# Global Dynamics of Modeling Floculation of Microorganism
## 2 The Global Existence, Nonnegativity and Boundedness of Solutions  
From the biological considerations, it is necessary to show that all of the solutions of Model (4) are nonnegative and bounded for all \(t\geq 0\). By using the basic theory of ordinary differential equations [45] and some simple calculations, it is not difficult to show the following result. **Theorem 1**. : _The solution \((C(t),N(t),X(t),P(t))\) of Model (4) with the initial Condition (5) is existent, unique and nonnegative for all \(t\geq 0\) and satisfies:_  
\[\limsup_{t\to\infty}C(t)\leq 1,\ \limsup_{t\to\infty}N(t)\leq 1,\ \limsup_{t\to\infty}M(t)\leq\alpha,\] \[\limsup_{t\to\infty}P(t)\leq 1,\ \liminf_{t\to\infty}C(t)\geq\underline{C},\ \liminf_{t\to\infty}N(t)\geq \underline{N},\ \liminf_{t\to\infty}P(t)\geq\underline{P},\]  
_where \(M(t)=\frac{r}{2r_{1}}C(t)+\frac{r}{2r_{2}}N(t)+X(t)\), \(\alpha=\frac{r}{2r_{1}}+\frac{r}{2r_{2}},\underline{C}=\frac{K_{1}(K_{2}+1)} {K_{1}(K_{2}+1)+r_{1}\alpha},\ \underline{N}=\frac{K_{2}(K_{1}+1)}{K_{2}(K_{1}+1)+r_{2} \alpha},\ \underline{P}=\frac{1}{1+m_{2}\alpha}.\)_  
Proof. : From the theory of the local existence of solutions for ordinary differential equations, it can be obtained that the solution \((C(t),N(t),X(t),P(t))\) of Model (4) is existent and unique for \(t\in[0,\delta)\). Here, \(\delta\) is some positive constant [29; 45; 46]. Furthermore, we also have that the solution \((C(t),N(t),X(t),P(t))\) is nonnegative for \(t\in[0,\delta)\). We can easily show that \(C(t)\), \(N(t)\) and \(P(t)\) are bounded on \(t\in[0,\delta)\). Let us further show that \(X(t)\) is also bounded on \(t\in[0,\delta)\). For \(t\geq 0\), define:  
\[M(t)=\frac{r}{2r_{1}}C(t)+\frac{r}{2r_{2}}N(t)+X(t).\]  
From Model (4), we obtain that, for \(t\geq 0\),  
\[\dot{M}(t)\leq\frac{r}{2r_{1}}+\frac{r}{2r_{2}}-M(t). \tag{6}\]  
From (6) and the well-known comparison principle, we have that \(M(t)\) is also bounded for \(t\in[0,\delta)\). Hence, by employing the continuation theorems of the solutions [29; 45; 46], the solution \((C(t),N(t),X(t),P(t))\) is existent and unique for any \(t\geq 0\). Similarly, the solution is nonnegative for any \(t\geq 0\). Thus, from the comparison principle, we have that:  
\[\limsup_{t\to\infty}C(t)\leq 1,\ \limsup_{t\to\infty}N(t)\leq 1,\ \limsup_{t\to\infty}M(t)\leq\frac{r}{2r_{1}}+\frac{r}{2r_{2}}=\alpha,\ \limsup_{t\to\infty}P(t)\leq 1.\]  
By the first equation of Model (4), we obtain that, for \(t\geq 0\),  
\[\dot{C}(t)=1-C(t)-\frac{r_{1}C(t)N(t)X(t)}{(K_{1}+C(t))(K_{2}+N(t))}\geq 1- \Big{(}1+\frac{r_{1}\alpha}{K_{1}(K_{2}+1)}\Big{)}C(t).\]Again, we can conclude that \(\liminf_{l\to\infty}C(t)\geq\underline{C}\). By using the technique similar above, we can show that \(\liminf_{t\to\infty}N(t)\geq\underline{N}\), \(\liminf_{l\to\infty}P(t)\geq\underline{P}\). We complete the proof of Theorem 1. Theorem 2: _The compact set:_  
\[\Omega=\{(C,N,X,P)\in R_{+}^{4}|\ \underline{C}\leq C\leq 1,\ \underline{N}\leq N \leq 1,\ 0\leq M\leq\alpha,\ \underline{P}\leq P\leq 1\}\]  
_attracts all of the solutions of Model (4) and is positively invariant with respect to Model (4)._  
According to Theorem 1, it only needs to be proven that \(\Omega\) is positively invariant with respect to Model (4). That is, it needs to be shown that \(\underline{C}\leq C(t)\leq 1,\ \underline{N}\leq N(t)\leq 1,\ M(t)\leq\alpha,\ \underline{P}\leq P(t)\leq 1\) for any \(t\geq 0\) if \((C(0),N(0),X(0),P(0))\in\Omega\). Let us show \(M(t)\leq\alpha\) for any \(t\geq 0\). In fact, if there exists some \(t_{1}>0\), such that \(M(t_{1})>\alpha\), then \(t_{1}^{*}=\sup\{t|M(t)=\alpha,t\in[0,t_{1}]\}\) is existent and \(t_{1}^{*}\geq 0\). Hence, we obtain that \(M(t_{1}^{*})=\alpha\), \(M(t_{1})>\alpha\) and \(M(t)>\alpha\) for \(t\in(t_{1}^{*},t_{1})\). By the Lagrange mean-value theorem, there exists some \(t_{2}\in(t_{1}^{*},t_{1})\), such that \(\dot{M}(t_{2})=\frac{M(t_{1})-M(t_{1}^{*})}{t_{1}-t_{1}^{*}}>0\).