# Global Dynamics of Modeling Floculation of Microorganism
## 1 Introduction
Motivated by the papers mentioned above, in this paper, we further consider a dynamic model describing the cultivation and flocculation of _Rhodopseudomonas palustris_, and the nutrients presented in [40] will be divided into carbon source and nitrogen source, which are perfectly complementary in the culture of _Rhodopseudomonas palustris_ (see, for example, [41; 42; 43; 5; 44]). We assume that the growth of _Rhodopseudomonas palustris_ is always co-limited by carbon and nitrogen for all possible nutrient conditions. We do not consider the case where only one of these nutrients limits growth, for example, in environmental scenarios of high carbon, but very low nitrogen loads, the growth of bacteria may be purely nitrogen limited. Let \(C(t)\) and \(N(t)\) denote the concentration of carbon source and nitrogen source, respectively, in the culture vessel at time \(t\). The constants \(C^{0}>0\) and \(N^{0}>0\) denote the input concentration of carbon source and nitrogen source, respectively. We assume that the conversion of nutrients to microorganism biomass occurs instantly. That is, the time delay \(\tau\) in Model (1) equals zero. Hence, we have the following dynamic model describing the cultivation and flocculation of a microorganism:  
\[\left\{ \begin{array}{l}\dfrac{dC(t)}{dt}=(C^{0}-C(t))D-\dfrac{X(t)}{\delta_ {1}}r_{1}\mu_{1}(C(t))\mu_{2}(N(t)),\\ \dfrac{dN(t)}{dt}=(N^{0}-N(t))D-\dfrac{X(t)}{\delta_{2}}r_{2}\mu_{1}(C(t))\mu_{ 2}(N(t)),\\ \dfrac{dX(t)}{dt}=r\mu_{1}(C(t))\mu_{2}(N(t))X(t)-DX(t)-m_{1}X(t)P(t),\\ \dfrac{dP(t)}{dt}=(P^{0}-P(t))D-m_{2}X(t)P(t).\end{array} \right. \tag{2}\]In Model (2), the parameters \(D,\ r,\ m_{1},\ m_{2}\) and \(P^{0}\) are the same as Model (1). The term \(r\mu_{1}(C(t))\mu_{2}(N(t))\) is the growth rate of _Rhodopseudomonas palustris_, and the terms \(r_{1}\mu_{1}(C(t))\mu_{2}(N(t))\) and \(r_{2}\mu_{1}(C(t))\mu_{2}(N(t))\) represent the quantity of the decreasing of the carbon source and nitrogen source, respectively, where \(r_{1}\) and \(r_{2}\) are non-negative constants; the functions \(\mu_{1}(C(t))\) and \(\mu_{2}(N(t))\) are nonnegative and continuous for \(C(t)\geq 0\), \(N(t)\geq 0\). For the simplicity of the theoretical analysis, in this paper, the functions \(\mu_{1}(C(t))\) and \(\mu_{2}(N(t))\) are chosen as Monod-type functions, i.e.,  
\[\mu_{1}(C(t))=\frac{C(t)}{K_{1}+C(t)},\ \ \mu_{2}(N(t))=\frac{N(t)}{K_{2}+N(t)},\]  
where \(K_{1}>0\) and \(K_{2}>0\) are the half-saturation constants with respect to the carbon source and nitrogen source, respectively. \(\delta_{i}\ (i=1,2)\) are yield coefficients (see, for example, [29]), which are defined as:  
\[\delta_{i}=\frac{\mbox{\it mass of }\mbox{\it organism }\mbox{\it formed}}{\mbox{ \it mass of substrate consumed}},\ (i=1,2).\]  
Therefore, the dynamic Model (2) can be rewritten in the following form:  
\[\left\{ \begin{array}{l}\frac{dC(t)}{dt}=(C^{0}-C(t))D-\frac{r_{1}C(t)N(t)X(t )}{\delta_{1}(K_{1}+C(t))(K_{2}+N(t))},\\ \frac{dN(t)}{dt}=(N^{0}-N(t))D-\frac{r_{2}C(t)N(t)X(t)}{\delta_{2}(K_{1}+C(t)) (K_{2}+N(t))},\\ \frac{dX(t)}{dt}=\frac{rC(t)N(t)X(t)}{(K_{1}+C(t))(K_{2}+N(t))}-DX(t)-m_{1}X( t)P(t),\\ \frac{dP(t)}{dt}=(P^{0}-P(t))D-m_{2}X(t)P(t).\end{array} \right. \tag{3}\]  
It is convenient to introduce dimensionless variables. In particular, we define:  
\[\overline{C}=\frac{C}{C^{0}},\ \overline{N}=\frac{N}{N^{0}},\ \overline{X}=X,\ \overline{P}=\frac{P}{P^{0}},\ \overline{K}_{1}=\frac{K_{1}}{C^{0}},\ \overline{K}_{2}=\frac{K_{2}}{N^{0}},\ \overline{t}=tD,\]  
\[\overline{r}_{1}=\frac{r_{1}}{\delta_{1}DC^{0}},\ \overline{r}_{2}=\frac{r_{2}}{ \delta_{2}DN^{0}},\ \overline{r}=\frac{r}{D},\ \overline{m}_{1}=\frac{m_{1}P^{0}}{D},\ \overline{m}_{2}=\frac{m_{2}}{D},\]  
and still denote \(\overline{C},\ \overline{N},\ \overline{X},\ \overline{P},\ \overline{K}_{1},\ \overline{K}_{2},\ \overline{t},\overline{r}_{1},\ \overline{r}_{2},\ \overline{r},\ \overline{m}_{1}\) and \(\overline{m}_{2}\) with \(C,\ N,\ X,\ P,K_{1},\ K_{2},\)\(t,r_{1},r_{2},r,\)\(m_{1}\) and \(m_{2}\), then Model (3) becomes:  
\[\left\{