## Network Mutualism

Turning now to the utility analysis, the net flow, utility matrix, D, can be used to determine quantitatively and qualitatively the relations between any two components in the network such as predation, mutualism, or competition. Entries in the direct utility matrix, D, or integral utility matrix, U, can be positive or negative (—1 < dj uij< 1). The elements of D represent the direct relation between that (i, j) pairing; for the example in Figure 1, this produces the following:

0 |
T2 |
T3 |
0 |
T5 | ||||||||||

T |
0 |
T3 |
T4 |
0 | ||||||||||

T3 |
T3 |
0 |
T4 |
T5 | ||||||||||

0 |
T4 |
T4 |
0 |
T5 | ||||||||||

h T |
0 |
T5 |
Ts |
The direct matrix D, being zero-sum, always has the same number of positive and negative signs: The elements of U provide the integral, system-determined relations. Continuing the example, and now including flow values derived from 10% transfer efficiency along each link (gj = 0.10, if aj = 1, and gj = 0 otherwise), we get the following integral relations between compartments:
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