Stability Analysis Of Several Dynamic Models

In this paper,we mainly study several classes of dynamic models.The main content is as follows.One is to investigate a class of SIRS epidemic model.Firstly,we modified a classical SIR model by taking into account of temporary immunity and the transfer from infectious class to the susceptible class.One feature of the model is that the nonlinear incidence rate is Sf(I),which makes the system more general.Secondly,the next generation matrix is used to obtain the basic reproduction number of the model and the existence of the equilibrium of the model.Thirdly,the global asymptotical stabilities of the equilibria are given by constructing Lyapunov function and using LaSalle's Invariance Principle.Finally,numerical simulations are carried out to verify the relevant conclusion of this chapter.Besides,the threshold dynamics of the model is completely determined by the basic reproduction number Ro.The obtained results show that the basic reproduction number R0 is the only indicator for the outbreak of the disease.Any efficient control measure for infection spread should aim to decrease the value of R0.The next one is to analyze a class of prey-predator model with Holling's type II functional response.Firstly,considering the carrying capacity of predator depending on the amount of prey,a prey-predator model with Holling's type II functional response is established and its global behavior is analyzed.Secondly,system may have positive equilibriuums of four types,the locally asymptotic stability of the equilibria of the model is proved by using the method of characteristic value.Thirdly,the different relations between the carrying capacity of predator and the number of prey was mumerically simulated,which show the complexity of global behavior of the model.Finally,the relations between vegetation and plateau pika in alpine meadow ecosystem can be described by this model.According to the actual data,the cause of alpine meadow degrading and recovery strategy of degraded alpine meadow are obtained in Application.The third one is to consider the relationship between the alpine meadow and effective hole and abandoned hole of plateau pika.Firstly,the plateau pika in alpine meadow feed on vegetation.At the same time,the soil that were dug by plateau pika formed the mound.However,the hole and mound of plateau pika lead to the decrease of carrying capacity of vegetation.The dynamic model is formulated to analyze the relationship between the alpine meadow,effective hole and abandoned hole.Secondly,the existence of the equilibrium of the model is obtained and the locally asjymptotic stability of the trivial and prey-only equilibrium are proved by using the method of characteristic value.In addition,we proved the locally asynptotic stability of positive equilibrium with the help of the Routh-Hurwitz criterion.Finally,numerical simulations were carried out to illustrate the main theoretical results.