Theorem 2

In an open system, if the ith tank is not an input one, then the diagonal entry of the ith row is aii < 0 and aii j j ¼ Ri being the sum of the non-diagonal entry modules of that row.

Proof

If Φai, Φbi, ⋯, Φni are the incoming fluxes from other tanks (a, b, ⋯, n) of the MP system, ΦiA, ΦiB, ⋯, ΦiJ are the outgoing fluxes to other tanks (A, B, ⋯, J), and Φ<sup>1</sup> <sup>i</sup> , Φ<sup>2</sup> <sup>i</sup> , ⋯, Φ<sup>s</sup> <sup>i</sup> are the fluxes from the ith tank to outside the system, then the corresponding ODE can be written as

$$V\_i \frac{d\mathbb{C}\_i}{dt} = \Phi\_{\text{il}}\mathbb{C}\_{\text{il}} + \dots + \Phi\_{\text{mi}}\mathbb{C}\_{\text{n}} - \left(\Phi\_{i\text{A}} + \dots + \Phi\_{\text{jl}}\right)\mathbb{C}\_i - \Phi\_i^1\mathbb{C}\_i - \dots - \Phi\_i^s\mathbb{C}\_i \tag{22}$$

This equation gives:

$$\frac{d\mathbb{C}\_{i}}{dt} = \frac{\Phi\_{ai}}{V\_{i}}\mathbb{C}\_{a} + \dots + \frac{\Phi\_{ni}}{V\_{i}}\mathbb{C}\_{n} - \frac{\sum \Phi\_{ij} + \sum \Phi\_{i}^{p}}{V\_{i}}\mathbb{C}\_{i} \tag{23}$$

Eq. (22) implies that the ith row of the MP-matrix has entries <sup>Φ</sup>ki Vi for k 6¼ i and � P<sup>Φ</sup>ij<sup>þ</sup> PΦ<sup>p</sup> i Vi for k ¼ i, and this equation does not contribute to the independent term.

$$\begin{aligned} \text{In this case a flux balance gives the following equation } \sum \Phi\_{li} &= \sum \Phi\_{\vec{\eta}} + \sum \Phi\_{i}^{p}, \text{ then,} \\\ a\_{\vec{u}} = -\frac{\sum \Phi\_{\vec{\eta}} + \sum \Phi\_{i}^{p}}{V\_{i}} < 0, \text{ and also } \mathbb{R}\_{i} = \frac{\sum \Phi\_{\vec{u}} - \sum \Phi\_{i}^{p}}{V\_{i}} = \frac{\sum \Phi\_{\vec{\eta}} + \sum \Phi\_{i}^{p}}{V\_{i}} = |a\_{i}|\_{\prime} \text{ and the theorem is proved.} \end{aligned}$$
