## 3 Results and Discussion
### Simulation Results
In BR\({}_{2}\), the dynamics of the state variable \(Q_{CH_{4}}\) predicted by GA, FA, CS, and COA are different. Although the resulting GA model has the lowest \(J\) value (\(J_{2}\) = 0.0761), the graphical comparison shows that the resulting FA model is better. The FA model describes the experimental data in the best way, following the trend of the \(Q_{CH_{4}}\) dynamics. The CS and COA models have similar \(J\) values, but the resulting models' dynamics do not follow the data trend very well. It should be mentioned that the experimental data are raw, unprocessed data. No filtering is applied in order to test the performance of the metaheuristics for a real nonlinear complex problem. The performance of the considered metaheuristic algorithms has been statistically evaluated by comparing the observed average value, standard deviation (SD), and the median of the estimated model parameters and the obtained objective function value \(J_{2}\). A statistical analysis of the suggested algorithms in terms of average, SD, and median on the results for the second identification step is presented in Table 3. Only the second set of estimated model parameters, observed in the case of modelling the processes in BR\({}_{2}\), is presented and discussed because of their crucial role and significance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Algorithm**} & \(\mathbf{J_{2}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{Y_{CH_{4}}}\) & \(\mu_{2 max}\) & \(\mathbf{K_{S_{2}}}\) \\ \hline \multirow{4}{*}{GA} & average & 0.09552 & 0.14501 & 5.32245 & 0.18386 & 0.36572 \\  & SD & 0.01732 & 0.26303 & 8.04380 & 0.14778 & 0.41212 \\  & median & 0.08833 & 0.05033 & 1.74911 & 0.14962 & 0.35478 \\ \hline \multirow{4}{*}{FA} & average & 0.11683 & 0.11048 & 1.10871 & 0.46463 & 0.94210 \\  & SD & 0.00936 & 0.01077 & 0.18236 & 0.03767 & 0.03814 \\  & median & 0.11686 & 0.10858 & 0.99504 & 0.44297 & 0.92456 \\ \hline \multirow{4}{*}{CS} & average & 0.10984 & 0.21904 & 4.26557 & 0.40936 & 0.26250 \\  & SD & 0.01361 & 0.53544 & 2.99461 & 0.28370 & 0.32967 \\  & median & 0.10693 & 0.00138 & 3.07739 & 0.45413 & 0.11308 \\ \hline \multirow{4}{*}{COA} & average & 0.11454 & 0.28767 & 6.18897 & 0.37877 & 0.21275 \\  & SD & 0.00741 & 0.23002 & 3.01675 & 0.22794 & 0.21626 \\ \cline{1-1}  & median & 0.11521 & 0.21474 & 6.85870 & 0.37701 & 0.14627 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical analysis of the results. Box plot diagrams are commonly used to present summary statistics as a standard technique for analyzing the distribution of the model estimates [58; 59]. Box plots of the obtained results are shown in Figure 3.  
As shown in Figure 3A, the median of the objective function values obtained from 30 runs of GA is better than that of the rest of the results, including the minimum achieved by the other algorithms. On the other hand, the SD in the case of GA is larger. The best SD is observed for the results of COA, followed by those of FA. The median of 30 runs of CS is better than the median of COA and FA. These observations are a reason to conclude that the advantages and disadvantages of the considered algorithms should be analyzed in depth. Subsequently, a hybrid metaheuristic can be designed to combine the advantages. It is shown that the distribution of FA estimates of the model parameters is smaller than that of the rest of the algorithms. The FA SD is times smaller than the SD of GA, CS, and COA. The model parameters \(\mu_{2 max}\) and \(K_{S_{2}}\) are more difficult to evaluate than the model parameters \(Y_{3}\) and \(Y_{CH_{4}}\). This is an interesting result, taking into account that the parameters \(\mu_{2 max}\) and \(K_{S_{2}}\) are usually the most sensitive ones [60]. It is worth investigating the sensitivity of the parameters of the new mathematical model proposed here. Moreover,  
Figure 3: Box plot for the results of the second identification step.