# Global Dynamics of Modeling Floculation of Microorganism
## 1 Introduction
In recent years, the dynamics of the chemostat models has received considerable attentions (see, for example, [29]). The article of [PERSON] and [PERSON] has played an important role in the development of the chemostat models [30]. From then on, much research on the chemostat models has been extensively studied by many authors. A model describing two populations of microorganisms competing for one single limiting nutrients was proposed in [31]. Later, the model was extended to an arbitrary number of populations in [32; 33]. These models, which were studied in articles [31; 32; 33], have proven that they all include a competitive exclusion effect. In the articles [34; 35; 36; 37; 38; 39], some further developments have been performed on the chemostat models to place the relevant models in a naturally more sensible manner. Let \(S(t)\) and \(X(t)\) denote the concentration of the nutrients and _Rhodopseudomonas palustris_, respectively, in the culture vessel at time \(t\). \(P(t)\) denotes the concentration of inorganic flocculants, which are used for harvesting _Rhodopseudomonas palustris_ (see Figure 1). The constants \(S^{0}>0\) and \(P^{0}>0\) denote the input concentration of the nutrients and flocculants, respectively. For simplicity, it is assumed that the input of the nutrients and flocculants is continuous. The constant \(D>0\) is the dilution rate of the chemostat. Figure 1: The device for collecting _Rhodopseudomonas palustris_ in the chemostat by inputting inorganic flocculants. The constant \(\tau\geq 0\) denotes the time delay involved in the conversion of nutrients to _Rhodopseudomonas palustris_. The flocculation rate of microorganisms is assumed to be a bilinear mass-action function response \(m_{1}X(t)P(t)\), where \(m_{1}\geq 0\) is the per capita contact rate. At the same time, flocculants produce loss or consumption [17], and the loss rate of flocculants is also assumed to be a bilinear mass-action function response \(m_{2}X(t)P(t)\), where \(m_{2}\geq 0\) is constant. Thus, in [40], the following dynamic model has been proposed:  
\[\left\{ \begin{array}{l}\dfrac{dS(t)}{dt}=(S^{0}-S(t))D-r_{1}S(t)X(t),\\ \dfrac{dX(t)}{dt}=rS(t-\tau)X(t-\tau)-DX(t)-m_{1}X(t)P(t),\\ \dfrac{dP(t)}{dt}=(P^{0}-P(t))D-m_{2}X(t)P(t).\end{array} \right. \tag{1}\]  
In Model (1), \(r\geq 0\) and \(r_{1}\geq 0\) are constants, and the bilinear mass-action uptake function \(S(t)X(t)\) has been used. It should be mentioned here that, the analysis reveals that Model (1) proposed in [40] exhibits the phenomenon of backward bifurcation for the existence of positive equilibria. Moreover, the local stability properties of the equilibria have been dealt with in detail. People found that the influence of different nutrients has played an important role in the culture of microorganisms. In order to take this into consideration, appropriate combinations of nutrients are considered in chemostat models. Models with two competitors and two perfectly-complemented growth-limiting nutrients are studied in [41; 42]. Local asymptotic conditions for the equilibria are derived. When there is a microorganism to compete for two or more resources, it may become necessary to consider how the resources, once consumed, interact to promote growth. In [41], the authors employ consumer needs to provide a criterion to classify resources and classify resources as perfectly complementary, perfectly substitutable or imperfectly substitutable. Perfectly-complementary resources are different essential substances that must be taken together. In this case, each substance fulfills different functions with respect to the growth of microorganisms. For example, carbon source and nitrogen source may be complementary for the growth of bacterium.