## 3 Results and Discussion
### Simulation Results  
A series of parameter identification procedures of the proposed model (Equations (1)-(9)) using tuned GA, FA, CS, and COA has been performed. Due to the stochastic nature of the algorithms, a meaningful statistical analysis requires each metaheuristic to be run at least 30 times. The best model parameter estimations obtained for models (1)-(9) with the corresponding values of the objective functions are summarized in Table 2.  
The observed results are very interesting. Four distinct models are obtained based on the applied metaheuristic algorithms. The parameter estimations presented in Table 2 are generally satisfying and relevant to their physical meaning. The estimates of the maximum specific growth rate of the microbial biomass concentration in BR\({}_{1}\)\(\mu_{1 max}\) as well as the saturation constant \(K_{S_{1}}\) are almost identical for the four algorithms. Yield coefficient \(Y_{H_{2}}\) is also estimated to the same order of magnitude by all algorithms. For BR\({}_{2}\), the results are different. Both GA and FA provide similar results, while the estimates of CS and COA, especially for the parameter \(K_{S_{2}}\), differ by order of magnitude. In the case of dynamics in BR\({}_{1}\), GA and FA result in similar models, while in the case of process dynamics in BR\({}_{2}\), similar models are obtained by CS and COA, except the estimates of parameter \(K_{S_{2}}\). Since there are no reported parameter values for the proposed model or similar, it is not possible to determine which parameters, i.e., which model is the best. In such cases, a graphical comparison can clearly show if the model follows the experimental data, i.e., the presence or absence of systematic deviations between the model predictions and the experimental measurements. Such a quantitative measure is also an important criterion for \begin{table}
\begin{tabular}{c c} \hline
**GA** & **Value** \\ \hline population size \(n\) & 100 \\ generation gap & 0.97 \\ crossover rate & 0.80 \\ mutation rate & 0.05 \\ \hline
**FA** & **Value** \\ \hline number of fireflies \(n\) & 20 \\ initial attractiveness \(\beta_{0}\) & 1 \\ light absorption coefficient \(\gamma\) & 1 \\ randomization parameter \(\alpha\) & 0.2 \\ \hline
**CS** & **Value** \\ \hline number of nests \(n\) & 20 \\ switching parameter \(p_{a}\) & 0.25 \\ Lévy exponent \(\lambda\) & 1.5 \\ \hline
**COA** & **Value** \\ \hline number of packs \(N_{p}\) & 30 \\ number of coyotes \(N_{c}\) & 100 \\ probability of eviction of a coyote \(p\_\)_leave_ & \(0.0005\times N_{c}{}^{2}\) \\ scatter probability \(P_{s}\) & \(1/D\) \\ association probability \(P_{a}\) & \((1{-}P_{s})/2\) \\ \hline \end{tabular}
\end{table} 
Table 1: The optimal values of the main parameters of the metaheuristic algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & \(\mu_{1 max}\) & \(K_{S_{1}}\) & \(Y_{1}\) & \(Y_{2}\) & \(Y_{H_{2}}\) & \(J_{1}\) & \(\mu_{2 max}\) & \(K_{S_{2}}\) & \(Y_{3}\) & \(Y_{CH_{4}}\) & \(J_{2}\) \\ \hline GA & 0.012 & 0.944 & 0.067 & 14.671 & 11.289 & 0.0035 & 0.517 & 0.812 & 0.029 & 0.543 & 0.0761 \\ FA & 0.017 & 1.100 & 0.222 & 10.276 & 8.117 & 0.0035 & 0.443 & 0.919 & 0.100 & 0.989 & 0.1075 \\ CS & 0.010 & 1.139 & 0.010 & 0.122 & 13.727 & 0.0036 & 0.077 & 0.001 & 19.781 & 2.298 & 0.0913 \\ COA & 0.012 & 1.004 & 12.137 & 5.702 & 11.598 & 0.0035 & 0.029 & 0.0001 & 24.272 & 7.933 & 0.0940 \\ \hline \end{tabular}
\end{table} 
Table 2: Model parameter estimations. the adequacy of the models proposed here. The model predictions of the state variables \(Q_{H_{2}}\) and \(Q_{CH_{4}}\), based on GA, FA, CS, and COA estimated set of model parameters, have been compared to the experimental data of the TSAD of corn steep liquor in Figure 2. \(J\) is the value of the mean least square error for each model (Equations (12) and (13)). As can be seen in Figure 2, the models' dynamics of the state variable \(Q_{H_{2}}\) in the BR\({}_{1}\) are very similar. Therefore, the best model has to be determined based on the results for BR\({}_{2}\).