, 17–24. Fath, B.D., Patten, B.C., 1999. Review of the foundations of network environ analysis. Ecosystems 2, 167–179. Heymans, J.J., Ulanowicz, R.E., Bondavalli, C., 2002. Network analysis of the South Florida Everglades graminoid marshes and comparison with nearby cypress ecosystems. Ecol. Model. 149, 5–23. Higashi, M., Patten, B.C., 1989. Dominance of indirect causality in ecosystems. Am. Nat. 133, 288–302. Hill, J. IV. On the largest real root of a polynomial. Unpublished manuscript. Jain, S., Krishna, S., 2003. Graph theory and the evolution of autocatalytic networks. In: Bornholdt, S., Schuster, H.G. (Eds.), Handbook of Graphs and Networks: From the Genome to the Internet. Wiley-VCH, Berlin, pp. 355–395. Leguerrier, D., Niquil, N., Boileau, N., Rzeznik, J., Sauriau, P.-G., Le Moine, O., Bacher, C., 2003. Numerical analysis of the food web of an intertidal mudflat ecosystem on the Atlantic coast of France. Mar. Ecol. Prog. Ser. 246, 17–37. Lindeman, R.L., 1942. The trophic dynamic aspect of ecology. Ecology 23, 399–418. Patricio, J., Ulanowicz, R., Pardal, M.A., Marques, J.C., 2004. Ascendency as an ecological indicator: a case study of the estuarine pulse eutrophiciation. Estuar. Coast. Shelf Sci. 60, 23–25. Patten, B.C., 1982. Envrions: relativistic elementary particles for ecology. Am. Nat. 119, 179–219. Patten, B.C., 1985. Energy cycling in the ecosystem. Ecol. Model. 28, 1–71. Pimm, S.L., Lawton, J.H., Cohen, J.E., 1991. Food web patterns and their consequences. Nature 350, 669–674. Pollis, G.A., Strong, D.R., 1996. Food web complexity and community dynamics. Am. Nat. 157, 813–846. Quince, C., Higgs, P.G., McKane, A.J., 2005. Topological structure and interaction strengths in model food webs. Ecol. Model. 187, 389–412. Sandberg, J., Elmgren, R., Wulff, F., 2000. Carbon flows in Baltic Sea food webs—a re-evaluation using a mass balance approach. J. Mar. Syst. 25, 249–260. Solow, A.R., Beet, A.R., 1998. On lumping species in food webs. Ecology 79 (6), 2013–2018. Ulanowicz, R.E., 1983. Identifying the structure of cycling in ecosystems. Bioscience 65, 219–237. Williams, R.J., Martinez, N.D., 2000. Simple rules yield complex food webs. Nature 404, 180–183.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
Structural food web regimes Brian D. Fath a,b,∗ a b
Biology Department, Towson University, Towson, MD 21252, USA Dynamic Systems, International Institute for Applied Systems Analysis, Laxenburg, Austria
a r t i c l e
i n f o
a b s t r a c t
Article history:
Ecological network analysis allows for an investigation of the structural and functional inter-
Received 27 March 2006
connectedness in ecosystems. Typically, these interactions are seen to comprise a food web
Received in revised form 2 June 2007
of “who eats whom”, but more generally applies to the transfer of energy-matter within
Accepted 12 June 2007
the biotic and abiotic ecosphere. This web of transactions can be depicted as a digraph or
Published on line 24 July 2007
an adjacency matrix in which the presence of direct transactions are represented as a 1 and no transactions as 0. Each transaction between system components leads to an over-
Keywords:
all network structural pattern. These structures cluster into different categories or regimes
Network analysis
based on their cyclic nature. This paper demonstrates threshold effects of the placement
Regime changes
or removal of links, such that certain changes essentially keep the structure in the same
Food webs
regime whereas others shift it to another regime in a non-linear manner. © 2007 Elsevier B.V. All rights reserved.
Cycling
1.
Introduction
One of the most important ecosystem properties is the presence of cyclic pathways (Lindeman, 1942; Ulanowicz, 1983; Patten, 1985; Burns, 1989); therefore, it is important to identify network structures that have this feature. Food webs provide a convenient way to depict who eats whom in an ecological community (Pimm et al., 1991; Pollis and Strong, 1996) and more generally assemble ecological data regarding trophic and non-trophic energy flow in the ecosphere. Structural analysis of ecological food webs has received considerable attention of late (Dunne et al., 2002; Borrett et al., 2007; Fath and Halnes, 2007). Properties such as link density, connectance, and the ratio of top predator-to-basal species have revealed new patterns and insights (e.g., Solow and Beet, 1998; Williams and Martinez, 2000; Quince et al., 2005). An older approach to investigating network structure uses matrix properties of the associated adjacency matrix, including identification of indi-
∗
Tel.: +1 410 704 2535; fax: +1 410 704 2405. E-mail address: [email protected]. 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.06.013
rect pathways and presence of cyclic pathways (Patten, 1982; Ulanowicz, 1983; Higashi and Patten, 1989; Burns et al., 1991; Fath and Patten, 1999). In particular, recent attention has been given to a straightforward measure for the presence of cyclic pathways in irreducible networks using the maximum eigenvalue (max ) of the adjacency matrix (Fath, 1998; Jain and Krishna, 2003; Fath and Halnes, 2007; in particular see Borrett et al., 2007 for a detailed explanation of the application of max ). This eigenvalue analysis produces three categories of networks, those in which the maximum eigenvalue is: 1) equal to zero (max = 0), 2) equal to one (max = 1), or 3) greater than one (max > 1). The systems that fall into the first category have no structural cycles present. Energy-matter flowing in such networks passes once through never returning to an originating compartment after its initial release. Those in the second category have weak cycling, meaning that closed loop pathways are present such that energy-matter may return to an originating compartment, but the number of
392
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
Table 1 – Number of structural three-compartment configurations for given cyclicity as links increase max = 0
max = 1
max > 1
2 3 4 5 6 7 8 9
12 6 0 0 0 0 0 0
0 50 72 36 6 0 0 0
0 0 39 87 78 36 9 1
Total
18
164
250
No. of links
pathways between two nodes does not increase geometrically as the path length between them increases. Networks in the third category exhibit strong cycling such that the number of pathways between two nodes increases geometrically as the path length between them increases. Networks with strong cycling contain a complex path structure with a large number of pathways between compartments not evident from simple inspection. In fact, in the limit as the path length, m, increases, the rate of pathway rise is equal to the maximum eigenvalue (m+1) (m) aij /aij = max as m → ∞1 (Hill, personal communication). Therefore, networks with a higher value of max correspond to a greater cycling potential. Here, this eigenvalue metric is used to identify the presence of structural cycling in basic network configurations and to show that the three categories produce a threshold effect such that certain links are more effective than others at determining the cycling regime.
2.
Structural regimes
2.1.
Three-compartment networks
Cycling presence presents a natural classification for networks. Starting with a simple three-node digraph representing a three-compartment model, there are a total of 432 irreducible distinct configurations,2 18 of which have no cycling pathways (max = 0), 164 with weak cycling (max = 1), and 250 configurations with strong cycling (max > 1). More refined inspection shows the number of typologies falling into each regime for a given number of links (Table 1). Fig. 1 shows the relation between number of links and cyclicity, where several points are evident. First, and obviously, it is impossible to connect three compartments with less than two links. Second, weak cycling first occurs with three links and strong cycling occurs with four links. Third, there is a general trend toward increasing cyclicity as the number of links increases, but there are many four-, five-, and even six-link configurations that do not have strong cycling. Other standard measures, such as connectance and link density are not dependent on the interrelations between nodes, but cyclicity directly relates to
1 Where a(m) are elements of the matrix Am , resulting from matrix multiplication. 2 Although there are 232ˆ = 512 possible combinations several of those do not link all three compartments, such as the null matrix, or A = [0 0 0; 1 0 0; 0 0 0], etc., and are not considered here.
Fig. 1 – Cyclicity vs. number of links for three-compartment network. In general, cyclicity increases as number of links increases, but there are three regimes in which the network typologies fall.
Fig. 2 – Six different configurations of the three-node network with six links and weak cycling.
the network organization. For example, a three-compartment network with six links has a link density = 2.0, however, the cyclicity of networks with the same link density could vary greatly depending on the placement of those links. Fig. 2 shows all six, six-link configurations that have weak cycling.3 These particular configurations represent networks in which links have not enabled the network to cross the threshold from weak to strong cycling. Therefore, certain links are less effective in the network typology at providing cyclic feed-
3 The MATLAB Spy plots a dot where a link exists and a blank otherwise. The matrices are oriented from columns to rows because in ecological networks the flow transaction is in the opposite direction of the ecological relations. For example, a flow in a three-chain module from i–j–k, has a grazing relation from j to i, and a predation relation from k to j. Therefore, to maintain the natural orientation for the relations requires transposing the flows.
393
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
Table 2 – Number of structural four-compartment configurations for given cyclicity as links increase max = 0
max = 1
max > 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0 128 186 108 24 0 0 0 0 0 0 0 0 0 0
0 0 758 2292 3091 2306 1006 240 23 0 0 0 0 0 0
0 0 0 864 3979 8630 11681 11160 7980 4368 1820 560 120 16 1
Total
446
9717
51179
No. of links
Fig. 3 – Cyclicity vs. number of links for four-compartment network. In general, cyclicity increases as number of links increases, but there are three regimes in which the network typologies fall.
2
back. Further note that three-node, six-link structures without strong cycling are poised for large increase in cyclicity with a further additional link. For example, considering the first structure of Fig. 2, in which max = 1:
⎡
1 A = ⎣0 0
1 1 0
⎤
1 1⎦ 1
A seventh link in either the a23 or a32 positions takes max to 2.0 and the seventh link in the a31 position gives max = 2.32.
2.2.
Four-compartment networks
As with the three-compartment network, a similar regime pattern emerges in which cyclicity is dependent heavily on the link placement (Fig. 3). In a four-compartment model, with a possibility of 61,342 irreducible structures,4 51,179 have strong cycling, 9717 have weak cycling, and 446 have no cycling. We see that four-compartment networks with one, two, three, or four links can all have zero cycling. Networks with anywhere from three to six links can have simple cycling, while four links, specifically placed, are sufficient to produce strong cycling (Table 2). In fact, a four-compartment network with five or six links could fall into any of the three regimes depending on where the links are in the network. The majority of typologies with five links have weak cycling and the majority with six links have strong cycling, but all three regimes are possible for the same number of links. Therefore, two networks each with the same link density or connectivity vary greatly in the network cyclicity. Furthermore, cyclicity is directly related to the performance of the network as characterized by its ability to transfer matter-energy and to provide indirect control. We could continue this exercise for larger and larger networks, but computationally the number of structural possi-
There are a total of 216 = 65,536 structural configurations including reducible and irreducible. 4
bilities increases as 2n , so even with just five compartments there are over 33 million distinct configurations. That alone should is pause for thought: only five nodes yield over 33 million ways to interact with each other. Therefore, let us turn to empirical food webs to investigate how they are structured. Although most earlier empirical food webs were constructed with only a few compartments, recently there have been several on the order of n = 100. Many of these networks exclude key elements of the web notably the detrital elements, but there is a subset of ecologists that construct energy-matter flow networks including all trophic and nontrophic pathways. A study of 16 empirically derived networks (e.g., Heymans et al., 2002; Sandberg et al., 2000; Leguerrier et al., 2003; Patricio et al., 2004) that included detrital links showed that strong cycling was a feature in all of them (Fath and Halnes, 2007), thereby indicating the ubiquity of this regime in real ecosystems. In another study, Borrett et al. (2007) showed that in 17 well-studied food webs, nine had strong cycling, five have weak cycling, and two had no cycling. In fact, the only reason those did not have greater cycling is because they systematically excluded the detrital loop from the structural network. Inclusion of the loop in the ten cases that had available data, as demonstrated in Fath and Halnes (2007), resulted in all ten also falling in the strong cycling regime.
3.
Conclusions and discussion
Network structures can be classified into one of three cycling regimes based on the maximum eigenvalue of the adjacency matrix, those with no cycling (max = 0), weak cycling (max = 1), and strong cycling (max > 1). There are thresholds regarding the number of links necessary for a network to fall into a particular regime, but there is also overlap meaning that certain links are more efficient at producing cycles than other. A wellplaced link can mean the difference in the network from no cycling to strong cycling. These cycling regimes are more relevant than other structural properties such as connectance or
394
e c o l o g i c a l m o d e l l i n g 2 0 8 ( 2 0 0 7 ) 391–394
link density because they reveal organizational properties of the network. Cycling is an important mechanism for feedback and self-regulation because it means that compartments are both reaching to and reachable from other compartments in the network (Fath, 2004). Since it is impossible for large-scale networks to physically “test” each configuration, network patterns emerge non-ergodically from the myriad of interactions for a given resource flow. Evidence shows that this self-organization of ecospheric webs results in networks that contain strong cycling pathways signifying that there is some advantage for systems to be in this regime. One may find that as system size increases the likelihood of strong cycling also increases. While it is clear that real ecosystems fall in the strong cycling regime, it is not yet clear whether or not there is an optimal level of cycling within this regime.
Acknowledgements The author thanks Audrey Mayer for organizing a workshop at EPA on Complex Systems that motivated this research and Stuart Borrett for review comments on the manuscript.
references
Borrett, S.R., Fath, B.D., Patten, B.C., 2007. Pathway proliferation and modularity in ecological networks. J. Theor. Biol. 245, 98–111. Burns, T.P., 1989. Lindeman’s contradiction and the trophic structure of ecosystems. Ecology 70, 1355–1362. Burns, T.P., Higashi, M., Wainright, S.C., Patten, B.C., 1991. Trophic unfolding of a continental shelf energy-flow network. Ecol. Model. 55, 1–26. Dunne, J.A., Williams, R.J., Martinez, N.D., 2002. Food-web structure and network theory: the role of connectance and size. Proc. Natl. Acad. Sci. U.S.A. 99, 12917–12922. Fath, B.D., 1998. Network analysis: foundations, extensions, and applications of a systems theory of the environment. Ph.D. Dissertation. University of Georgia, Athens, Georgia. Fath, B.D., 2004. Distributed control in ecological networks. Ecol. Model. 179, 235–246.
Fath, B.D., Halnes, G., 2007. Cyclic energy pathways in ecological food webs. Ecol. Model. 208, 17–24. Fath, B.D., Patten, B.C., 1999. Review of the foundations of network environ analysis. Ecosystems 2, 167–179. Heymans, J.J., Ulanowicz, R.E., Bondavalli, C., 2002. Network analysis of the South Florida Everglades graminoid marshes and comparison with nearby cypress ecosystems. Ecol. Model. 149, 5–23. Higashi, M., Patten, B.C., 1989. Dominance of indirect causality in ecosystems. Am. Nat. 133, 288–302. Hill, J. IV. On the largest real root of a polynomial. Unpublished manuscript. Jain, S., Krishna, S., 2003. Graph theory and the evolution of autocatalytic networks. In: Bornholdt, S., Schuster, H.G. (Eds.), Handbook of Graphs and Networks: From the Genome to the Internet. Wiley-VCH, Berlin, pp. 355–395. Leguerrier, D., Niquil, N., Boileau, N., Rzeznik, J., Sauriau, P.-G., Le Moine, O., Bacher, C., 2003. Numerical analysis of the food web of an intertidal mudflat ecosystem on the Atlantic coast of France. Mar. Ecol. Prog. Ser. 246, 17–37. Lindeman, R.L., 1942. The trophic dynamic aspect of ecology. Ecology 23, 399–418. Patricio, J., Ulanowicz, R., Pardal, M.A., Marques, J.C., 2004. Ascendency as an ecological indicator: a case study of the estuarine pulse eutrophiciation. Estuar. Coast. Shelf Sci. 60, 23–25. Patten, B.C., 1982. Envrions: relativistic elementary particles for ecology. Am. Nat. 119, 179–219. Patten, B.C., 1985. Energy cycling in the ecosystem. Ecol. Model. 28, 1–71. Pimm, S.L., Lawton, J.H., Cohen, J.E., 1991. Food web patterns and their consequences. Nature 350, 669–674. Pollis, G.A., Strong, D.R., 1996. Food web complexity and community dynamics. Am. Nat. 157, 813–846. Quince, C., Higgs, P.G., McKane, A.J., 2005. Topological structure and interaction strengths in model food webs. Ecol. Model. 187, 389–412. Sandberg, J., Elmgren, R., Wulff, F., 2000. Carbon flows in Baltic Sea food webs—a re-evaluation using a mass balance approach. J. Mar. Syst. 25, 249–260. Solow, A.R., Beet, A.R., 1998. On lumping species in food webs. Ecology 79 (6), 2013–2018. Ulanowicz, R.E., 1983. Identifying the structure of cycling in ecosystems. Bioscience 65, 219–237. Williams, R.J., Martinez, N.D., 2000. Simple rules yield complex food webs. Nature 404, 180–183.