*2.5. Quantitative Simulations*

Simulations using *A<sup>P</sup>* further enabled to quantitatively assess the stability of the community and the direction of change of its variables relative to the qualitatively specified matrix with same structure and sign of interactions. Simulations were performed by assigning interaction strengths between 0.01 and 1.0 to non-zero elements using a pseudorandom number generator from a uniform distribution [31]. The overall direction of change and determinacy in a quantitatively-specified ecosystem was derived from the proportional net change for each community variable (Δ*p*ˆ):

$$
\Delta \hat{p}\_{\vec{i}\vec{k}} = \frac{f(+Adj - A\_{i\vec{k}})^\* - f(-Adj - A\_{i\vec{k}})^\*}{N} \tag{7}
$$

where *f*(+*Adj* − *Aik*)\* and *f*(−*Adj* − *Aik*)\* respectively denote the absolute frequencies of positive and negative *Adj* − *Aik* in stable matrices, and *N* is the total number of simulations for each perturbation scenario, including stable and non-stable matrices (*N* = 10,000). As in the case of *Ws*, the value of Δ*p*<sup>ˆ</sup> provided both the overall direction of change and its uncertainty for each community variable, with extreme potential values (Δ*p*<sup>ˆ</sup> = −1, negative response and Δ*p*<sup>ˆ</sup> = 1, positive response) both indicating no uncertainty in the direction of change among simulations, Δ*p*<sup>ˆ</sup> ≥ |0.5| and < |1|indicating low uncertainty, and Δ*p*<sup>ˆ</sup> = 0 denoting total uncertainty. Quantitative simulations were performed in a program written in C# version 3.0 [96].
