# Global Dynamics of Modeling Floculation of Microorganism
## 7 Discussion and Conclusions  
In the paper, based on some biological considerations and chemostat models, a dynamic model governed by ordinary differential equations with four variables (carbon source, nitrogen source, _Rhodopseudomonas palustris_ and flocculants) is presented. There is a boundary equilibrium and at most five positive equilibria for the proposed model. To give a theoretical analysis for the existence of all of the positive equilibria of Model (4), the method of the Descartes rule of signs is applied to the classifications of the positive roots of a fifth order algebraic equation. The local and global stability properties of the boundary equilibrium of Model (4) have been studied in detail. An interesting phenomenon of backward and forward bifurcations is observed. That is, there may exist two positive equilibria even if the condition \(R_{0}<1\) holds. Hence, sufficient Condition (10) to ensure the global stability of the boundary equilibrium is reasonable in mathematics. The local stability of the positive equilibrium of Model (4) is also carried out. From Condition (11), we have that the positive equilibrium is locally asymptotically stable when the flocculation coefficient \(m_{1}\) is small enough. Hence, Condition (11) is also reasonable in biology. Uniform persistence of Model (4) has also been completely studied under the condition \(R_{0}>1\). Uniform persistence has very important significance both in mathematics and biology, and it characterizes the long-term survival of some microorganisms [45]. Finally, some control strategies are provided by simple theoretical analysis. From Theorem 5, we have that _Rhodopseudomonas palustris_ in the chemostat will tend to extinction if \(R_{0}<1\)  
Figure 5: **(a)**_Rhodopseudomonas palustris_ in the chemostat will tend to be constant if the flocculation effect (\(m_{1}\)) is reduced. **(b)**_Rhodopseudomonas palustris_ in the chemostat will tend to be constant if the Michaelis–Menten constant of carbon (\(K_{1}\)) is reduced. The initial conditions are \(C_{0}=2\), \(N_{0}=4\), \(X_{0}=1\), \(P_{0}=3\). Figure 6: _Rhodopseudomonas palustris_ in the chemostat will tend to be constant if the Michaelis–Menten constant of nitrogen (\(K_{2}\)) is reduced. The initial conditions are \(C_{0}=2\), \(N_{0}=4\), \(X_{0}=1\), \(P_{0}=3\). In this case, these control strategies can be applied to remove \(Cyanobacteria\), which are well known to produce a variety of toxins and have serious harm on human health. From Theorem 7, we have that _Rhodopseudomonas palustris_ in the chemostat will tend to be positive constant if \(R_{0}>1\). In this case, these control strategies can be widely used for the collection of useful microorganisms. It is well-known that the existence of time delays is inevitable in biology. For example, in the cultivation of microorganisms, there are always time delays in the process of transferring nutrients and the uptake of nutrients. Hence, chemostat models with time delays that account for the time lapsing between the uptake of nutrients by cells and the incorporation of these nutrients as biomass have been given much attention [40; 55; 56; 57]. Based on Model (3), it may have the following more general form with time delays,  
\[\left\{ \begin{array}{l}\frac{dC(t)}{dt}=(C^{0}-C(t))D-\frac{r_{1} C(t)N(t)X(t)}{\delta_{1}(K_{1}+C(t))(K_{2}+N(t))}+\rho_{1}X(t-\sigma),\\ \frac{dN(t)}{dt}=(N^{0}-N(t))D-\frac{r_{2}C(t)N(t)X(t)}{\delta_{2}(K_{1}+C(t))( K_{2}+N(t))}+\rho_{2}X(t-\sigma),\\ \frac{dX(t)}{dt}=\frac{re^{-d_{1}\tau}C(t-\tau)N(t-\tau)X(t-\tau)}{(K_{1}+C(t- \tau))(K_{2}+N(t-\tau))}-DX(t)-d_{1}X(t)-m_{1}X(t)P(t),\\ \frac{dP(t)}{dt}=(P^{0}-P(t))D-m_{2}X(t)P(t).\end{array} \right. \tag{14}\]  
In Model (14), the constants \(\rho_{1}\geq 0\) and \(\rho_{2}\geq 0\) are the rate constants at which the carbon source and nitrogen source are recycled because of the death of _Rhodopseudomonas palustris_.