# Global Dynamics of Modeling Floculation of Microorganism
## 6 Control Strategies  
In this section, some control strategies are provided by suitable theoretical analysis. If \(R_{0}<1\) holds, then, we have from Theorem 5 that the boundary equilibrium \(E_{0}\) of Model (4) is locally asymptotically stable. We note that the condition \(R_{0}<1\) is equivalent to the following inequality,  
\[R_{0}=\dfrac{r}{(K_{1}+1)(K_{2}+1)(m_{1}+1)}<1. \tag{12}\]  
All of the parameters \(r\), \(K_{1}\), \(K_{2}\) and \(m_{1}\) in (12) are defined as in Model (4). (12) can be further written as the following form,  
\[\dfrac{r}{(\frac{K_{1}}{C^{0}}+1)(\frac{K_{2}}{N^{0}}+1)(m_{1}P^{0}+D)}<1. \tag{13}\]Here, all of the parameters \(r\), \(K_{1}\), \(K_{2}\), \(m_{1}\), \(C^{0}\), \(N^{0}\), \(P^{0}\) and \(D\) in (13) are defined as in Model (3).  
In view of the biological meanings of the parameters in Model (3) and Condition (13), Theorem 5 indicates that the concentration of _Rhodopseudomonas palustris_ in the chemostat tends to zero, and the concentration of the carbon source, nitrogen source and flocculants may tend to the constant values \(C^{0}\), \(N^{0}\) and \(P^{0}\), respectively, as time \(t\) increases, if one of the following two cases occurs: (a) reducing the absorption of _Rhodopseudomonas palustris_ or the carbon input concentration, or the nitrogen input concentration; (b) improving the velocity or the flocculation effect or flocculant input concentration. These cases are reasonable, since they imply the insufficient sources for _Rhodopseudomonas palustris_ to grow. Hence, in the environmental science field, it can be used to remove algae and heavy metals.  
From Theorem 7, we have that Model (3) is uniformly persistent if \(R_{0}>1\). This means that the concentrations of the carbon source, nitrogen source, _Rhodopseudomonas palustris_ and flocculants in the chemostat may be ultimately maintained at some positive constant values, as time \(t\) increases, if one of the following two cases occurs: (a) improving the absorption of _Rhodopseudomonas palustris_, or the carbon input concentration, or the nitrogen input concentration; (b) reducing the velocity or the flocculation effect or flocculant input concentration.  
These control strategies can be performed by numerical simulations.  
In the following, for convenience, we simulate the extinction or persistence of microorganism (\(X(t)\)) numerically by using (12) and Model (4).  
If the parameters are chosen as in Table 11 (see Appendix A), _Rhodopseudomonas palustris_ in the chemostat will tend to extinction (see Figure 4a).  
If the absorption of _Rhodopseudomonas palustris_ (\(r\)) is improved from \(r=5.8\) to \(r=8\) and the other parameters are the same as Table 11, _Rhodopseudomonas palustris_ in the chemostat will tend to be constant (see Figure 4b).  
If the flocculation effect (\(m_{1}\)) is reduced from \(m_{1}=1\) to \(m_{1}=0.1\), _Rhodopseudomonas palustris_ in the chemostat will tend to be constant (see Figure 5a).  
If Michaelis-Menten constant of carbon (\(K_{1}\)) is reduced from \(K_{1}=1\) to \(K_{1}=0.5\), _Rhodopseudomonas palustris_ in the chemostat will tend to be constant (see Figure 5b).  
Figure 4: (**a**) _Rhodopseudomonas palustris_ in the chemostat will tend to extinction with the parameters in Table 11. (**b**) _Rhodopseudomonas palustris_ in the chemostat will tend to be constant if the absorption of _Rhodopseudomonas palustris_ (\(r\)) is improved. The initial conditions are \(C_{0}=2\), \(N_{0}=4\), \(X_{0}=1\), \(P_{0}=3\).  
If Michaelis-Menten constant of nitrogen (\(K_{2}\)) is reduced from \(K_{2}=1\) to \(K_{2}=0.25\), _Rhodopseudomonas palustris_ in the chemostat will tend to be constant (see Figure 6).