# Global Dynamics of Modeling Floculation of Microorganism
## 1 Introduction  
Photosynthetic bacteria, which are common microorganisms in the natural environment, have been applied in the field of environmental protection, such as in the treatment of sewage, domestic wastewater and the bioremediation of sediment mud polluted with organic matter (see, for example, [1; 2; 3; 4]). On the other hand, photosynthetic bacteria can produce relatively large amounts of physiologically-active substances, such as vitamin \(B_{12}\),ubiquone (coenzyme \(Q10\) ), 5-aminolevulinic acid (ALA) and RNA (see, for example [5]). In particular, vitamin \(B_{12}\) has been used in treating anemia and as an eye lotion. Recently, applications as health food supplements have received considerable attention. Coenzyme \(Q10\) has been used in treating heart diseases for many years. Further, coenzyme \(Q10\) has been used not only as a medicine, but also as some food supplements, because of its physiological activities. One of the developments of ALA applications is in the area of photodynamic diagnosis. RNA is an attractive source of 5\({}^{\prime}\)-ribonucleotides for use as a flavor enhancer in the food industry. In recent years, the production of RNA has been used as a dietary source of pyrimidine for human immune functions (see, for example [6; 7]). Some photosynthetic bacteria, such as _Rhodopseudomonas__palustris_, are extensively used in the production of lycopene, aquaculture, and so on [8]. It can use sunlight, inorganic and organic compounds for energy. Further, _Rhodopseudomonas__palustris_ can have practical value for removing microcystin from the water body during algal blooms [9]. It can also degrade 2,4,6-trinitrotoluene (TNT), which has negative effects on the human body and aquatic life, resulting ina major threat to drinking and irrigation water supplies, as well as the recreational use of surface waters worldwide. Moreover, _Rhodopseudomonas palustris_ is regarded as the most promising microbial system for the biological production of hydrogen, which has been extensively developed because of its high-energy content and clean product after combustion [10]. However, the concentration of _Rhodopseudomonas palustris_ is very low under anaerobic light culture conditions (see, for example, [11; 12]). Therefore, cost-efficient harvesting of _Rhodopseudomonas palustris_ is a new challenge. In order to harvest _Rhodopseudomonas palustris_ from the liquid, it is necessary to flocculate the single cells into large cell aggregates. Flocculation is a chemically-based separation process that requires less energy than centrifugation and ultrafiltration and, thus, is regarded as the most promising means for degrading microorganisms. Since algal toxins of blooms have happened occasionally in recent years, the problems of degrading microorganisms have received wide attentions (see, for example, [13; 14; 15; 16]). Flocculants are a kind of important water treatment reagent, which can be divided into organic flocculants and inorganic flocculants according to the chemical compositions [17]. Although organic flocculants, such as polyacrylamide, are frequently used in wastewater treatment and industrial downstream processes because of their high efficiency, some of them are not easily degraded in nature [18; 19], and some of the monomers derived from synthetic polymers are harmful to the human body (see, for example, [20; 21]). To solve these environmental problems, inorganic flocculants are increasingly being seen as an alternative in the settlement of microorganisms, more specifically in wastewater treatment owing to their inexpensive and nontoxic characteristics. Thus, inorganic flocculants may be used as nontoxic, cost-effective and widely-available flocculants for harvesting _Rhodopseudomonas palustris_ (see, for example, [22; 23; 24; 25; 26; 27] and the references therein). Mathematical models have played an important role in better understanding microbiology and population biology (see, for example, [28; 29]).