Stochastic Dynamics Analysis Of Mutualism Model And Rumor Propagation Model

In this thesis,we discuss the stochastic dynamical behavior of the mutualism model and the rumor propagation model,respectively.Firstly,we discuss the permanence and extinction of species to the stochastic regime-switching mutualism model.Secondly,we study the asymptotic stability of the solution to the stochastic rumor propagation model.The main content of the thesis is divided into the following two aspects:The first part is the stochastic regime-switching mutualism model.According to considering the factors of the environment switching and the noise perturbation,we establish a stochastic regime-switching mutualism model which is more consistent with the ecological environment than the one without regime-switching.Firstly,utilizing the basic knowledge of Markov chain and ergodicity,the unique stationary distribution of stochastic switching process is obtained.Secondly,we construct two appropriate auxiliary processes and we can judge their recurrence and persistence.Therefore,the recurrence and persistence of stochastic switching process are decided.Finally,the sufficient conditions of the permanence and extinction of species are derived by using It?o's formula.Moreover,compared with the system without regime-switching,the advantages of the stochastic regime-switching mutualism model are given.The second part is the stochastic rumor propagation model.Firstly,the asymptotic stability of the solution to the stochastic rumor propagation model is discussed by using the integrable Markov semigroup and by constructing a Khasminski?? function.Then,constructing an appropriate auxiliary stochastic differential equation and applying the ergodicity,we obtain that the distribution of the solution converges to a measure.