# Global Dynamics of Modeling Floculation of Microorganism
## Appendix B  
In Section 3, we have discussed the phenomena of backward and forward bifurcations by numerical simulations. In this section, the center manifold theory is used on Model (4) to obtain the rigorous result (see, for example, [40; 47; 48]).  
Let \(C=x_{1}\), \(N=x_{2}\), \(X=x_{3}\), \(P=x_{4}\), so that Model (4) can be re-written in the following form:  
\[\left\{\begin{array}{l}\frac{dx_{1}(t)}{dt}=1-x_{1}-\frac{r_{1}x_{1}x_{2}x_{3 }}{(K_{1}+x_{1})(K_{2}+x_{2})}=g_{1},\\ \frac{dx_{2}(t)}{dt}=1-x_{2}-\frac{r_{2}x_{1}x_{2}x_{3}}{(K_{1}+x_{1})(K_{2}+x_ {2})}=g_{2},\\ \frac{dx_{3}(t)}{dt}=\frac{rx_{1}x_{2}x_{3}}{(K_{1}+x_{1})(K_{2}+x_{2})}-x_{3} -m_{1}x_{3}x_{4}=g_{3},\\ \frac{dx_{4}(t)}{dt}=1-x_{4}-m_{2}x_{3}x_{4}=g_{4}.\end{array}\right. \tag{11}\]The Jacobian matrix of Model (15) at \(E_{0}(1,1,0,1)\) is given by:  
\[J(E_{0})=\left(\begin{array}{cccc}-1&0&\dfrac{-r_{1}}{(K_{1}+1)(K_{2}+1)}&0\\ 0&-1&\dfrac{-r_{2}}{(K_{1}+1)(K_{2}+1)}&0\\ 0&0&\dfrac{r}{(K_{1}+1)(K_{2}+1)}-1-m_{1}&0\\ 0&0&-m_{2}&-1\end{array}\right).\]  
Suppose \(r\) is chosen as a bifurcation parameter. Solving \(R_{0}=1\) gives:  
\[r=r^{*}=(K_{1}+1)(K_{2}+1)(m_{1}+1).\]  
**Eigenvectors of \(J(E_{0})|_{r=r^{*}}\)**  
It can be shown that the Jacobian matrix of Model (15) at \(r=r^{*}\) has a right eigenvector (corresponding to the zero eigenvalue) given by \(\omega=(\omega_{1},\omega_{2},\omega_{3},\omega_{4})^{\mathrm{T}}\), where:  
\[\omega_{1}=\dfrac{-r_{1}}{(K_{1}+1)(K_{2}+1)}\omega_{3},\ \ \ \omega_{2}=\dfrac{-r_{2}}{(K_{1}+1)(K_{2}+1)}\omega_{3},\ \ \ \omega_{3}=\omega_{3}>0,\ \ \ \omega_{4}=-m_{2}\omega_{3}.\]  
Further, the Jacobian matrix of Model (15) at \(r=r^{*}\) has a left eigenvector (associated with the zero eigenvalue) given by \(v=(v_{1},v_{2},v_{3},v_{4})^{\mathrm{T}}\), where:  
\[v_{1}=0,\ \ \ v_{2}=0,\ \ \ v_{3}=v_{3}>0,\ \ \ v_{4}=0.\]  
**Computations of \(\hat{a}\) and \(\hat{b}\)**  
For Model (15), the associated non-zero partial derivatives of \(g=(g_{1},g_{2},g_{3},g_{4})^{\mathrm{T}}\) (at \(E_{0}\)) are given by:  
\[\dfrac{\partial^{2}g_{3}}{\partial x_{1}\partial x_{3}}= \dfrac{rK_{1}}{(K_{1}+1)^{2}(K_{2}+1)},\ \ \ \dfrac{\partial^{2}g_{3}}{\partial x_{2}\partial x_{3}}=\dfrac{rK_{2}}{(K_{1}+1 )(K_{2}+1)^{2}},\] \[\dfrac{\partial^{2}g_{3}}{\partial x_{3}\partial x_{4}}= -m_{1},\ \ \ \dfrac{\partial^{2}g_{3}}{\partial x_{3}\partial r^{*}}= \dfrac{1}{(K_{1}+1)(K_{2}+1)}.\]  
It follows from the above expressions that:  
\[\hat{a}= v_{3}\sum_{i=1,j=1}^{4}\dfrac{\omega_{i}\omega_{j}\partial^{2}g_{3} }{\partial x_{i}\partial x_{j}}\] \[= 2v_{3}\Big{(}\omega_{1}\omega_{3}\dfrac{rK_{1}}{(K_{1}+1)^{2}(K_ {2}+1)}+\omega_{2}\omega_{3}\dfrac{rK_{2}}{(K_{1}+1)(K_{2}+1)^{2}}-\omega_{3} \omega_{4}m_{1}\Big{)}\] \[= 2v_{3}\omega_{3}^{2}\Big{(}m_{1}m_{2}-\dfrac{rr_{1}K_{1}}{(K_{1} +1)^{3}(K_{2}+1)^{2}}-\dfrac{rr_{2}K_{2}}{(K_{1}+1)^{2}(K_{2}+1)^{3}}\Big{)},\]  
from which it can be shown that \(\hat{a}>0\) if:  
\[m_{1}m_{2}>\dfrac{rr_{1}K_{1}}{(K_{1}+1)^{3}(K_{2}+1)^{2}}+\dfrac{rr_{2}K_{2}} {(K_{1}+1)^{2}(K_{2}+1)^{3}}.\]  
For the sign of \(\hat{b}\), it can be shown that the associated non-vanishing partial derivatives of \(g\) are:  
\[\hat{b}= v_{3}\sum_{i=1}^{4}\dfrac{\omega_{i}\partial^{2}g_{3}}{\partial x_{i }\partial r^{*}}\ =\dfrac{2v_{3}\omega_{3}}{(K_{1}+1)(K_{2}+1)}>0.\]  
Thus, we have established Theorem 4 in view of [48]. The proof is completed.