# Global Dynamics of Modeling Floculation of Microorganism
## 4 The Global Stability of the Boundary Equilibrium  
Global stability properties of the equilibria \(E_{0}\) or \(E^{*}\) imply that the asymptotic properties of the carbon source, nitrogen source, _Rhodopseudomonas palustris_ and flocculants in the culture vessel are not dependent on the initial values \(C_{0},\ N_{0},\ X_{0}\) and \(P_{0}\). For the global stability property of the boundary equilibrium \(E_{0}\) of Model (4), we have the following result. **Theorem 5**. : _If \(R_{0}<1\), then the boundary equilibrium \(E_{0}(1,1,0,1)\) of Model (4) is locally asymptotically stable. Further, if:_  
\[R_{0}\leq\frac{m_{1}\underline{P}+1}{m_{1}+1}<1, \tag{10}\]  
_then the boundary equilibrium \(E_{0}\) of Model (4) is globally asymptotically stable._  
Proof. : It is easy to show that the boundary equilibrium \(E_{0}(1,1,0,1)\) of Model (4) is locally asymptotically stable by the characteristic equation of the linearization of Model (4). Next, we prove the global asymptotic stability of the boundary equilibrium \(E_{0}\) of Model (4). Since \(\Omega\) is attractive and positively forward invariant for Model (4), hence it just considers Model (4) in \(\Omega\). Define:  
\[V_{1}=X.\]  
Apparently, \(V_{1}(1,1,0,1)=0\) and \(V_{1}\) is continuous on \(\Omega\). If \(R_{0}\leq\frac{m_{1}\underline{P}+1}{m_{1}+1}<1\), then the derivative of \(V_{1}\) along the solutions of Model (4) is:  
\[\dot{V}_{1} =\frac{rC(t)N(t)X(t)}{(K_{1}+C(t))(K_{2}+N(t))}-X(t)-m_{1}X(t)P(t)\] \[\leq (\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t))}-1-m_{1}\underline{P})X (t)\] \[\leq (m_{1}+1)(\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t))}\frac{1}{m_{1 }+1}-R_{0})X(t)\] \[\leq (m_{1}+1)(\frac{r}{(K_{1}+1)(K_{2}+1)}\frac{1}{m_{1}+1}-R_{0})X(t )=0\]for any \(t\geq 0\). Hence, \(V_{1}\) is a Lyapunov function of Model (4) on \(\Omega\). Define \(E=\{(C,N,X,P)|(C,N,X,P)\in\Omega,\ \dot{V}_{1}=0\}.\) We have that:  
\[E\subset\{(C,N,X,P)|(C,N,X,P)\in\Omega,\ X=0,\ \text{or}\ C=1\ \text{and}\ N=1\}.\]  
Let \(\hat{M}\) be the largest set in \(E\), which is invariant with respect to Model (4). Clearly, \(\hat{M}\) is not empty, since \((1,1,0,1)\in\hat{M}.\) For any \((C_{0},N_{0},X_{0},P_{0})\in\hat{M}\), let \((C(t),N(t),X(t),P(t))\) be the solution of Model (4) with the initial Condition (5). From the invariance of \(\hat{M}\), we get \((C(t),N(t),X(t),P(t))\in\hat{M}\subseteq E\) for any \(t\in R\). Thus, we get, for each \(t,\)\(X(t)=0\), or \(C(t)=1\), and \(N(t)=1\). If for some \(\widetilde{t}\), \(C(\widetilde{t})=1\) and \(N(\widetilde{t})=1\), then we have that \(\hat{C}(\widetilde{t})=\hat{N}(\widetilde{t})=0\). Hence, from the first or second equation of Model (4), we obtain that \(X(\widetilde{t})=0\). Thus, for any \(t\in R\), we have that \(X(t)\equiv 0\). Subsequently, from the first, second and forth equations of Model (4), we have that, for any \(t\in R\),  
\[\hat{C}(t)=1-C(t),\ \ \dot{N}(t)=1-N(t),\ \ \dot{P}(t)=1-P(t).\]  
Furthermore, for any \(t\in R\), we have that:  
\[C(t)=1-(1-C(0))e^{-t},\ \ N(t)=1-(1-N(0))e^{-t},\ \ P(t)=1-(1-P(0))e^{-t}.\]  
If \(C(0)<1\) or \(N(0)<1\) or \(P(0)<1\), then \(C(t)\), \(N(t)\) and \(P(t)\) become negative values \((t\to-\infty)\). Then, we obtain that \(C(0)=1\), \(N(0)=1\) and \(P(0)=1\). Hence, we obtain that, for any \(t\in R\), \(C(t)=N(t)=P(t)\equiv 1\). Therefore, we obtain \(\hat{M}=\{(1,1,0,1)\}.\) The classical Lyapunov-LaSalle invariance principle shows that \(E_{0}\) is globally attractive. Since it has been shown that, if \(R_{0}\leq\frac{m_{1}P+1}{m_{1}+1}<1\), the boundary equilibrium \(E_{0}\) of Model (4) is locally asymptotically stable. Hence, the boundary equilibrium \(E_{0}\) of Model (4) is globally asymptotically stable. **Remark 2**. : _Let us use some numerical simulations to check the correctness of the theoretical analyses.