# Global Dynamics of Modeling Floculation of Microorganism
## 5 Stability and Uniform Persistence
### The Global Asymptotic Stability of the Positive Equilibrium  
In this subsection, the global asymptotic stability of the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) is studied. **Theorem 9**. : _Assume that \(R_{0}>1\) and Model (4) has a unique positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\). Let:_  
\[D_{1} =r_{1}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2}N^{*},\] \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\] \[D_{3} =b_{13}^{2}+b_{23}^{2}+4b_{33},\] \[D_{4} =4b_{33}b_{44}+(b_{23}^{2}+b_{13}^{2})b_{44}+(b_{14}^{2}+b_{24}^{2 })b_{33}-b_{34}^{2}+b_{23}b_{24}b_{34}+b_{13}b_{14}b_{34}.\]  
_If \(D_{1}\geq 0\), \(D_{2}\geq 0\), \(D_{3}<0\) and \(D_{4}>0\) hold, then the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) is globally asymptotically stable. Here:_  
\[b_{13} =\frac{2r_{1}+r_{1}m_{1}P^{*}}{r}+\frac{r}{(K_{1}+1)(K_{2}+1)(K_{ 1}+C^{*})}-\frac{rN^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})},\] \[b_{14} =\frac{r_{1}m_{1}\alpha}{r},\ b_{23}=\frac{2r_{2}+r_{2}m_{1}P^{* }}{r},\ b_{24}=\frac{r_{2}m_{1}\alpha}{r},\ b_{33}=-\frac{(r_{1}^{2}+r_{2}^{2} )(1+m_{1}P^{*})}{r^{2}},\] \[b_{34} =\frac{(r_{1}^{2}+r_{2}^{2})m_{1}\alpha}{r^{2}}+m_{1}+m_{2},\ b_{ 44}=-(1+m_{2}X^{*}).\]  
Proof. : Let us consider the following Lyapunov function on \(A_{0}\),  
\[V_{2} =\frac{1}{2}\Big{(}C-C^{*}+\frac{r_{1}}{r}(X-X^{*})\Big{)}^{2}+ \frac{1}{2}\Big{(}N-N^{*}+\frac{r_{2}}{r}(X-X^{*})\Big{)}^{2}\] \[\quad+X-X^{*}-X^{*}\ln\frac{X}{X^{*}}+P-P^{*}-P^{*}\ln\frac{P}{P^ {*}}.\]  
Clearly, \(V_{2}(C^{*},N^{*},X^{*},P^{*})=0\), and \(V_{2}\) is positive definite with respect to \(E^{*}(C^{*},N^{*}\,X^{*},P^{*})\). The derivative of \(V_{2}\) along the solutions of Model (4) is:  
\[\dot{V}_{2} =\Big{(}C(t)-C^{*}+\frac{r_{1}}{r}(X(t)-X^{*})\Big{)}\Big{(} \dot{C}(t)+\frac{r_{1}}{r}\dot{X}(t)\Big{)}\] \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]0 \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]1 \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]2 \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]3 \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]4 \[D_{2} =r_{2}(2+m_{1}P^{*})(K_{1}+C^{*})(K_{2}+N^{*})-r^{2},\]5  
for any \(t\geq 0\). Noting that:  
\[\left\{ \begin{array}{l}1-C^{*}-\frac{r_{1}C^{*}N^{*}X^{*}}{(K_{1}+C^{*})(K_{ 2}+N^{*})}=0,\\ 1-N^{*}-\frac{r_{2}C^{*}N^{*}X^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}=0,\\ \frac{rC^{*}N^{*}X^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}-X^{*}-m_{1}X^{*}P^{*}=0, \\ 1-P^{*}-m_{2}X^{*}P^{*}=0.\end{array} \right.\]Thus, we have that:  
\[\dot{V}_{2} =\Big{(}C(t)-C^{*}+\frac{r_{1}}{r}(X(t)-X^{*})\Big{)}\Big{(}C^{*}-C(t )+\frac{r_{1}}{r}(X^{*}-X(t))+\frac{r_{1}m_{1}}{r}(X^{*}P^{*}-X(t)P(t))\Big{)}\] \[\quad+\Big{(}N(t)-N^{*}+\frac{r_{2}}{r}(X(t)-X^{*})\Big{)}\Big{(} N^{*}-N(t)+\frac{r_{2}}{r}(X^{*}-X(t))+\frac{r_{2}m_{1}}{r}(X^{*}P^{*}-X(t)P(t)) \Big{)}\] \[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\] \[\quad+m_{1}(X(t)-X^{*})(P^{*}-P(t))+\frac{P(t)-P^{*}}{P(t)}\Big{(} P^{*}-P(t)+m_{2}(X^{*}P^{*}-X(t)P(t))\Big{)}\]  
for any \(t\geq 0\).