Dynamic Analysis Of Two Classes Of Species Models

Differential equations and dynamical systems are important mathematical theories for studying the laws of motion and evolution of things and phenomena in nature and society.Mathematical models of differential equations play an important role in describing population dynamic behaviors.In this thesis,based on existing research works,we propose two classes of species models.One is three-species delayed food chain model,the other is epidemic model on complex networks.Some important dynamics such as permanence,extinction and stability of the equilibria are studied by applying comparison theorem in differential equations,mean-field approximation and Kirchhoff's matrix three theorem.The effect of various immunization strategies is also investigated.This thesis consists of three parts.In Chapter 1,the research background,significance and development of issues in this paper are introduced.The main research contents of the thesis are addressed as well.In Chapter 2,we discuss a delayed three-species food chain model with stage structure and time-varying coefficients.It is assumed that both the predator and the top predator may belong to one of two classes:the immatures and the matures.By employing the comparison theorem in differential equations and inequality analytical techniques,some sufficient condi-tions to ensure the permanence and partial extinction of the system are derived respectively.In addition,we analyze the effect of the delays on permanence of the system.Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior.In Chapter 3,we present an SIVRS epidemic model with drug-resistant variation on complex networks.Then the basic reproduction number of the model is derived by analysis the existence of the endemic equilibrium.By using the method of characteristic root analy-sis,Lyapunov functions,Kirchhoff's matrix three theorem and Lasalle Invariance principle,we discuss the stability of the disease-free equilibrium and the endemic equilibrium.We also compare the effect of two immunization strategies,and do sensitivity analysis in terms of model parameters.Moreover,numerical simulations are arranged to demonstrate our theoretical results.Last but not least,our research works and the prospect of our future research are summarized.