## 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:

D =

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:

 + - - + + + + - - + + - + - - + + + + - + + Unlike, the direct relations, this is not zero-sum. Instead, we see that there are 17 positive signs (including the diagonal) and eight negative signs. If there are a greater number of positive signs than negative signs in the integral utility matrix, then network mutualism is said to occur. Network analysis demonstrates the positive mutualistic relations in the system. Specifically, here, we can identify two cases of indirect mutualism, seven of exploitation, and one of competition (Table 2). 