## 2 Materials and Methods
### Mathematical Models
#### 2.2.3 Modeling the Microbial Interaction in Co-Cultures  
To directly construct the tertiary model of the simultaneous growth of _S. aureus_, _E. coli_, and LAB in milk, the H-GD model [25], with the competition coefficients and secondary models (Equations (2) and (3)), was used to describe interspecies competition growth for all isothermal growth curves. The suggested model, as a system of the ordinary differential equations with the initial conditions, can be written as follows:  
\[\frac{dx_{Ec}}{dt}=\frac{\frac{\mu_{\text{max}}^{Ec}}{\ln 10}}{1+10^{-a(t-t_{ \lambda}^{Ec})}}\Big{[}\Big{(}1-10^{(x_{Ec}-x_{\text{max},Ec})}\Big{)}\Big{(}1 -10^{(x_{Sa}-x_{\text{max},Ec})}\Big{)}\Big{(}1-10^{(x_{LAB}-x_{\text{max},LB} )}\Big{)}\Big{]}I_{ESL} \tag{11a}\] \[\frac{dx_{Sa}}{dt}=\frac{\frac{\mu_{\text{max}}^{Sa}}{\ln 10}}{1+10^{-a(t-t_{ \lambda}^{Sa})}}\Big{[}\Big{(}1-10^{(x_{Sa}-x_{\text{max},Sa})}\Big{)}\Big{(}1 -10^{(x_{Ec}-x_{\text{max},Ec})}\Big{)}\Big{(}1-10^{(x_{LAB}-x_{\text{max},LB} )}\Big{)}\Big{]}I_{SEL}\] (11b) \[\frac{dx_{LAB}}{dt}=\frac{\frac{\mu_{\text{max}}^{LAB}}{\ln 10}}{1+10^{-a (t-t_{\lambda}^{LAB})}}\Big{[}\Big{(}1-10^{(x_{LAB}-x_{\text{max},LAB})} \Big{)}\Big{(}1-10^{(x_{Ec}-x_{\text{max},Ec})}\Big{)}\Big{(}1-10^{(x_{Sa}-x_ {\text{max},Sa})}\Big{)}\Big{]}I_{LES}\] (11c) \[t=0\ \ \ x_{Ec}=x_{Ec,0}\ \ \ x_{Sa}=x_{Sa,0}\ \ \ x_{LAB}=x_{LAB,0}\]  
where \(I_{ESL}\), \(I_{SEL}\), and \(I_{LES}\) are the competition coefficients representing the effects of _S. aureus_ and LAB (Fresco) on _E. coli_, _E. coli_ and LAB (Fresco) on _S. aureus_, and _E. coli_ and _S. aureus_ on LAB (Fresco), respectively. The following secondary square root models were used to incorporate the temperature effect on the growth parameter in the above equations  
\[\mu_{\text{max}}^{Ec}=\left[b_{T,Ec}\cdot(T-T_{\text{min},Ec}) \right]^{2};\ \mu_{\text{max}}^{Sa}=\left[b_{T,Sa}\cdot(T-T_{\text{min},Sa}) \right]^{2};\ \mu_{\text{max}}^{LAB}=\left[b_{T,LAB}\cdot(T-T_{\text{min},LAB}) \right]^{2};\] \[t_{\lambda}^{Ec}=\frac{1}{\left[b_{\lambda,Ec}\cdot(T-T_{\text{ min},Ec})\right]^{2}};\ t_{\lambda}^{Sa}=\frac{1}{\left[b_{\lambda,aa}\cdot(T-T_{\text{min},Sa}) \right]^{2}};\ t_{\lambda}^{LAB}=\frac{1}{\left[b_{\lambda,LAB}\cdot(T-T_{ \text{min},LAB})\right]^{2}}.\]  
The tertiary H-GD model (Equation (11a,b,c)) contains 15 parameters (Equation (12)). \[p_{H-GD}=\left\{ \begin{array}{l}I_{ESL};I_{SEL};I_{LES};x_{\text{max},Ec};x_{ \text{max},Sa};x_{\text{max},LAB};\\ b_{T,Ec};b_{T,Sa};T_{\text{min},Ec};T_{\text{min},Sa};T_{\text{min},LAB};b_{ \lambda,Ec};b_{\lambda,Sa};b_{\lambda,LAB}\end{array} \right\} \tag{12}\]  
where \(p_{H\text{-GD}}\) is the vector of the evaluated parameters from the entire growth curves for the simultaneous competitive bacterial growth in milk at different temperatures and Fresco inoculum levels. The tertiary H-GD model was tested for four cases (H-GD\({}_{1}\), H-GD\({}_{2}\), H-GD\({}_{3}\), and H-GD\({}_{4}\)). In H-GD\({}_{1}\), only the interaction coefficients (\(I_{ESL}\), \(I_{SEL}\), and \(I_{LES}\)) were evaluated by applying one-step kinetic data analysis to predict the simultaneous growth of _S. aureus_, _E. coli_, and LAB in milk. The remaining parameters in Equation (12), optimized by nonlinear regression analysis for the tertiary HM\({}_{1}\)-S for single cultures, were fixed constant. Then,the first six parameters were evaluated \(p_{\text{H-GD}}=\{I_{\text{{\it{LSEL}}}}\); \(I_{\text{{\it{SEL}}}}\); \(I_{\text{{\it{LES}}}}\); \(x_{\text{max},\text{\it{EC}}}\); \(x_{\text{max},\text{\it{Sa}}}\); \(x_{\text{max},\text{\it{LAB}}}\) in H-GD\({}_{2}\). Next, in H-GD\({}_{3}\), all parameters (Equation (12)) of the tertiary H-GD model (Equation (11a-c)) were optimized. Finally, in H-GD\({}_{4}\), the interaction coefficients were excluded (\(I_{\text{{\it{LSEL}}}}\), \(I_{\text{{\it{SEL}}}}\), and \(I_{\text{{\it{LES}}}}\) equal one). The practical usefulness of this model is that all model parameters for the prediction of the competition between _E. coli_, _S.