Research On Dynamics Models Of Stochastic Population And Epidemic Systems

In the real ecosystem,there exists all kinds of disturbances which can greatly influence the dynamics of the biological populations.In this dissertation,considering the effects of seasonal fluctuations and capture,we focus on constructing stochastic non-autonomous?impulsive population models and stochastic epidemic models,studying the dynamics properties,obtaining the control strategies for the coexistence of the population,and providing constructive suggestions for sustainable development and utilization of the ecological reserve by virtue of the theory and methods of stochastic differential equation?impulsive differential equation and functional differential equation.The maim contents and contributions of the dissertation are as follows:1.Based on the theory of non-autonomous differential equations,a non-autonomous model for migratory birds with Leslie-Gower Holling-type II schemes and saturated recovery rate is proposed to discuss the role of migratory birds in disease spreading.Sufficient conditions for the extinction and permanence of diseases are established on the basis of relatively weak assumptions.The global attractiveness of the model is also given through spectral analysis and Lyapunov function.Our results reveal that predation is beneficial to control diseases and enhance permanence in a predator–prey model.The predator may also be an effective biocontrol agent to prevent the prevalence of diseases.2.Considering the effects of environmental noises to the survival of predator and prey populations,a non-autonomous predator-prey model with Crowley-Martin functional response is proposed and analyzed.The existence of a global positive solution and stochastically ultimate boundedness are obtained.Sufficient conditions for extinction,persistence in the mean,and stochastic permanence of the system are established.We also derive conditions to guarantee the global attractiveness and stochastic persistence in probability of the model.Numerical simulations are carried out to investigate the effects of white noise and functional response on the species.3.Considering the effects of generalized nonlinear harvesting for prey and predator populations,stochastic non-autonomous predator–prey models with and without impulses are investigated.For the stochastic system without impulses,we prove that the positive solution exists and is unique.Sufficient conditions that guarantee the extinction and persistence of the population in the mean are achieved.We show the existence of a nontrivial positive periodic solution by constructing appropriate Lyapunov functions and using Khasminskii's theory.Moreover,the global attractiveness and stochastic persistence in probability of the stochastic model are discussed.Results show that the stronger noises and nonlinear harvesting component can significantly influence the dynamics of the system and lead to the extinction of the predator population.Additionally,for the stochastic predator–prey system with impulsive effect,we prove that there exists a positive periodic solution.Results show that if the impulses are sufficiently large,then the predator will eventually tend to exhibit periodicity.4.In consideration of the effects of white noises,two types of epidemic models with saturated incidence rate are proposed.First,a stochastic SIR epidemic model with saturated treatment and incidence rates is proposed.The existence and uniqueness of the global positive solution are achieved.By constructing suitable Lyapunov functions,appropriate conditions are established to prove that the stochastic model has a unique stationary distribution and the ergodicity holds.Moreover,the extinction of the epidemic diseases is given.Second,considering the effects of environment fluctuations and media coverage,a stochastic SIRS epidemic model is discussed.The stochastic endemic dynamics and the stationary distribution of the system are derived.The results reveal that the maximum reduced contact rate,due to media coverage,will accelerate the extinction of infective populations and reduce the risk of epidemic prevalence.