Stationary Distribution And Quasi Stationary Distribution Of Several Stochastic Biological Models

As important research fields in biomathematics,the studies of infectious disease dynamics and population ecology have attracted more and more attention.Due to the diversity and complexity of random phenomena,many meaningful problems remain to be solved.This dissertation considers several specific stochastic biological models in infectious disease dynamics and population ecology,and focuses on the corresponding stationary or quasi-stationary distributions to study their long-time dynamic behavior as follows:For the stochastic SIS(Susceptible Infected Susceptible)epidemic model with demographic stochasticity,we study the existence,uniqueness,domain of attraction as well as convergence rate of the quasi-stationary distribution.Specifically,according to the classification of the boundary of the state space in the Feller sense,we show the existence and uniqueness theorem of the quasi-stationary distribution.In particular,for the case that the right boundary is a regular boundary,we prove that the exponential convergence rate of the quasi-stationary distribution equals to the difference between the second eigenvalue and the first eigenvalue of the associated infinitesimal generator.In addition,the assumptions used in the analysis of the hitting probability of the boundaries are weakened compared with(Discrete Contin.Dyn.Syst.Ser.B 20(2015)2859–2884),and the same conclusion is obtained.For the stochastic SEIR(Susceptible Exposed Infectious Recovered)epidemic model with random immunization and nonlinear incidence,we investigate the dynamics of the infectious diseases.Firstly,we prove the existence and uniqueness of the global positive solution of the underlying model by using the stopping time.Secondly,we find that the solution fluctuates around the disease-free equilibrium of the corresponding deterministic model under certain conditions.And,the intensity of the fluctuation is positively correlated with the intensity of the white noise,that is to say,as the intensity of the white noise decreases,the solution of the stochastic model will be close to the disease-free equilibrium of the corresponding deterministic system,which partly reflects the disappearance of infectious diseases.Moreover,we indicate that the system is persistent in the time average sense under certain conditions.To be specific,the solution fluctuates around the endemic equilibrium of the corresponding deterministic model,i.e.,the stochastic model has a unique stationary distribution,which reveals that the disease will prevail.Finally,several numerical tests are reported for verification of the theoretical findings.For the stochastic Holling-III predator-prey model with Markovian switching in an impulsive polluted environment,we discuss the survival analysis of the species.Firstly,we provide the existence and uniqueness result of the positive solution to the underlying model.And then we study the nonpersistence in the mean,weak persistence in the mean and extinction of two species,respectively.The results imply that both increasing the intensity of the white noise and decreasing the value of the impulsive input period can increase the extinction risk of the two species.The distribution of Markov chains also plays an important role in their survival.Furthermore,we establish the sufficient criteria of the existence and uniqueness of the stationary distribution with the aid of Lyapunov functions,which implies the predator and prey can coexist for a long time under certain conditions.Particularly,when the considered model of this thesis reduces to that of(Discrete Contin.Dyn.Syst.Ser.B23(2018)3275–3296),in the analysis of the weak persistence in the mean of predator species,the sufficient conditions proposed in this thesis are more convenient than those given by(Discrete Contin.Dyn.Syst.Ser.B 23(2018)3275–3296).Finally,the theoretical analysis are verified by performing some numerical simulations.