Ecosystem Flow Analysis - Ecological Systems - page 72

Table 5.1 Network representation of flow partitioning into five modes for any (i, j) pair in
a system

Pair-wise interaction System-wide contribution

f ð 0 Þ ¼ P f ð 0 Þ
j 0

Mode 0 (boundary input)

f ð 0 Þ
j 0

¼ z j

z j

f ð 1 Þ ¼ PP f ð 1 Þ
ij

Mode 1 (first passage)

f ð 1 Þ
ij

n ij
n ii d ij

¼

f ð 2 Þ ¼ PP f ð 2 Þ
ij

Mode 2 (cyclic)

f ð 2 Þ
ij

n ij
n ii ð

¼ n ii 1 Þ z j

z j

f ð 3 Þ ¼ PP f ð 3 Þ
ij

Mode 3 (dissipation)

f ð 3 Þ
ij

n ij
n ii d ij

¼

f ð 4 Þ ¼ P f ð 4 Þ
0 j

Mode 4 (boundary output)

f ð 4 Þ
0 j

¼ y j

steady state is actually composed of flow elements of all orders, m = 1, 2, .In

fact, a major result of this flow analysis is that indirect flows can dominate direct

...

flows: P m¼ 2 G m

G. In the above developments F, T, z, and y represent matter or
energy fluxes, and G and N are dimensionless intensive flows.
Finn [ 7 ] developed a cycling index using this basic approach and, Higashi et al. [ 8 ]
described a three-mode partition of the flows, expanded by Fath et al. [ 9 ]intofive
modes ( Table 5.1 ). Mode 0 is the boundary input into the system. Mode 1 accounts for
all flow in which substance moves from node j to a terminal node i for the first time
only without cycling. Mode 2 is flow cycled at terminal nodes i of each (i, j) pair. Mode
3 is component-wise dissipative flow in the sense that it exits from node i never to
return again to i. Mode 4 is the boundary output from i constituting systemically
dissipative flows exiting the system ( Fig. 5.2 ). d ij is the Kronecker delta defined by d ij
= 1 for i = j and d ij =0fori

>

j.
Note, the symmetry in that quantitatively Mode 0 = Mode 4, and Mode 1 =
Mode 3. This is due to the conservation of mass/energy and at steady state what
comes in must go out. Mode 2 represents the cycled flow which has additional
impact on the system by staying in the system longer, increasing the residence
time, and returning to its source of emanation. Therefore,

6¼

total system

throughflow can be written as:

f ð 0 Þ þ f ð 1 Þ þ f ð 2 Þ ¼ f ð 2 Þ þ f ð 3 Þ þ f ð 4 Þ TST ¼

And, on a nodal basis, throughflow is:

z j þ

n ij
n ii d ij

n ij
n ii ðn ii 1 Þz j

T ij ¼ f ð 0 Þ
ij

þf ð 1 Þ
ij

þf ð 2 Þ
ij

¼ z j d ij þ

z j

n ij
n ii d ij

n ij
n ii

¼ d ij þ þ n ij ¼ n ij z j

The mode partition designation clearly shows the contribution of flow within the
entire system of interactions.