## 3 Results and Discussion
### Application of One-Step Kinetic Data Analysis to Predict the Growth of Co-Cultures in Milk  
Another challenge to determine the suitability of the one-step modeling approach was the application to the simultaneous prediction of competitive bacteria growth in milk at different temperatures and LAB Fresco inoculum levels. The H-GD model (Equation (11a-c)) with incorporated interaction coefficients values and secondary square models for \(\mu\) and \(t_{\lambda}\) was chosen in this work. The model was tested for four cases, as described above. In H-GD1, only the interaction coefficients (\(I_{FSL}\), \(I_{SEL}\), and \(I_{LES}\)) were evaluated. The remaining parameters were replaced with those optimized by the one-step kinetic data analyses for single cultures (Equation (12)) and fixed. Similarly, in H-GD2, only the first six parameters (\(I_{FSL}\); \(I_{SEL}\); \(I_{LES}\); \(x_{\text{max},Ec}\); \(x_{\text{max},Sa}\); \(x_{\text{max},LAB}\)) were evaluated; finally, all parameters in Equation (11a-c) were optimized in H-GD3. The last case, H-GD4 did not consider any interactions among populations, as the coefficients \(I_{ESI}\), \(I_{SEL}\), and \(I_{LES}\) were fixed to 1. The practical usefulness of this option is that all parameters for prediction of the \(E\). _coli_, \(S\). _aureus_, and LAB in co-cultures in milk could be estimated from the individual growth of a single species. The results of the one-step model parameter estimations in the four cases of the H-GD model and their goodness-of-fit indices are summarized in Tables 7 and 8, respectively. Based on the goodness-of-fit indices of the tertiary H-GD models provided in Table 8, it logically follows that the lowest values of the global \(SSE\) and \(RMSE\) were found for the H-GD3 model case, for which all parameters used (Equation (12)) were optimized. The global \(SSE\) and \(RMSE\) of the H-GD models that included the interaction coefficients (H-GD1, H-GD2, and H-GD3), even H-GD1, in which only three interaction coefficients were optimized as model parameters, were obviously lower than those of H-GD4, which indicated significantly improved accuracy between the observed and calculated values, as well as improved prediction performance [3]. However, it is disputable whether it is necessary to evaluate all parameters of the H-GD models with interaction coefficients using one-step analysis, as indicated by the negligible differences between the \(RMSE\) values and between the errors of the parameters estimated. It is clear that the fewer the parameters optimized, the lower the error values of the parameters. As such, the H-GD1 model with only three parameters optimized appears to be a good option for growth prediction. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factor** & **HM1-5a** & **HM1-5c** \\ \hline \(B_{f}\) & 1.076 & 0.911 \\ \(A_{f}\) & 1.170 & 1.200 \\ \hline \hline \(A_{f}\) is the accuracy factor, \(B_{f}\) is the bias factor. & & \\ \hline \end{tabular}
\end{table} 
Table 6: Bias and accuracy of independent data sets of \(S\). _aureus_ (Sa), \(E\). _coli_ (Ec) during their growth as single cultures in milk. The prediction ability of the cases of the tertiary H-GD model were tested using the independent data sets of the simultaneous competitive bacteria growth in milk at different temperatures and Fresco inoculum levels that were not used in the one-step modeling. The values of \(B_{f}\) and \(A_{f}\) calculated on the above-mentioned independent data sets (\(n=136\) for each microorganism) are summarized in Table 9.  
It is difficult to judge which of the H-GD competition models provides the most accurate prediction for describing the co-culture growth of _E. coli_, _S. aureus_, and Fresco culture in milk from the results shown in Table 9. All the factors are very close to 1.0 and all tested model cases have comparable predictive ability.