Dynamic Analysis Of Some Classes Stochastic Delayed Chemostat Models

In this paper,we mainly study the dynamical behaviors of three classes single-population stochastic delayed chemostat models and a stochastic delayed two-species competition chemostat model.The article includes five chapters.The preface is in chapter 1,we introduce the research background of this article,the main task and some important preliminaries.In Chapter 2,we formulate a stochastic delayed chemostat model with monotone response function,and the nutrient conversion process involves time delay.First,we show that this model has a unique global positive solution.Furthermore,using the classical approach of Lyapunov function analysis and Ito’s formula,we investigate the asymptotic behavior of this solution.We find that the solution of stochastic system will oscillate around the equilibriums of the corresponding deterministic model,moreover,when the time delay is small,microorganism is persistent;when the time delay is large,microorganism will be extinct.Finally,numerical simulations are presented to illustrate the main findings,in addition,we can find by the computer simulation that large noise may lead to microorganism become extinct,although microorganism is persistent in the deterministic systems when the time delay is small.In Chapter 3,a stochastic delay differential equations chemostat model with nonmonotone response function is considered,and the nutrient conversion process involves time delay.First,we verify that there is a unique global positive solution of the stochastic system.Second,using the classical approach of Lyapunov function analysis and Ito’s formula,we find that the solutions of stochastic system will oscillate around the equilibriums of the corresponding deterministic model,moreover,when the time delay is small,microorganism is persistent;when the time delay is large,microorganism will be extinct.Finally,computer simulations are carried out to illustrate the obtained results and the existence of bistability is observed,in addition,we can find by the computer simulation that large noise may lead to microorganism become extinct,although microorganism is persistent in the deterministic systems when the time delay is small.In Chapter 4,a stochastic delayed chemostat model with nutrient storage is proposed and investigated,and delay is considered in the storage process of nutrients.First,we verify that there is a unique global positive solution for this stochastic system.Second,using the classical approach of Lyapunov function analysis,Ito’s formula and strong law of large numbers of Brownian motion,this stochastic delayed chemostat model is discussed in detail.We establish some sufficient conditions for the extinction of the microorganism,furthermore,we prove that the microorganism will become stochastic persistent in the chemostat under some conditions.Finally,the obtained results are illustrated by computer simulations,and simulation results reveal the effects of time delay on the extinction and persistence of the microorganism.In Chapter 5,we consider a stochastic delayed two-species competition chemostat model with Monod growth response function,and the nutrient conversion process involves time delay.First,we verify that there is a unique global positive solution for this stochastic delayed system.Second,we study the asymptotic behavior of the solution of stochastic delay system around the equilibriums of deterministic system,in particular,we discuss the competition exclusion and coexistence of microorganisms x1 and x2.Finally,computer simulations are carried out to illustrate the obtained results,moreover,results show that time delay has critical effects on the extinction and persistence of the microorganisms.