along with the following information: Finn Cycling Index is 0.0919, meaning about 9% of flow is cycled flow, the ratio of direct to indirect flow is 0.9126, meaning that almost half of all total system throughflow traveled along indirect paths (note, this is actually a rare case exception when indirect flow contribution is not a majority), and the network evenness measure (homogenization) is 2.0360, meaning that the values in integral flow matrix, N, are twice as evenly distributed than the values in the direct flow matrix G— which is evident just from eyeballing the two matrices. Another, important analysis possible with the flow data, but not displayed here is the calculation of the actual unit environs, i.e., flow decompositions showing the amount of flow within the system "environ" needed to generate one unit of input or output at each compartment. Unfortunately, this particular example is one in which the powers of the utility matrix, D, do not guarantee convergence (since the maximum eigenvalue of D is greater than 1); and therefore, we cannot present the synergism metrics. Although this is an area of ongoing research, we can speculate about the ecological relationships in such cases by looking at the signs of the utility matrices:
The direct utility matrix is zero-sum in that for every donor there is a receiver of the flow. In ecological terms with think of a (+, —) relationship as predation or exploitation, but more generally it represents transfer from one compartment to another. Matching compartments pair-wise across the main diagonal gives the relationship type as shown in Table 5.2.
Notice, first that in the integral consideration that all compartments interact with each other, not just directly—there are no zero elements in the matrix. Next, notice that while two of the neutral direct relations became exploitation in one direction or the other, two others became mutualistic relations. Furthermore, note that one direct relations flipped when viewed in light of the system interactions—the relation between compartments 2 and 5 was antagonistic in the direct sense, but mutualistic in the holistic evaluation. The presence of each compartment benefits each other. Lastly, note that overall the integral matrix has more positive signs than negative signs leading to a holistic emergence of network mutualism. See previous literature cited (Patten, 1991, 1992; Fath and Patten, 1998; Fath, 2007) for other examples of utility analysis calculations. Let us now turn to and end with a qualitative interpretation of these network properties.
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