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## An elementary dynamic model for non-binary food webs (2003)

Venue: | Ecological Modelling |

Citations: | 5 - 0 self |

### Citations

3928 | Emergence of scaling in random networks - Barabási, Albert - 1999 |

3321 |
Collective dynamics of ‘small-world’ networks
- Watts, Strogatz
- 1998
(Show Context)
Citation Context ...ism described above (here denoted as mech1), with cut = cutlin (see Fig. 4A). For the sake of comparison, columns 4 and 5 contain the values for random and regular networks with the same connectance (=-=Watts and Strogatz, 1998-=-). For column 6 the basic mechanism 150 T. Wilhelm / Ecological Modelling 168 (2003) 145–152 Table 1 Comparison of empirical and simulated food webs Measure Empirical websa Mech1 Rand1 Reg1 Mech2 Rand... |

672 |
Exploring complex networks
- Strogatz
- 2001
(Show Context)
Citation Context ...alled “scale-free” structure, where the probability of the number of connections per node declines in a power-law like manner. It was found that “The scale-free property is common but not universal” (=-=Strogatz, 2001-=-). Food webs seem to possess quite skewed distributions (Montoya and Solé, 2002; Matutinovic, 2002). The present model, like the SW model, generally yields Poisson distributions, but for high cut-valu... |

166 | The diversity-stability debate. - McCann - 2000 |

156 |
Weak trophic interactions and the balance of nature.
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Citation Context ...d maturity of ecosystems, an interesting contradiction arise. Some authors state that the more connections there are in a food web, the more stable the corresponding ecosystem (Rutledge et al., 1976; =-=McCann et al., 1998-=-; Polis, 1998). (Some of the confusion about stability has been clarified by Pimm (1991) who has given precise definitions for resilience, persistence, resistance, variability and stability in the mat... |

124 |
Small Worlds.
- Watts
- 1999
(Show Context)
Citation Context ...rresponding random networks, but the diameter D is not. It has been speculated that a small D has a big influence on the dynamic behaviour of the networks, for example, facilitating high homeostasis (=-=Watts, 1999-=-). In contrast to the SW model (Watts and Strogatz, 1998) which always starts with regular networks, our models simulate true self-organisation processes, where the final structure is independent of t... |

110 |
The Balance of Nature
- Pimm
- 1991
(Show Context)
Citation Context ... to assimilate much complex information rapidly” (Paine, 1988). This could help to better understand the often dominating role of indirect effects in the ecosystem answer to manipulation experiments (=-=Pimm, 1991-=-). Most food web models use only binary information: linkage or not (Cohen and Newman, 1985; Williams and Martinez, 2000; Pimm et al., 1991). The cascade model (Cohen and Newman, 1985) assigns each sp... |

94 | Will a Large Complex System be Stable? Nature, - May - 1972 |

93 |
Ecology: The Ascendent Perspective.
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(Show Context)
Citation Context ...he small fluxes and equalising the remaining ones. Therefore, a subtle analysis should use all the information (empirical examples in Riemann et al., 1986; de Ruiter et al., 1995; Lyche et al., 1996; =-=Ulanowicz, 1997-=-) by analysing the corresponding statistics (Fig. 2) and appropriate measures (such as MA). The most restricting assumption in the presented model is that of a constant number of nodes n. However, thi... |

91 | Alice's adventure in Wonderland - Carroll - 1876 |

77 |
Small-world patterns in food webs
- Montoya, Solé
- 2002
(Show Context)
Citation Context ...ly random, for non-binary food webs by Ulanowicz and Wolff (1991), and later for binary food webs by Solow and Beet (1998). Recently, small-world patterns have been identified in empirical food webs (=-=Montoya and Solé, 2002-=-), although in contrast to our approach, all links were taken as undirected. A comparison of mech1-networks with the corresponding random networks shows that the binary measures are not sensitive enou... |

61 |
Self-similarity in the distribution and abundance of species.
- Harte, Kinzig, et al.
- 1999
(Show Context)
Citation Context ... relationship between non-binary webs and the corresponding binary webs. Our model can be extended to the most general case of variable C and n, to cover questions such as species–area relationships (=-=Harte et al., 1999-=-) or Elton’s hypothesis “that ecological stability should depend on biological diversity” (Tilman et al., 1998), where the focus is upon discussion of the number of nodes itself (Tilman et al., 1998; ... |

58 |
Comparing five modelling techniques for predicting forest characteristics
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- 2002
(Show Context)
Citation Context ...equations often the simpler cellular automata models are used (Spencer, 1997; Keitt, 1997). Recently, five different modelling techniques have been compared to make predictions on a short time scale (=-=Moisen and Frescino, 2002-=-). However, all these dynamical models are dynamic only on a short time scale, but static on a longer time, for instance, the time scale of evolution. On such longer time scales typically also the top... |

51 |
Food webs from the small to the large.
- Schoener
- 1989
(Show Context)
Citation Context ...mber of cycles of length k/maximal number of length-k cycles), MAn: medium articulation/maximal possible medium articulation, Gen: generality (normalised average number of prey species of a predator (=-=Schoener, 1989-=-)), Vul: vulnerability (normalised average number of predators of a prey species (Schoener, 1989)), GenSd and VulSd: standard deviation of normalised generality and vulnerability, respectively (Willia... |

48 | Energetics, patterns of interaction strengths, and stability in real ecosystems. - Ruiter, Neutel, et al. - 1995 |

46 |
Diversity-stability relationships: Statistical inevitability or ecological consequence
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- 1998
(Show Context)
Citation Context ...t general case of variable C and n, to cover questions such as species–area relationships (Harte et al., 1999) or Elton’s hypothesis “that ecological stability should depend on biological diversity” (=-=Tilman et al., 1998-=-), where the focus is upon discussion of the number of nodes itself (Tilman et al., 1998; Harte et al., 1999; McCann, 2000). Acknowledgements I thank R. Brüggemann, J. Friedrich, J. Hochschild, and S.... |

43 |
Food web patterns and their consequences.
- Pimm, Lawton, et al.
- 1991
(Show Context)
Citation Context ...direct effects in the ecosystem answer to manipulation experiments (Pimm, 1991). Most food web models use only binary information: linkage or not (Cohen and Newman, 1985; Williams and Martinez, 2000; =-=Pimm et al., 1991-=-). The cascade model (Cohen and Newman, 1985) assigns each species a “niche value” and allows only predation of species with a lower niche value. Therefore, this does not yield a food web which expres... |

41 |
Topological evolution of dynamical networks: Global criticality from local dynamical rules, Phys
- Bornholdt, Rohlf
- 2000
(Show Context)
Citation Context ...urthermore, the connectance C is a crucial parameter in these models and in other network models, such as the SW model (Watts and Strogatz, 1998; Watts, 1999). Models also exist with variable C (e.g. =-=Bornholdt and Rohlf, 2000-=-), but these are not discussed in ecology. A binary network model with a growing number of nodes has been proposed (Barabási, and Albert, 1999), here L grows linearly with n. This model yields a so-ca... |

40 |
Constant connectance in community food webs.
- Martinez
- 1992
(Show Context)
Citation Context ... the square of the nodes (minimally articulated webs), and the contradicting linear “link-species scaling law”, which asserts that links increase linearly with the nodes (maximally articulated webs) (=-=Martinez, 1992-=-; Solow and Beet, 1998). Martinez (1992) underlined this contradiction and found by linear regression of empirical food web data points a medium exponent of 1.54. Food webs with MA = MAmax scale with ... |

37 | Improving food webs. - Cohen - 1993 |

35 |
A stochastic theory of community food webs I. Models and aggregated data.
- Cohen, Newman
- 1985
(Show Context)
Citation Context ... to better understand the often dominating role of indirect effects in the ecosystem answer to manipulation experiments (Pimm, 1991). Most food web models use only binary information: linkage or not (=-=Cohen and Newman, 1985-=-; Williams and Martinez, 2000; Pimm et al., 1991). The cascade model (Cohen and Newman, 1985) assigns each species a “niche value” and allows only predation of species with a lower niche value. Theref... |

31 |
Integration of Ecosystem Theories: a Pattern.
- Jørgensen
- 1992
(Show Context)
Citation Context ...es in the number of nodes and links). The problem “What happens to food webs (or generally to ecosystems) on longer time scales?” has been tackled under the question for goal functions of ecosystems (=-=Jørgensen, 1997-=-). We have argued that no universal goal function can exist that describes the direction of development of all natural systems (Wilhelm and Brüggemann, 2000). However, to investigate basic mechanisms ... |

28 |
Food webs: road maps of interactions or grist for theoretical development?
- Paine
- 1988
(Show Context)
Citation Context ...conservation” (Cohen et al., 1993) are examples for practical problems which should benefit from improved food web theory. A food web “allows a reader to assimilate much complex information rapidly” (=-=Paine, 1988-=-). This could help to better understand the often dominating role of indirect effects in the ecosystem answer to manipulation experiments (Pimm, 1991). Most food web models use only binary information... |

23 |
Stochastic structure and nonlinear dynamics of food webs: qualitative stability in a Lotka-Volterra cascade model
- Cohen, Luczak, et al.
- 1990
(Show Context)
Citation Context ...iven topology are often simulated dynamically using differential equations. Two recent examples are van Nes and Scheffer (2003) and Jordáne et al. (2003). The well-known Lotka-Volterra cascade model (=-=Cohen et al., 1990-=-) also fits into this category. It deduces the topology of links from a refined version of the cascade model (Cohen and Newman, 1985), and models the dynamics of the interactions with Lotka-Volterra t... |

21 |
A statistical measure of complexity. Phys
- Lopez-Ruiz, Mancini, et al.
- 1992
(Show Context)
Citation Context ... new information theoretic measure, called “medium articulation” (MA) which resolves this contradiction (Wilhelm and Brüggemann, 2001). In the same way as has been done for other complexity measures (=-=Lopez-Ruiz et al., 1995-=-; Shiner et al., 1999), we combined a measure which is zero in the minimally articulated case, i.e. each node has inputs from and outputs to each other node (the mutual information I = ∑ij Tij log[Tij... |

18 |
Simple measure for complexity.
- Shiner, Davison, et al.
- 1999
(Show Context)
Citation Context ...c measure, called “medium articulation” (MA) which resolves this contradiction (Wilhelm and Brüggemann, 2001). In the same way as has been done for other complexity measures (Lopez-Ruiz et al., 1995; =-=Shiner et al., 1999-=-), we combined a measure which is zero in the minimally articulated case, i.e. each node has inputs from and outputs to each other node (the mutual information I = ∑ij Tij log[Tij/( ∑ k Tkj ∑ l Til)] ... |

17 |
Ecological stability: An information theory viewpoint.
- Rutledge, Basore, et al.
- 1976
(Show Context)
Citation Context ...cussion of stability and maturity of ecosystems, an interesting contradiction arise. Some authors state that the more connections there are in a food web, the more stable the corresponding ecosystem (=-=Rutledge et al., 1976-=-; McCann et al., 1998; Polis, 1998). (Some of the confusion about stability has been clarified by Pimm (1991) who has given precise definitions for resilience, persistence, resistance, variability and... |

15 | Scale invariance in food web properties.
- Sugihara, Schoenly, et al.
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(Show Context)
Citation Context ... ∼ T−αij (α = f(a)) for Tij > 2a/n2. For a > 1, one obtains the power-law P(Tij) ∼ T−αij co with the cutoff co = 2a/n2 − Tij, because usually no flux can be larger than 2a/n2. resolution and lumping (=-=Sugihara et al., 1989-=-; Solow and Beet, 1998). Fig. 3A shows the complexity measure MA during evolution of the network. In the present model the amount of a flux after a mutation is independent of its previous amount, thus... |

14 |
Stability and complexity on a lattice: coexistence of species in an individual-based food web model.
- KEITT
- 1997
(Show Context)
Citation Context ...logy remains fixed. In order to incorporate also spatial dependencies, instead of studying the corresponding differential equations often the simpler cellular automata models are used (Spencer, 1997; =-=Keitt, 1997-=-). Recently, five different modelling techniques have been compared to make predictions on a short time scale (Moisen and Frescino, 2002). However, all these dynamical models are dynamic only on a sho... |

13 |
On lumping species in food webs.
- SOLOW, BEET
- 1998
(Show Context)
Citation Context ...he nodes (minimally articulated webs), and the contradicting linear “link-species scaling law”, which asserts that links increase linearly with the nodes (maximally articulated webs) (Martinez, 1992; =-=Solow and Beet, 1998-=-). Martinez (1992) underlined this contradiction and found by linear regression of empirical food web data points a medium exponent of 1.54. Food webs with MA = MAmax scale with an exponent of 1.5 (Wi... |

12 | Ecosystem flow networks: loaded dice?. - Ulanowicz, Wolff - 1991 |

10 |
Stability is woven by complex webs.
- Polis
- 1998
(Show Context)
Citation Context ...ems, an interesting contradiction arise. Some authors state that the more connections there are in a food web, the more stable the corresponding ecosystem (Rutledge et al., 1976; McCann et al., 1998; =-=Polis, 1998-=-). (Some of the confusion about stability has been clarified by Pimm (1991) who has given precise definitions for resilience, persistence, resistance, variability and stability in the mathematical sen... |

10 |
The balance between adaptability and adaptation.
- Ulanowicz
- 2002
(Show Context)
Citation Context ...b interaction strengths are indeed characterised by many weak interactions and a few strong interactions”. Moreover, there is some evidence that the amounts of fluxes in food webs follow a power-law (=-=Ulanowicz, 2002-=-). 148 T. Wilhelm / Ecological Modelling 168 (2003) 145–152 1s100 Mutation number 0 0.05 T i j (A) (B) 1s2000 Mutation number 0 0.05 T i j Fig. 1. Evolution of all n2 fluxes for n = 20 (“random initia... |

9 |
Information theoretical analysis of ecological networks.
- Hirata, RE
- 1984
(Show Context)
Citation Context ...to be unstable. In this sense it has been argued that in ecosystems “autocatalytic cycles” arise so that food webs of mature undisturbed ecosystems are characterised by very “articulated” structures (=-=Hirata and Ulanowicz, 1984-=-). However, “real systems may be expected to range somewhere between these extremes” (Pahl-Wostl, 1995). Recently, we proposed a new information theoretic measure, called “medium articulation” (MA) wh... |

7 | Persistence and flow reliability in simple food webs. - Jordán, Scheuring, et al. - 2003 |

7 |
The e!ects of habitat size and energy on food web structure: An individual-based cellular automata model.
- SPENCER
- 1997
(Show Context)
Citation Context ...y that the topology remains fixed. In order to incorporate also spatial dependencies, instead of studying the corresponding differential equations often the simpler cellular automata models are used (=-=Spencer, 1997-=-; Keitt, 1997). Recently, five different modelling techniques have been compared to make predictions on a short time scale (Moisen and Frescino, 2002). However, all these dynamical models are dynamic ... |

6 | Mesocosm tracer studies. 1. Zooplankton as sources and sinks in the pelagic phosphorus cycle of a mesotrophic lake - Lyche, Andersen, et al. - 1996 |

5 |
The Dynamic Nature of Ecosystems
- Pahl-Wostl
- 1995
(Show Context)
Citation Context ...bs of mature undisturbed ecosystems are characterised by very “articulated” structures (Hirata and Ulanowicz, 1984). However, “real systems may be expected to range somewhere between these extremes” (=-=Pahl-Wostl, 1995-=-). Recently, we proposed a new information theoretic measure, called “medium articulation” (MA) which resolves this contradiction (Wilhelm and Brüggemann, 2001). In the same way as has been done for o... |

4 | Alternative attractors may boost uncertainty and sensitivity in ecological models - Nes, Scheffer - 2003 |

4 |
Goal functions for the development of natural systems
- Wilhelm, Brüggemann
- 2000
(Show Context)
Citation Context ...led under the question for goal functions of ecosystems (Jørgensen, 1997). We have argued that no universal goal function can exist that describes the direction of development of all natural systems (=-=Wilhelm and Brüggemann, 2000-=-). However, to investigate basic mechanisms of evolution, the corresponding dynamic models should be characterised by food webs with variable topology and variable connectance C (number of actual link... |

2 | Carbon metabolism and community regulation in eutrophic, temperate lakes - Riemann - 1986 |

2 |
Information theoretic measures for the maturity of ecosystems. In
- Wilhelm, Brüggemann
- 2001
(Show Context)
Citation Context ...expected to range somewhere between these extremes” (Pahl-Wostl, 1995). Recently, we proposed a new information theoretic measure, called “medium articulation” (MA) which resolves this contradiction (=-=Wilhelm and Brüggemann, 2001-=-). In the same way as has been done for other complexity measures (Lopez-Ruiz et al., 1995; Shiner et al., 1999), we combined a measure which is zero in the minimally articulated case, i.e. each node ... |

1 | Ecological Modelling 168 (2003) 145–152 - Wilhelm - 2000 |

1 |
Organizational patterns of economics: an ecological perspective
- Matutinovic
- 2002
(Show Context)
Citation Context ...nes in a power-law like manner. It was found that “The scale-free property is common but not universal” (Strogatz, 2001). Food webs seem to possess quite skewed distributions (Montoya and Solé, 2002; =-=Matutinovic, 2002-=-). The present model, like the SW model, generally yields Poisson distributions, but for high cut-values, distributions are also shifted to highly skewed ones. Three points, listed with descending imp... |