The Research On Extinction And Persistence Of Several Stochastic Ecosystems

This paper mainly considers the effect of the environmental pollution on a single-species under perturbation,also focuses on the dynamical properties of a couple of stochastic epidemic models.Throughout the quantitative analysis to the stochastic ecosystems,the sufficient conditions of the extinction and the persistence of the solutions are derived,then the change of the density for the ecosystems could be predicted,the better protection strategy and management could be made for the government.This paper consists of four chapters.The research background and the current investigations are introduced for the further use,the known related results are stated herewith.A single-species population model with psychological effect in the polluted environment is proposed and formulated in Chapter 2.The sufficient condition of weakly persistence in the mean for the deterministic model is obtained.For the stochastic single-species model,the existence and uniqueness of the solution is proved after introducing the perturbation,and the weakly persistence in the mean under the expectation and the stochastic permanence are followed.Some numerical simulations definitely support the main results of this chapter.A stochastic susceptible-infected-removed epidemic model with varying population size and standard incidence rate(short for SIR model)is investigated in Chapter 3.Under the uniform transformation,the existence and uniqueness of the positive global solution is derived,and sufficient condition that the density of the infected declines to zero exponentially is investigated,the persistence of the infected and the removed is also considered.Under the moderate intensity conditions,the stochastic epidemic SIR model admits a stationary distribution,and the solution is ergodic.Several examples and their simulations could support the main results of this chapter.Chapter 4 considers a stochastic susceptible-infected-susceptible model with partial vaccination and nonlinear incidence rate(short for SIS model).The stochastic SIS model with perturbed efficient contact rate,by constructing a suitable Lyapunov function,admits the existence and uniqueness ofintensity the global solution with probability one.Together with the generalized Ito's formula,the density of the infected decreases exponentially to zero is observed when the meets some sufficient condition.Further,by Has'minskii ergodic theory and Markov semigroup theory,an important result is derived.That is,the stochastic SIS model admits a stationary distribution and the solution is ergodic.Correspondingly,numerical simulations show the efficiency of the main results.The research demonstrates that the thresholds of the stochastic models are definitely dependent on the intensity of the white noise.When the white noise is small enough,the stochastic model behaves the similar properties of the corresponding deterministic model.When the white noise is large,then the properties of the solution for the deterministic model are destroyed,and the properties of the solution for the stochastic model will depend on the intensity of the white noise.