*2.4. Qualitative Analyses*

The modeled subsystem at each X2 position was considered stable if the two Routh–Hurwitz (R–H) criteria [94] were met. The first R–H criterion requires all roots of the characteristic polynomial be different from zero and have the same sign. The second R–H criterion requires all Hurwitz determinants be positive, resulting in stronger feedback at lower feedback levels than at higher levels. The effect of sustained input (*Ch*), on the rate of change of a *h*th variable corresponding to the *j*th variable in the community matrix was predicted assuming an ecosystem in moving equilibrium [29,71]. Such input could result in positive, negative, or no effect on the equilibrium magnitude or carrying capacity (*Ni\**) of a species. The inverse of the negative community matrix (−*Aij*−1) provides an estimate of change in the equilibrium level of variable *Ni\** carrying capacity due to change in parameter *Ch* [95].

$$\frac{dN\_i^\*}{d\mathbb{C}\_{\mathbb{I}}} = \frac{-\mathbb{C}\_{ji}^T}{\left| -A\_{ij} \right|} \tag{3}$$

where −*Cji<sup>T</sup>* is the adjoint of the negative community matrix (*Adj* − *Aij*), and |−*Aij*| is the determinant of the negative community matrix. Since |−*Aij*| is constant and positive for each element of −*A*−<sup>1</sup> in stable ecosystems, the predicted response under local equilibrium was inferred from the *Adj* − *A*:

$$Adj - A\_{ij} = \left(-A\_{i\bar{j}}^{-1}\right)^{T} \tag{4}$$

which corresponds to the transpose of the negative cofactor matrix [33]. The absolute number of positive and negative complementary feedback cycles (*Tc*) which contributes to each element of the *Adj* − *A* was used to derive model uncertainty. Because *Tc* differed from zero in all present models, it was deemed practical to modify the weighted prediction matrix [47] to express both the direction of change in response to input and its uncertainty by defining a signed weighted prediction matrix (*Ws*):

$$\text{Ws} = \frac{\overrightarrow{Adj} - A}{Tc} \tag{5}$$

where *Adj* − *A* is as defined in Equation (4), and "→" is a vectorized matrix operator denoting element-by-element division for each element of the *Ws* matrix [47], and *Tc* is as defined earlier and - 0. Hence, both the lowest *Ws* value (−1, negative response) and highest *Ws* value (+1, positive response) indicate unconditional sign determinacy and *Ws* = 0 denote total uncertainty in the direction of change, while absolute values of *Ws* ≥ 0.5 and < 1 are indicative of high level of sign determinacy, after [27].

To estimate the overall influence of a press input (*Pk*) due to a disturbance *k* acting directly upon two or more variables, the combined effect of *Pk* on the direction of change for community variable *i*

and their corresponding uncertainty level was computed from the corresponding matrix element *Wsik* (Equation (5)) and these were derived from a modified community matrix specifying press inputs (*A<sup>P</sup>*):

$$A^P = \begin{bmatrix} a\_{11} & \dots & a\_{1n} & P\_{1k} \\ \cdot & \cdot & \cdot & \cdot \\ a\_{n1} & \cdot & a\_{nn} & P\_{nk} \\ 0 & 0 & 0 & -1 \end{bmatrix} \tag{6}$$

where *a*11 to *ann* represent the interaction elements *aij* of community matrix *Ao*, and *Pik* denotes the press input *k* on community variable *i*, which in a qualitatively defined ecosystem is specified as having positive (1), negative (−1) or no effect (0). The last row of matrix *A<sup>P</sup>* includes zero elements to denote no effect of community variables on *Pk*, and a negative element (−1) to preserve the stability of matrix *A<sup>P</sup>* provided the community matrix *Ao* is stable. Qualitative analyses were conducted in Maple 18 [47].
