## 3 Results and Discussion
### Application of One-Step Kinetic Data Analysis to Predict the Growth of Co-Cultures in Milk
However, the validation factors also confirm the advantage of the H-GD\({}_{1}\) case, in which only the interaction coefficients were evaluated, and the other parameters (Equation (12)) from one-step kinetic data analysis of  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **H-GD\({}_{1}\)** & **H-GD\({}_{2}\)** & **H-GD\({}_{3}\)** & **H-GD\({}_{4}\)** \\ \hline \(I_{\textit{ESL}}\) & 0.760 \(\pm\) 0.007 & 0.761 \(\pm\) 0.007 & 0.818 \(\pm\) 0.014 & 1 \\ \(I_{\textit{SEL}}\) & 1.061 \(\pm\) 0.007 & 1.060 \(\pm\) 0.016 & 0.973 \(\pm\) 0.020 & 1 \\ \(I_{\textit{LES}}\) & 0.906 \(\pm\) 0.004 & 0.921 \(\pm\) 0.019 & 0.934 \(\pm\) 0.033 & 1 \\ \(x_{\max,\textit{Ec}}\) [log CFU mL\({}^{-1}\)] & * 8.36 & 7.25 \(\pm\) 0.57 & 7.23 \(\pm\) 0.78 & * 8.36 \\ \(x_{\max,\textit{Sa}}\) [log CFU mL\({}^{-1}\)] & * 8.12 & 7.21 \(\pm\) 0.72 & 7.24 \(\pm\) 1.34 & * 8.12 \\ \(x_{\max,\textit{LAB}}\) [log CFU mL\({}^{-1}\)] & * 9.19 & 8.93 \(\pm\) 0.07 & 8.90 \(\pm\) 0.06 & * 9.19 \\ \(b_{\textit{F,Ec}}\) [\({}^{\circ}\)C\({}^{-1}\) h\({}^{-0.5}\)] & * 0.0421 & * 0.0421 & 0.0410 \(\pm\) 0.0  
single cultures were fixed. Referring to the validation data sets, the _RMSE_ of the predictions of the H-GD\({}_{1}\), H-GD\({}_{2}\), and H-GD\({}_{3}\) options of the one-step approach were 0.325, 0.333, and 0.345, respectively. They are close to the _RMSE_ of the nonlinear regression in Table 8, which is why the overall predictions may be considered as reasonably accurate. Based on the results obtained for the pure cultures and simultaneous competitive bacteria growth in milk at different temperatures and Fresco inoculum levels, we recommend the following procedure to predict their competitive growth: Apply one-step analysis to construct directly the tertiary model to determine the parameters for the prediction of the growth of single cultures and then, from the practical point of view, only evaluate the interaction coefficients to describe interspecies competition growth. The H-GD\({}_{1}\) one-step approach model was chosen to visually demonstrate the prediction of the simultaneous growth of _E. coli_, _S. aureus_, and a starter culture of LAB Fresco. As shown in Figure 1, the predicted co-culture growth curves using the H-GD\({}_{1}\) model are sufficiently in agreement with the observed growth curves data in milk. Optimizing the interaction coefficients provided good performance under all tested conditions. From the application of this model, we found that the presence of LAB clearly induced the early stationary phase of _E. coli_, _S. aureus_; the larger the initial population of LAB Fresco, the greater the suppression of _E. coli_ and _S. aureus_ in the co-culture [3]. For example, the maximum densities reached by _E. coli_ and _S. aureus_ at 12 and 30 \({}^{\circ}\)C were kept under 5.1 and 3.8 log CFU mL\({}^{-1}\) and 5.1 and 5.2 log CFU mL\({}^{-1}\), respectively, in co-cultures with Fresco inoculum at 10\({}^{6}\) CFU mL\({}^{-1}\). Thus, inocula of Fresco higher than 5 log CFU mL\({}^{-1}\) managed to control the outgrowths by _E. coli_ and _S. aureus_. Figure 1: Co-culture growth of _E. coli_, _S. aureus_, and four different inocula of LAB Fresco in milk at 12 \({}^{\circ}\)C (**upper left**, \(N_{0,LAB}\)\(\sim\) 5 log CFU mL\({}^{-1}\)), 15 \({}^{\circ}\)C (**upper right**, \(N_{0,LAB}\)\(\sim\) 3 log CFU mL\({}^{-1}\)), 18 \({}^{\circ}\)C (**lower left**, \(N_{0,LAB}\)\(\sim\) 6 log CFU mL\({}^{-1}\)), and 21 \({}^{\circ}\)C (**lower right**, \(N_{0,LAB}\)\(\sim\) 4 log CFU mL\({}^{-1}\)). The continuous lines represent the growth predicted by the H-GD1-model and dots represent the actual logarithmic counts (log \(N\)) in selected time intervals. The analysis of the residuals between the observed and predicted values in co-culture growth also showed that 83.1% of the _E. coli_ data (113/136), 96.3% of the LAB Fresco data (131/136) and 88.2% of the _S. aureus_ data (120/136) were in the range of the narrow acceptable values, namely, between +0.5 and \(-\)0.5 log CFU mL\({}^{-1}\)[31].