# Global Dynamics of Modeling Floculation of Microorganism
## 5 Stability and Uniform Persistence
### The Global Asymptotic Stability of the Positive Equilibrium
Thus, we obtain that, for any \(t\geq 0\),  
\[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\]5 \[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\]6 \[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\]7 \[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\]8 \[\quad+(X(t)-X^{*})\Big{(}\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t) )}-\frac{rC^{*}N^{*}}{(K_{1}+C^{*})(K_{2}+N^{*})}\Big{)}\]9 \[\quad+m_{1}(X(t)-X^{*})(P^{*}-P(t))+\frac{P(t)-P^{*}}{P(t)}\Big{(} P^{*}-P(t)+m_{2}(X^{*}P^{*}-X(t)P(t))\Big{)}\]0  
where \(Y(t)=(|C(t)-C^{*}|\,,|N(t)-N^{*}|\,,|X(t)-X^{*}|\,,|P(t)-P^{*}|)\). \(W\) is a symmetric \(4\times 4\) matrix and \(W=\{b_{ij}\}_{1\leq i,j\leq 4}\) with:  
\[W=\left(\begin{array}{cccc}b_{11}&\frac{1}{2}b_{12}&\frac{1}{2}b_{13}&\frac {1}{2}b_{14}\\ \frac{1}{2}b_{12}&b_{22}&\frac{1}{2}b_{23}&\frac{1}{2}b_{24}\\ \frac{1}{2}b_{13}&\frac{1}{2}b_{23}&b_{33}&\frac{1}{2}b_{34}\\ \frac{1}{2}b_{14}&\frac{1}{2}b_{24}&\frac{1}{2}b_{3 the real quadratic form \(Y^{T}(t)WY(t)\) is negative definite. Then, by using some classical analysis techniques of differential equations, the positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\) of Model (4) is globally asymptotically stable. **Remark 3**. : _Obviously, it is easy to obtain that \(b_{11}=-1<0\), and:_  
\[ \begin{vmatrix}b_{11}&\dfrac{1}{2}b_{12}\\ \dfrac{1}{2}b_{12}&b_{22}\end{vmatrix} =1>0.\]  
_If the conditions \(D_{3}<0\) and \(D_{4}>0\) hold, we obtain that:_  
\[ \begin{vmatrix}b_{11}&\dfrac{1}{2}b_{12}&\dfrac{1}{2}b_{13}\\ \dfrac{1}{2}b_{12}&b_{22}&\dfrac{1}{2}b_{23}\\ \dfrac{1}{2}b_{13}&\dfrac{1}{2}b_{23}&b_{33}\end{vmatrix} <0,\]  
_and:_  
\[ \begin{vmatrix}b_{11}&\dfrac{1}{2}b_{12}&\dfrac{1}{2}b_{13}&\dfrac{1}{2}b_{14} \\ \dfrac{1}{2}b_{12}&b_{22}&\dfrac{1}{2}b_{23}&\dfrac{1}{2}b_{24}\\ \dfrac{1}{2}b_{13}&\dfrac{1}{2}b_{23}&b_{33}&\dfrac{1}{2}b_{34}\\ \dfrac{1}{2}b_{14}&\dfrac{1}{2}b_{24}&\dfrac{1}{2}b_{34}&b_{44}\end{vmatrix} >0.\]  
_Next, we illustrate that the conditions given in Theorem 9 are reasonable. We set \(r=5.8\), \(K_{1}=1\), \(K_{2}=0.6\), \(m_{1}=0.0001\), \(m_{2}=1\), \(r_{1}=6\), \(r_{2}=10\). It is easy to see that there exists a unique positive equilibrium \(E^{*}(C^{*},N^{*},X^{*},P^{*})\approx(0.6709,0.4515,0.3181,0.7587)\), and the conditions \(D_{1}\)-\(D_{4}\) hold. By Theorem 9, the positive equilibrium of Model (4) is globally asymptotically stable (see Figure 3b)._  
_In general, it is very difficult to obtain the global stability properties of the positive equilibrium of Model (4). In this paper, we have obtained the sufficient conditions to ensure the global stability properties of the positive equilibrium by constructing a suitable Lyapunov function. Because of the complexity of the conditions, it is difficult to grasp biological intuition. However, from the numerical simulations, we have found some interesting biological phenomena. The conditions \(D_{1}\)-\(D_{4}\) can be satisfied if the flocculation rate of microorganisms (\(m_{1}\)) is small enough (see, Figure 3b). From a biological point of view, these conditions are reasonable. For more detailed biological considerations, we will leave it for further investigation._