## 4 Discussion  
In this paper we have introduced a system of two delay differential Equation (8) for quorum sensing of _Pseudomonas putida_ in a continuous culture. Motivated by experimental data, a more detailed mathematical model (4) was previously proposed in [18]. Though the system (4) describes the regulatory network in greater detail, in the long run, bacteria reach a saturation level and the model can be reduced to two governing Equation (8), as we have shown in Section 2.1. Surprisingly, even a simple model such as (8) can be used to explain experimental data (Figure 3), maintaining parameter values from a previous fit [18] for almost all model parameters. However, one should take into account that this is valid only from the moment the bacteria population has reached its saturation level. If one is interested in understanding quorum sensing in the initial phases (lag and exponential phase) of bacterial population, then it is convenient to use a more detailed model, such as (4) or (5). The advantage of system (8) is that it can be investigated thoroughly thanks to well-established methods. We have shown existence and uniqueness of solutions and, more importantly, we could guarantee preservation of positivity. This property is often violated in systems of delay differential equations. We have studied linearized stability of non-negative equilibria and proved that the delay system (8) might show stability switches as the delay increases. On the other side, the Lactonse activity (\(\delta\)) can induce Hopf-bifurcation in the associated ODE system (12). For simplicity of computation in the analysis of the system, we have considered only the case of small Hill coefficients (\(n=m=2\)), which corresponds to a maximum of three biologically-relevant stationary states. This assumption is, however, not as restrictive as it seems. Three stationary states, with an intermediate unstable one, are the basis for the bistability situation already discovered in analogous regulatory networks [11; 17]. Moreover, similar small values for the Hill coefficients were found to fit experimental data (Table 1) and correspond well to the biological assumption of a dimer being relevant for the positive feedback in the quorum sensing system of _Pseudomonas putida_[16]. With the fitted parameter values, we do not find delay-induced stability switches. This is not a hint of the delay being not relevant. Though a positive time lag might not change the main qualitative  
Figure 3: Comparison between the numerical solution of the dynamical systems and experimental data. Red curve: solution of the reduced system (8); Blue curve: solution of the full model (4) in [18]. Initial data for the reduced system are \(x(t)=x_{0}(t),y(t)=y_{0}(t)\), \(t\in[18,20]\), where \(y_{0}(t)\equiv 5.2\times 10^{-13}\) was fitted and \(x_{0}(t)\equiv 5.4044\times 10^{-7}\) is the mean value of ELISA and UHPLC measurements at 19 h from the beginning of the experiment. When the cell population has reached its stationary level, the reduced model provides a good approximation of the dynamics. Parameter values used for the reduced model are given in Table 1.  
behavior of the system, the DDE model still describes the experimentally-determined data and their time course quantitatively better than the associated ODE system, in particular when the bacteria population is in the lag or exponential phase. Stability switches and periodic oscillatory behavior might appear for a different choice of the parameters in system (8). As the main focus of this work was to provide a description for a real biological process, we decided to omit further numerical investigation on the qualitative behavior of (8). Delay equations have been previously used in mathematical models of continuous cultures. Commonly, a time lag was included to describe the time necessary for the bacteria to convert nutrients in new biomass [26; 27]. Being interested in the long term dynamics with bacteria being at an equilibrium, we have chosen not to consider such reproduction lags in our model.