## 2 Material and Methods
### Mathematical Models
#### 2.4.2 Parameter Determination and Evaluation of Model Performance  
The one-step kinetic data analysis method described by [PERSON] [35] was applied for parameter optimisation from the given isothermal growth curves of the co-culture microbial populations. To construct the tertiary H-GD model and minimise the global sum of squared errors (\(SSE\)), they were analysed concurrently using the H-GD models (Equations (1)-(4)) and secondary square root models (Equations (5)-(7)). In the beginning, the tertiary H-GD model had 19 parameters (Equations (1)-(8)): three average maximum values of \(x_{\text{max}}\), competition coefficients \(I_{LP}\), \(I_{PL}\); and reduction coefficient \(k_{\text{Cc}}\) in the H-GD model type R and reduction coefficient \(k_{\text{Cc}}\) (Equations (1)-(4)) and parameters \(b_{\text{T}}\), \(T_{\text{min}}\), \(b_{\lambda}\) from the square-root models (Equations (5)-(7)). The parameters \(N_{\text{res}}\), \(k_{\text{ref}}\), and \(z\) are derived from the reparametrized Gompertz-inspired survival model (Equation (3)), as well as the Bigelow secondary model (8), respectively. \[p_{{}_{H-GD}}=\left\{ \begin{array}{l}x_{\text{max},Lab};x_{\text{max},P};x_{ \text{max},Gc};I_{LP};I_{PL};k_{Gc}\\ b_{T,Lab};b_{T,P};b_{T,Gc};T_{\text{min},Lab};T_{\text{min},P};T_{\text{min},Gc };b_{\lambda,Lab};b_{\lambda,P};b_{\lambda,Gc}\\ k_{ref},p;N_{res,P};z_{P}\end{array} \right\} \tag{9}\]  
\(p_{\text{H-GD}}\) is the vector of parameters of H-GD models for the simultaneous competitive growth of the co-culture series. The prediction ability of the tertiary H-GD model was tested for a reduced number of parameters (\(p_{\text{E}}\)): the average maximum density counts of microorganisms (\(x_{\text{max},\text{Lab}}\), \(x_{\text{max},\text{P}}\), \(x_{\text{max},\text{G}}\)), competition coefficients (\(I_{\text{LP}}\), \(I_{\text{PL}}\)), reduction coefficient \(k_{\text{Cc}}\), regression coefficient, \(b_{\lambda,Gc}\), maximum declination rates at a reference temperature \(T_{\text{ref}}\) (\(k_{\text{ref},\text{Ec}}\), \(k_{\text{ref},\text{Sa}}\)), and \(z\)-values (\(z_{\text{EC}}\), \(z_{\text{Sa}}\)). Regression coefficients \(b_{\lambda,Ec}\) for the isolate _E. coli_ PS2 were also evaluated by using one-step kinetic data analysis. The remaining parameters in Equation (8), which were previously optimised by nonlinear regression analysis for single cultures [19] or taken from the following scientific articles [24; 36], were fixed as constants. This approach has the advantage that some parameters for co-culture growth prediction in milk could be estimated from the growth of individual species. The goodness of fit of the tertiary H-GD model was evaluated with the global sum of squared errors (\(SSE\)), the root mean square error (\(RMSE\)), and the determination coefficient (\(R^{2}\)) to evaluate its suitability to fit the whole set of observation points according to Equations (1)-(8). \[SSE=\sum_{i=1}^{n}\Bigl{(}x_{i}^{\text{exp}}-x_{i}^{cal}\Bigr{)}^{2} \tag{10}\]\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(x_{i}^{\text{exp}}-x_{i}^{\text{cal}} \right)^{2}}{n-p}} \tag{11}\]  
\[R^{2}=1-\frac{SSE}{SST} \tag{12}\]  
where \(x_{i}^{\text{exp}}\) and \(x_{i}^{\text{cal}}\) correspond to the observed and predicted values, respectively, \(n\) is the total number of data points, \(p\) is the number of estimated parameters, and \(SST\) is the total sum of squared errors. The prediction capability of the H-GD tertiary model was tested through the bias (\(B_{\text{f}}\)) and accuracy (\(A_{\text{f}}\)) factors [37] on the datasets of different _E. coli_ (BR and PS2) and _S.