# Global Dynamics of Modeling Floculation of Microorganism
## 5 Stability and Uniform Persistence
### Uniform Persistence  
As pointed out in [29; 45; 50], uniform persistence is an important concept in the cultivation of microorganisms. From a biological perspective, one basic question about biological models involves the long-time survival of the species. To this end, we may want to know whether the constructed model is uniformly persistent with respect to one or more species. That is, the number of the species will keep positive and bounded away from zero for any positive time. In recent years, this kind of research has been particularly common in microorganisms (see, for example, [45]), especially in the simulation of harmful algae growth, where we usually want to predict if the harmful algae will go towards extinction or if blooms of the harmful algae will remain, which directly impacts human health and food webs in aquatic ecosystems. From a mathematical point of view, uniform persistence can give sufficient criteria for the existence of a positive equilibrium for the dissipative system [51]. Model (4) is said to be uniformly persistent if there are positive constants \(n_{1},\ n_{2},\ n_{3},\ n_{4},\ N_{1},\ N_{2},\ N_{3},\ N_{4}\), such that each positive solution \((C(t),N(t),X(t),P(t))\) of Model (4) satisfies:  
\[n_{1}\leq\liminf_{t\to\infty}C(t) \leq\limsup_{t\to\infty}C(t) \leq N_{1},\] \[n_{2}\leq\liminf_{t\to\infty}N(t) \leq\limsup_{t\to\infty}N(t) \leq N_{2},\] \[n_{3}\leq\liminf_{t\to\infty}X(t) \leq\limsup_{t\to\infty}X(t) \leq N_{3},\] \[n_{4}\leq\liminf_{t\to\infty}P(t) \leq\limsup_{t\to\infty}P(t) \leq N_{4}.\]  
Now, we give a result on the uniform persistence of Model (4). To proceed, we introduce the following notation and terminology. Denote by \(R(t)\ (t\geq 0)\) the family of solution operators corresponding to Model (4). The \(\omega\)-limit set \(\omega(x)\) of \(x\) consists of \(y\), such that there exists a sequence \(t_{n}\to\infty\) as \(n\to\infty\) with \(R(t_{n})x\to y\) as \(n\to\infty\). Define:  
\[G=\{(C,N,X,P)\in R_{+}^{4}\ |\ \underline{C}\leq C\leq 1,\ \underline{N}\leq N \leq 1,\ X\geq 0,\ \underline{P}\leq P\leq 1\},\]  
\[G_{0}=\{(C,N,X,P)\in G\ |\ \underline{C}\leq C\leq 1,\ \underline{N}\leq N \leq 1,\ X>0,\ \underline{P}\leq P\leq 1\},\]  
\[\partial G_{0}=G\setminus G_{0},\]  
\[M_{\partial}=\{(C,N,X,P)\in\partial G_{0}\ |\ R(t)(C,N,X,P)\ \text{satisfies Model (4) and }R(t)(C,N,X,P)\in\ \partial G_{0},\ \forall t\geq 0\},\]  
\[\Omega(M_{\partial})=\cup_{x\in M_{\partial}}\omega(x).\]  
From Theorem 2, we obtain that \(G\) is positively invariant corresponding to \(R(t)\). It is easy to see that \(G_{0}\) is also positively invariant. Clearly, \(\partial G_{0}\) is relatively compact in \(G\). Now, we show that \(\Omega(M_{\partial})=\{(1,1,0,1)\}\). In fact, \(\{(1,1,0,1)\}\subseteq\Omega(M_{\partial})\). For any \((C(0),N(0),X(0),P(0))\in M_{\partial}\), it has that, for all \(t\geq 0\), \(X(t)\equiv 0\) and \(\lim_{t\to+\infty}C(t)=\lim_{t\to+\infty}N(t)=\lim_{t\to+\infty}P(t)=1\). Thus, \(\Omega(M_{\partial})=\{(1,1,0,1)\}\). **Theorem 7**. : _If \(R_{0}>1\), then Model (4) is uniformly persistent._  
Proof. : By Theorem 1, it can be obtained that nonnegative solutions of Model (4) are point dissipative. According to the definitions above, it suffices to show that \(\partial G_{0}\) repels uniformly nonnegative solutions of Model (4). It is obvious that there is only one equilibrium \(E_{0}\) in \(M_{\partial}\). We now show that \(W^{s}(E_{0})\cap G_{0}=\emptyset\). Assume \(W^{s}(E_{0})\cap G_{0}\
eq\emptyset\), then there exists a positive solution of Model (4), such that \(\lim_{t\to\infty}(C(t),N(t),X(t),P(t))=(1,1,0,1)\). Assume:  
\[U(C(t),N(t),P(t))=\frac{rC(t)N(t)}{(K_{1}+C(t))(K_{2}+N(t))}-1-m_{1}P(t),\]  
then:  
\[n_{2}\leq\liminf_{t\to\infty}N(t) \leq\limsup_{t\to\infty}N(t) \leq N_{2},\]0  
Then, \(r>(K_{1}+1)(K_{2}+1)(1+m_{1})\) is equivalent to \(U(1,1,1)>0\).