# Global Dynamics of Modeling Floculation of Microorganism
## 5 Stability and Uniform Persistence
### The Local Asymptotic Stability of the Positive Equilibrium  
For the local asymptotic stability of the positive equilibrium of Model (4), we have the following result.  
**Theorem 6**: _If the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) exists and the condition:_  
\[m_{1}<\frac{K_{1}(1-C^{*})}{P^{*}C^{*}(K_{1}+C^{*})}+\frac{K_{2}(1-N^{*})}{P^ {*}N^{*}(K_{2}+N^{*})} \tag{11}\]  
_holds, then the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) is locally asymptotically stable._  
**Proof.** For convenience, we assume that:  
\[\Delta=\frac{K_{1}r_{1}N^{*}X^{*}}{(K_{1}+C^{*})^{2}(K_{2}+N^{*})}+\frac{K_{2} r_{2}C^{*}X^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})^{2}}.\]  
For the linearized system, the corresponding characteristic equation of Model (4) can be expressed as follows:  
\[\lambda^{4}+A_{1}\lambda^{3}+A_{2}\lambda^{2}+A_{3}\lambda+A_{4}=0,\]  
where:  
\[A_{1} =3+m_{2}X^{*}+\Delta,\] \[A_{2} =\Delta(m_{1}P^{*}+2+m_{2}X^{*})+1+\Delta+2(1+m_{2}X^{*})-m_{1}m _{2}P^{*}X^{*},\] \[A_{3} =\Delta(2m_{1}P^{*}+3+2m_{2}X^{*})-2m_{1}m_{2}P^{*}X^{*}+1+m_{2}X ^{*},\] \[A_{4} =m_{2}X^{*}(\Delta-m_{1}P^{*})+(1+m_{1}P^{*})\Delta.\]  
According to the Routh-Hurwitz criterion, we need to show that:  
\[A_{1}A_{2}-A_{3}>0,\ \ A_{3}(A_{1}A_{2}-A_{3})-A_{1}^{2}A_{4}>0,\ \ A_{4}>0.\]  
For the sake of simplicity, we define:  
\[\Delta_{1}=A_{1},\ \Delta_{2}=A_{1}A_{2}-A_{3},\ \Delta_{3}=A_{3}(A_{1}A_{2}-A_{3})-A_{1}^{2}A_{4},\ \Delta_{4}=A_{4}\Delta_{3}.\]By computation, it can be obtained that:  
\[\Delta_{1} =A_{1}>0,\] \[\Delta_{2} =m_{2}X^{*}(1+\Delta+m_{2}X^{*})(\Delta-m_{1}P^{*})+(2+2m_{2}X^{*}+ \Delta)(3+m_{2}X^{*})\] \[\quad+\Delta(6+m_{1}P^{*})+\Delta(m_{2}X^{*}+\Delta)(3+m_{1}P^{*}),\] \[\Delta_{3} =P_{1}(\Delta-m_{1}P^{*})+Q,\] \[P_{1} =2m_{2}X^{*}\Delta(1+\Delta)(3+m_{1}P^{*}+m_{2}X^{*})+4m_{2}X^{* }\Big{(}1+\Delta(1+m_{2}X^{*})\Big{)}\] \[\quad+4m_{2}X^{*}(1+m_{2}X^{*})(m_{2}X^{*}+3)+2(m_{2}X^{*})^{2} \Delta(1+m_{2}X^{*})\] \[\quad+m_{2}X^{*}\Delta(1+m_{2}X^{*})(2m_{1}P^{*}+3+2m_{2}X^{*}),\] \[Q =a_{1}(1+m_{1}P^{*})+a_{2}(3+m_{1}P^{*}+m_{2}X^{*})+a_{3}(1+m_{2} X^{*})+a_{4}m_{1}P^{*}m_{2}X^{*}\Delta\] \[\quad+2a_{5}m_{1}P^{*}m_{2}X^{*}+a_{6}m_{2}X^{*}\Delta+\Delta^{2} (2+3m_{2}X^{*})(3+2m_{2}X^{*})+14\Delta+4\Delta^{2},\] \[a_{1} =2\Delta^{3}+8\Delta^{2}+7\Delta+m_{2}X^{*}\Delta(m_{2}X^{*}+7)+m _{1}P^{*}\Delta^{2}(\Delta+2),\] \[a_{2} =\Delta(1+\Delta)(1+m_{2}X^{*})+\Delta^{3}(2+m_{1}P^{*})+\Delta^ {2},\] \[a_{3} =2\Big{(}1+(1+m_{2}X^{*})(3+m_{2}X^{*})\Big{)},\] \[a_{4} =2(m_{2}X^{*})^{2}+(1+\Delta)^{2},\] \[a_{5} =4+(m_{2}X^{*})^{2}+m_{1}P^{*}m_{2}X^{*}+2m_{2}X^{*},\] \[a_{6} =4m_{2}X^{*}+4\Delta+13,\] \[\Delta_{4} =A_{4}\Delta_{3}.\]  
If:  
\[\Delta=\frac{K_{1}r_{1}N^{*}X^{*}}{(K_{1}+C^{*})^{2}(K_{2}+N^{*})}+\frac{K_{2 }r_{2}C^{*}X^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})^{2}}>m_{1}P^{*},\]  
that is,  
\[m_{1}<\frac{K_{1}(1-C^{*})}{P^{*}C^{*}(K_{1}+C^{*})}+\frac{K_{2}(1-N^{*})}{P^{ *}N^{*}(K_{2}+N^{*})},\]  
we obtain that \(\Delta_{1}>0\), \(\Delta_{2}>0\), \(\Delta_{3}>0\), and \(\Delta_{4}>0\). Hence, from the Routh-Hurwitz criterion, we obtain that the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) is locally asymptotically stable. This completes the proof of Theorem 6.