# Global Dynamics of Modeling Floculation of Microorganism
## 3 The Existence of the Equilibria and Its Classification  
Let \((C,N,X,P)\) be any equilibrium of Model (4). Then, \((C,N,X,P)\) satisfies the following nonlinear algebraic equations,  
\[\left\{ \begin{array}{l}1-C-\frac{r_{1}CNX}{(K_{1}+C)(K_{2}+N)}=0, \\ 1-N-\frac{r_{2}CNX}{(K_{1}+C)(K_{2}+N)}=0,\\ \frac{rCNX}{(K_{1}+C)(K_{2}+N)}-X-m_{1}XP=0,\\ 1-P-m_{2}XP=0.\end{array} \right. \tag{7}\]  
Model (4) always has the boundary equilibrium \(E_{0}(1,1,0,1)\). The existence of \(E_{0}\) indicates that, if there is no _Rhodopsudomonas palustris_ to be added into the culture vessel at the beginning of the culture, the concentrations of the carbon source, nitrogen source and flocculants always maintain the constant values 1, 1 and 1, respectively. The equilibrium \(E_{0}(1,1,0,1)\) is also called _Rhodopseudomonas palustris_-free equilibrium. Define the basic bifurcation parameter as:  
\[R_{0}=\frac{r}{(K_{1}+1)(K_{2}+1)(m_{1}+1)}.\]  
Let \((C^{*},N^{*},X^{*},P^{*})\) be any positive equilibrium of Model (4). From (7), we have that:  
\[\left\{ \begin{array}{l}(K_{1}+C^{*})(K_{2}+N^{*})(1+\frac{m_{1}}{1+m_{2}X^{* }})-rC^{*}N^{*}=0,\\ P^{*}=\frac{1}{1+m_{2}X^{*}},\\ C^{*}=\frac{-r_{1}m_{2}(X^{*})^{2}+(m_{2}r-r_{1}(m_{1}+1))X^{*}+r}{r(1+m_{2}X^{ *})},\\ N^{*}=\frac{-r_{2}m_{2}(X^{*})^{2}+(m_{2}r-r_{2}(m_{1}+1))X^{*}+r}{r(1+m_{2}X^{ *})}.\end{array} \right. \tag{8}\]  
Clearly, \(X^{*}\) should satisfy the following conditions:  
\[ \begin{array}{l}-r_{1}m_{2}(X^{*})^{2}+\Big{(}m_{2}r-r_{1}(m_{1}+1)\Big{)}X^ {*}+r>0,\\ -r_{2}m_{2}(X^{*})^{2}+\Big{(}m_{2}r-r_{2}(m_{1}+1)\Big{)}X^{*}+r>0.\end{array}  \tag{9}\]  
Substituting the second, third and forth equations of (8) into the first equation gives a fifth order algebraic equation,  
\[f(X^{*})= a(X^{*})^{5}+b(X^{*})^{4}+c(X^{*})^{3}+d(X^{*})^{2}+eX^{*}+f=0,\]  
where:  
\[a=r_{1}r_{2}m_{2}^{3}(1-r),\] \[b=r_{1}m_{2}^{3}r\Big{(}r-(K_{2}+1)-\frac{r_{2}(m_{1}+2)}{m_{2}} \Big{)}+3r_{1}r_{2}m_{2}^{2}(m_{1}+1)\] \[\quad+r_{2}m_{2}^{3}r\Big{(}r-(K_{1}+1)-\frac{r_{1}(m_{1}+1)}{m_ {2}}\Big{)},\] \[c=m_{2}^{3}r^{2}\Big{(}(K_{1}+1)(K_{2}+1)+\frac{(r_{1}+r_{2})(m _{1}+3)}{m_{2}}-r\Big{)}+q,\] \[d=m_{2}^{2}rR_{0}(K_{1}+1)^{2}(K_{2}+1)^{2}(m_{1}+1)^{2}(1-R_{0} )+d_{1},\] \[e=r_{1}r(m_{1}+1)\Big{(}r-(K_{2}+1)(m_{1}+1)\Big{)}+r_{2}r(m_{1} +1)\Big{(}r-(K_{1}+1)(m_{1}+1)\Big{)}\] \[\quad+m_{2}r^{2}\Big{(}(K_{1}+1)(K_{2}+1)(2m_{1}+3)-3r\Big{)},\] \[f=rR_{0}(K_{1}+1)^{2}(K_{2}+1)^{2}(m_{1}+1)^{2}(1-R_{0}),\] \[q=r_{1}r_{2}m_{2}(3-r)(1+m_{1})^{2}-2q_{1}r(1+m_{1})-rm_{2}^{2}( K_{1}r_{2}+K_{2}r_{1}),\] \[a=r_{1}r_{2}m_{2}^{3}(1-r),\]0 \[a=r_{1}r_{2}m_{2}^{3}(1-r),\]1 \[a=r_{1}r_{2}m_{2}^{3}(1-r),\]2 \[a=r_{1}r_{2}m_{2}^{3}(1-r),\]3 \[a=r_{1}r_{2}m_{2}^{3}(1-r),\]4Let us consider the necessary condition for the existence of the positive equilibria of Model (4). From the first equation in (8), we obtain the following function:  
\[a=r_{1}r_{2}m_{2}^{3}(1-r),\]5  
which implies that \(r>1\) is a necessary condition for a positive equilibrium to exist. Using the methods similar to [47], we can give the sufficient conditions of the existence of the positive equilibria of Model (4). The following results (Theorem 3) follow from the various possibilities enumerated in Table A1 (see Appendix A):  
**Theorem 3**. : _(i) Model (4) has a unique positive equilibrium if \(R_{0}<1\) and Conditions (9) hold and whenever Cases 1, 9, 13, 15 and 16 in Table A1 are satisfied;_  
_(ii) Model (4) could have more than one positive equilibrium if \(R_{0}<1\) and Conditions (9) hold and whenever Cases 2-8, 10-12 and 14 in Table A1 are satisfied;_  
_(iii) Model (4) could have five positive equilibria at most if \(R_{0}>1\) and Conditions (9) hold and whenever Cases 1-16 in Table A1 are satisfied._  
Hence, under suitable conditions, there may be at most five different positive roots for the fifth order algebraic equation. Let \(X=X^{*}\) be any such positive root, which also satisfies Conditions (9). Thus, from (8), \(C=C^{*}>0\), \(N=N^{*}>0\) and \(P=P^{*}>0\) can be obtained.