## 2 Material and Methods
### Mathematical Models  
#### 2.4.1 Modelling the Microbial Interaction in Co-Cultures  
The primary models of [PERSON] [31] and [PERSON] and [PERSON] [5] combined with secondary square root models as applied for growth rate and lag time were used to describe competitive growth of the co-cultures series in milk for all isothermal growth curves. The suggested interaction H-GD model with competition coefficients describing the growth of LAB, _G. candidum_ and behaviour of _E. coli_ and _S. aureus_ in inter-species competition were used in this study according to A\(\cai\) et al. [19]. Thus, the system of the ordinary differential equations with the initial conditions applied for the growth phase and a mixed system of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Microorganism** & **Isolate** & **Origin** \\ \hline DVS\({}^{\text{\text{\textregistered}}}\) Fresco\({}^{\text{\textregistered}}\) 1000 NG & - & commercial LAB culture; [PERSON], Hoersholm, Denmark \\ \hline _G. candidum_ & J & Slovakian traditional cheese “Bryndza” \\ \hline \multirow{3}{*}{_E. coli_} & Br & Slovakian traditional cheese “Bryndza” \\ \cline{2-3}  & PS2 & laboratory-produced pasta-filata cheese from raw cows’ milk \\ \hline \multirow{3}{*}{_S. aureus_} & 2064 & Slovakian ewes’ lump cheese \\ \cline{2-3}  & 14733 & milk vending machine biofilm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Microbial cultures and their origin. differential equations and nonlinear algebraic equation (G-B model) was used for survivors of _S. aureus_ (_Sa_) and _E. coli_ (_Ec_) in the death phase, denoted here as index \(P\) (_P_ = _Ec_ or _Sa_). The equations can be written as follows:  
1. H-GD model with the competition coefficients Growth of _LAB_; \(t\in\left\langle 0,t_{\mathrm{t}}\right\rangle\) \[\frac{dx_{Lab}}{dt}=\left[\frac{\frac{\mu_{\mathrm{max}}^{Lob}}{\ln 10}}{1+10^{- \alpha(t-t_{\mathrm{\lambda}}^{Lob})}}\Big{(}1-10^{(x_{Lab}-x_{\mathrm{max},Lob })}\Big{)}\Big{(}1-10^{(x_{P}-x_{\mathrm{max},P})}\Big{)}\right]I_{LP}\] (1) Growth of \(P\) = _Ec_ or _Sa_; \(t\in\left\langle 0,t_{\mathrm{t}}\right\rangle\) \[\frac{dx_{P}}{dt}=\left[\frac{\frac{\mu_{\mathrm{max}}^{P}}{\ln 10}}{1+10^{- \alpha(t-t_{\mathrm{\lambda}}^{P})}}\Big{(}1-10^{(x_{P}-x_{\mathrm{max},P})} \Big{)}\Big{(}1-10^{(x_{Lab}-x_{\mathrm{max},Lob})}\Big{)}\right]I_{PL}\] (2) Survival \(P\) = _Ec_ or _Sa_; \(t\in\left\langle t_{\mathrm{t}},t\right\rangle\) \[x=x_{\mathrm{max},P}+(x_{\mathrm{res},P}-x_{\mathrm{max},P})\exp\left\{-\exp \left[\left(\frac{\frac{k_{\mathrm{max},P}}{\ln 10}\cdot e}{(x_{\mathrm{res},P}-x_{ \mathrm{max},P})}\right)(t_{\mathrm{\lambda},s}-t)+1\right]\right\}\] (3) \[\frac{dx_{Gc}}{dt}=\frac{k_{\mathrm{Gc}}\frac{\mu_{\mathrm{max}}^{Gc}}{ \ln 10}}{1+10^{-\alpha(t-t_{\mathrm{\lambda}}^{Gc})}}\Big{(}1-10^{(x_{Gc}-x_{ \mathrm{max},Gc})}\Big{)}\] (4) \[t=0\quad x_{Lab}=x_{Lab,0}\quad x_{P}=x_{P,0}\quad x_{Gc}=x_{Gc,0}\]  
where \(\mu_{\mathrm{max}}^{\mathrm{i=Lab},P,Gc}\) are the maximum specific growth rates of the LAB, _E. coli_ or _S. aureus_ and _G. candidium_ (_Gc_), respectively, \(t_{\mathrm{\lambda}}^{\mathrm{i}}\) are the lag times of microorganisms, \(\alpha\) is the lag phase transition coefficient, taking a value of 4 [31]. Concentrations \(x\), which include \(x_{\mathrm{i}}\) = log \(N_{\mathrm{i}}\), \(x_{0\mathrm{i}}\) = log \(N_{\mathrm{0},\mathrm{i}}\), \(x_{\mathrm{max},\mathrm{i}}\) = log \(N_{\mathrm{max},\mathrm{i}}\), \(x_{\mathrm{res},\mathrm{i}}\) = log \(N_{\mathrm{res},\mathrm{i}}\) represent the real, initial, maximum and residual (or tail) cell density, \(N_{\mathrm{i}}\), \(N_{\mathrm{0},\mathrm{i}}\) = \(A_{\mathrm{max},\mathrm{i}}\) and \(N_{\mathrm{res},\mathrm{i}}\). \(I_{LP}\), \(I_{PL}\) are the competition coefficients representing the effects of LAB (Fresco) on _E. coli_ or _S. aureus_ and _E. coli_ or _S. aureus_ on LAB (Fresco), respectively, in the H-GD model type R. \(k_{\mathrm{max},P}\) is the maximum death rate of _E. coli_ or _S.