Qualitative Analysis For An Annular Model In An Unstirred Chemostat

Chemostat is a kind of industrial reactor.It is not only used in the chemical engineering,but also used in microbial continuous culture,waste treatment,biology pharmacy and food processing etc.Chemostat model is a widely applied mathematical models.It uses differential equation(s) to describe the variety laws of the concentration of microorganisms and post the relation between nutrient and microorganisms by study the characters of differential equation(s) during continuous culturing microorganisms. Now,using Chemostat to continuous culture microorganisms has been an important research instrument in study microorganisms,and a great intermedium between principles and applications.Chemostat model has been widely applied to the study of the increase in different populations of microorganisms and their interactive law.In addition,it also has been applied to the prediction and management of the marine ecosystem,and the control of the environmental pollution.A lot of Chemostat models are studied,but most of them focus on two populations competition model and the single food chain model,And lately start to study multiple food chain(food nets) models.Actually these models can not describe nature phenomena entirely.In the nature,the food chains relation between populations are greatly complex,and one kind just is annular food chain in which the relations between microorganisms exist predator-prey relation and also exist competition relation.In this paper,an annular model with Beddington-DeAngelis functional response in an unstirred Chemostat is discussed.In chapter one,the historic background and current situation of Chemostat model are introduced.In chapter two,existence and stability of coexistence solution of the annular model in an unstirred Chemostat is discussed.The significant conditions of existence for coexistence solution and the correspongding parameter regions for this system are established,and local stability for the coexistence solutions are obtained,and take some numerical simulation to illustrate the existence of the coexi- stence solutions.In chapter three,the asymptotic behavior of solutions to the systems is discussed,and get the conditions of both surviving populations.In the last,we take some numerical simulation to illustrate the asymptotic behavior of solutions in this section.