Dynamic Models Of Degradation Of Microorganism And Theory Analysis

In this thesis,the dynamic modeling of microbial degradation is studied,and by analyzing the dynamic properties(such as the stability of the equilibrium,the permanence of the system,the existence of Hopf branches and periodic solutions(periodic oscillations)),the interaction among nutrients,microorganisms,floccu-lants/degrading enzymes is studied,thus some feasible theoretical references for the problem of microbial degradation are provided.Several main theories and re-search methods on nonlinear ordinary differential dynamic systems and time de-lay/stochastic differential equations,such as Lyapunov stability theory.Lyapunov-LaSalle invariance principle,permanence theory,Hopf bifurcation theory,center manifold theorem and normal form method,coincidence degree theory,strong law of numbers and Ito formula,have been applied in the dissertation.The main innovations of this thesis are as follows:1.Based on the actual problem of degradation of harmful microorganisms in the ecological environment treatment,a new class of nonlinear ordinary differential equation dynamic model of microorganisms and their metabolites with the char-acteristics of degrading harmful microorganisms is constructed and the sufficient conditions for the global stability of equilibria and the estimation of the attractive region are given.2.New sufficient conditions are given for the global dynamics of the equilibrium of a class of nonlinear ordinary differential equation dynamics model describing the biodegradation of microcystins.It is found that the change of parameters of the dynamic model can cause Hopf bifurcation.At the same time,the related work is further extended to a more general nonlinear differential equation dynamic model with time delay.3.Generally,the growth and degradation process of microorganisms are closely related to the change of time.Based on the research work in innovation point 2,for a more general non-autonomous nonlinear delay differential equation dynamic model describing the biodegradation of microcystins,suffcient conditions of the global asymptotics,the existence and attraction of periodic solutions(periodic oscillations)are given.4.Considering the influence of environmental noise in the process of microbial growth and degradation,a kind of nonlinear stochastic differential equation dynamic model describing the biodegradation of microcystins is further constructed,and the conclusions of the persistence and periodic solution(periodic oscillation)of the dynamic model are obtained.The specific research contents of this dissertation are as follows:In chapter 3,considering that the metabolites of some microorganisms have the important characteristics of degrading harmful microorganisms in sewage,a kind of nonlinear ordinary differential equation dynamic model is proposed to de-scribe the degradation characteristics of microorganisms and their metabolites.By constructing appropriate Lyapunov function and using the classical Lyapunov sec-ond method,the principle of Lyapunov LaSalle invariance in the theory of motion stability of ordinary differential equations,the global asymptotic stability of the e-quilibrium state of the model is proved.At the same time,the domain of attraction estimation of the boundary equilibrium state of harmless microorganisms is studied,and the control strategy of microbial degradation process is analyzed.In Chapter 4,by constructing appropriate Lyapunov function,the global sta-bility of equilibria of a kind of nonlinear ordinary differential equation dynamic model describing the biodegradation of microcystins is studied,and new sufficient conditions are given.Furthermore,it is found that the dynamic model has more complex dynamic behavior:the change of system parameters can also causes Hopf bifurcation.At the same time,the persistence of the dynamic model is discussed completely,and the exact analytical expression of the lower limit of solution of the model is given.In Chapter 5,based on the nonlinear ordinary differential equation dynam-ic model of biodegradation of microcystins in Chapter 4,considering some actual factors such as time delay in the process of microbial growth and biomass transfor-mation,a more general nonlinear delay differential equation dynamic model with coefficient dependent time delay is constructed.By constructing appropriate Lya-punov functional,analyzing the null distribution of transcendental function and us-ing the normal method and central manifold theorem of delay differential equation theory,the global stability of boundary equilibrium,the local stability of positive equilibrium,the existence of periodic solutions(including stability and direction)of Hopf bifurcation and the persistence of the dynamic model are studied.In Chapter 6,considering that the growth and degradation of microorganisms are closely related to the change of time,the research work in Chapter 5 is further extended to a more general non-autonomous nonlinear delay differential equation dynamic model describing the biodegradation of microcystins.By analyzing the asymptotic behavior of the solution of the dynamic model and constructing appro-priate Lyapunov function,the persistence and extinction of microorganisms and metabolites described in the dynamic model are studied.Meanwhile,by construct-ing appropriate function space and corresponding mapping operators,the existence and global attractiveness of periodic solutions(periodic oscillations)are studied when the dynamic model is periodic system by employing the famous coincidence degree theory.In Chapter 7,considering the influence of environmental noise in the process of microbial growth and degradation,the main research work in Chapter 3 is further extended to a kind of stochastic differential equation dynamic model to describe the biodegradation of microcystins.The existence of global positive solution of the stochastic dynamic model,persistence and the asymptotic behavior of the solution near the equilibrium state(without random disturbance)and the existence of pe-riodic solution(periodic oscillation)are discussed by using the relevant theories of the stability of stochastic differential equation.