Ecological networks are food webs where the edges are quantified and represent exchanges of nutrients (usually grams of carbon per m2 per year, but also nitrogen or phosphorous) or energy. Moreover, inputs to the system and outputs from the system are explicitly represented by flows involving 'special compartments' (i.e., nodes that act as a source (imports) or sink (exports and respirations) for the system). Besides the graph representation, a system can be described using the so-called flow matrix T, where each coefficient tj describes the flow of energy-matter from the row-compartment (i) to the column compartment (j). An example of network and its matrix representation is given in Figure 4.

In order to show the computation of the Finn's cycling index, it is necessary to introduce the concept ofpower of adjacency matrices. Take as an example the matrix introduced in the first section. If we square it, we obtain

0 0 1 0 1 2 1 0 0 0 1 1 0 10 0 1 0 0 10 0 \0 0 0 0 0/

This matrix shows the pathways of length 2 that connect to two nodes. For example, there is just one path connecting node 1 to node 4 in two steps (the path 1 ! 3 ! 4), while there are two pathways connecting 1 to 5 (1 ! 4 ! 5 and 1 ! 3 ! 5).

-11 184

-11 184

8881

Detritus

2309

5205 1600

3109

2309

5205 1600

Carnivores

3275

0 |
11184 |
635 |
0 |
0 |
0 |
0 |
0 |
Imports | |

0 |
0 |
8881 |
0 |
0 |
0 |
300 |
2003 |
Plants | |

0 |
0 |
0 |
2309 |
5205 |
0 |
860 |
3109 |
Detritus | |

T = |
0 |
0 |
200 |
0 |
0 |
370 |
0 |
1814 |
Detritus feeders |

0 |
0 |
1600 |
75 |
0 |
0 |
255 |
3275 |
Bacteria | |

0 |
0 |
167 |
0 |
0 |
0 |
0 |
203 |
Carnivores | |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Exports | |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Dissipations |

Figure 4 Schematic representation of cone spring ecosystem (a). There are two imports (to Plants and Detritus), three exports (from Plants, Detritus, and Bacteria) and five Dissipations (dashed arrows). The network can be associated with a matrix of transfers (b). The first row represents imports, the last two columns stand for exports and dissipations, and the internal 5 x 5 part depicts intercompartment flows.

In the same way, if we multiply this matrix with the adjacency matrix we get A , which describes all the pathways connecting two nodes in three steps; A4 will similarly contain all the pathways of length 4, and so forth. The power Ax will contain all the pathways of length x. If the food web contains no cycles, then for some x < n (where n is the number of species) the matrix will contain just zeros. If the food web contains cycles, on the other hand, the powers never converge to 0. The pathways enumerated in these matrices belong to all the different types that we illustrated in the first section. Now we can see how these considerations apply to quantified networks.

Dividing each coefficient tj for the row sum produces the coefficients g^j (of matrix G), which describe the fraction of flow leaving each compartment:

sj sr t

For example, the G matrix for the network in Figure 4 would be

0 |
0 |
0.794 |
0 |
0 |
0 |
0.027 |
0.179 |

0 |
0 |
0 |
0.201 |
0.453 |
0 |
0.075 |
0.271 |

0 |
0 |
0.084 |
0 |
0 |
0.155 |
0 |
0.761 |

0 |
0 |
0.307 |
0.014 |
0 |
0 |
0.049 |
0.629 |

0 |
0 |
0.451 |
0 |
0 |
0 |
0 |
0.549 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

Multiplying G by itself, one obtains the fraction of flow leaving the row compartment and reaching the column compartment in two steps (i.e., passing by an intermediate compartment). G3 will describe the exchanges in three steps, and so forth. Summing over all possible powers of G, one obtains the average number of visits a quantum of matter leaving the row compartment will pay to the column compartment. This computation is made possible by the fact that the power series of G converges to the so-called Leontief matrix L. G is defined as the identity matrix I

The Leontief matrix for the network in Figure 4 would be 0 1 0.946 0.946 0.202 0.440 0.031 0.120 0.8

A particle entering compartment k will be recycled ¡¡j— 1 times. The fraction of flow recycled is therefore lkk -1

0 |
1 |
0.958 |
0.199 |
0.434 |
0.031 |
0120 |
0.880 |

0 |
0 |
1.207 |
0.251 |
0.547 |
0.039 |
0.117 0.883 | |

0 |
0 |
0.186 |
1.039 |
0.084 |
0.161 |
0.018 |
0.982 |

0 |
0 |
0.374 |
0.092 |
1.169 |
0.014 |
0.085 |
0.915 |

0 |
0 |
0.545 |
0.113 |
0.247 |
1 .018 |
0.053 |
0.947 |

0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |

0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |

In an acyclic network, the maximum coefficient of L will be 1 (i.e., a quantum of matter can visit another compartment maximum once). This is because a particle of matter leaving a compartment will never be recycled to the same compartment again. This is not true when cycles are present. In fact, when matter cycles in the network, a particle can be recycled into the same compartment many times, raising the maximum value of the coefficients of the Leontief matrix. Therefore, the Leontief matrix of an acyclic network would contain unitary coefficients on the diagonal for all compartments (a particle starting at any compartment will never come back). Consequently, a simple way of estimating the cycled fraction would be to see how much these coefficients deviate from 1. This is at the heart of the so-called 'Finn's cycling index' (FCI). There are various formulations for this index, but here we present the simplest one, adapted from the one developed in 1980 by J. T. Finn; the reader is referred to the 'Further reading' section for a complete account of the possible variations. The following computation is valid for steady-state network only, that is, for networks where the input to any node equals the output from the same node.

We will call Tk the sum of all flows entering the compartment k:

For example, in Figure 4 the sum of the flows to the 'Plants' compartment T1 would be 11 184.

lkk k

The recycled fraction for 'Bacteria' (fourth compartment) would be (1.018 - 1)/1.018 = 0.0172. The total flow cycled C will be

C = ]T RkTk k which, computed for the example, will result in 2777.23 units recycled.

The total fraction of recycled flow for the whole system will therefore be

which, for the network in Figure 4, would be 0.0654.

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