## Total System Throughput

There are various definitions of TST. In what follows we analyze the most intuitive one, which has been usually associated with the computation of ascendency.

We can sketch the flows of energy (or nutrients) inside an ecosystem using a matrix of flows T[ty]: ty will represent fluxes from compartment i to compartment j, that is, from row to column. If there are N compartments in the ecosystem, the matrix T will be composed by an N x N submatrix accommodating flows between compartments, plus the zeroth row and column including flows of energy (or nutrients) into the system, the (N + 1)th row and column accounting for exports from the system (usable energy/nutrients), and the (N + 2)th row and column storing dissipative flows (unusable energy/nutrients).

An example of such matrix is represented in Table 1. The correspondence between labels of rows and columns and functional compartments is explained in the caption.

The row sums for the compartments A, B, C, and D match the corresponding column sums, meaning that what exits a compartment (row sum) balances what enters in it (column sum). If this property holds, and the sum of imports equals the sum of exports and dissipations, then the system is said to be in steady state.

Table 1 has a pictorial counterpart in a network. Compartments will be represented by nodes, while weighted edges (arrows) depict exchanges of medium between compartments (Figure 1).

In an ecosystem the simplest possible measure of TST is simply the sum of all its flows. This gives an idea of how much medium is processed by the system and, as such, it

Table 1 Example of T matrix in a simple ecosystem composed by four compartments A, B, C, and D. Imp. stands for imports to the system, Exp. and Diss. for exports and dissipations from the system, respectively

 Imp. 0 100 0 0 0 0 0 A 0 0 50 30 0 0 20 B 0 0 0 5 25 10 10 C 0 0 0 0 20 0 15 D 0 0 0 0 0 20 25 Exp. 0 0 0 0 0 0 0 Diss. 0 0 0 0 0 0 0
Figure 1 Network representation of matrix T (Table 1).

is a measure of the system size. Using the T matrix, the TST is easily computed by adding up all the coefficients in the matrix.

In what follows, we will often use contractions to shorten the formulas. That is to say t.. will stand for summation across all rows (first dot) and columns (second dot). In the same way, t. will be the sum of the ith row and ty the sum of the yth column. For the network above, the TST, computed according to [1], is 330.

This is but one of the possible measures of system activity that has been put forward by researchers. In particular, the difference between system throughput and throughflow has been highlighted as a source of confusion. In its seminal paper, J. T. Finn considered the level of activity of each compartment as the sum of all incoming flows (that is equal, if the balancing of flows hold, to the sum of outgoing flows). The sum of compart-mental throughflows is usually defined total system throughflow. The total system throughflow can be computed as

TST throughflow = ^ X ty = X t, = X X ty = X ty i=i y=o i=i i=o y=i y=i

Ascendency has usually been associated with the formula presented in [1]. We will therefore use this measure in what follows.

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