Global Dynamics Analysis Of Several Biological Models With Time Delays

With the progress of science and technology,mathematical model is finding wider and wider application in many fields,especially in population ecology,epi-demiology and neural network.Because of time-delay phenomena,the state of system not only depends on the current state,but also depends on the previous states,and delay differential equation model is more accordant with reality than differential equation model.Through using the theory of delay differential equa-tion,fixed point theorem,fluctuation lemma,Lyapunov's functional method and inequality technique,this Ph.D thesis qualitatively studies the dynamic behaviors of population model with delay,infection model with delay,neural networks model with delay,which mainly include the attractivity of equilibrium,the existence and stability of(pseudo)almost periodic solution,the existence and stability of anti-periodic solution and the effects of time delay on the dynamic behavior of multiple group model,and the results obtained in this Ph.D thesis extend and improve some known results.The dissertation is divided into six chapters.The main contents are as follows:In Chapter 1,a review on the research background and status quo of problems to be studied is presented.The research work of this thesis is briefly addressed as well,and the motivations and significance of this work are also described.At the end of this chapter,the basic notations,definitions and preliminary lemma are listed.In Chapter 2,we first study a class of Lasota-Wazewska model with distributed delays and establish new criteria for the existence and global asymptotic stability of positive pseudo almost periodic solutions by using the fixed point method and the properties of pseudo almost periodic function,together with constructing a suitable Lyapunov function.Next,a Nicholson's blowflies model with an oscil-lating death rate and multiple time-varying delays is concerned.By constructing a integral inequality of exponential function,we establish a criterion on the global exponential convergence of the zero equilibrium point for this model,which con-tains the related results of the existing literature on the case of non-oscillating death rate.Finally,under pseudo almost periodic environment,a class of neoclas?sical growth model wit.h time-varying delay is analyzed.Combining the theory of pseudo almost periodic,fixed point theorem and inequality technique,we obtain new criteria for the existence and global exponential stability of positive pseudo almost periodic solutions,which improves and extends some relevant results of the latest literature.Moreover,we also use numerical simulations to demonstrate our theoretical results respectively.In Chapter 3,we investigate the effect of delay on the asymptotic behavior of Nicholson' s blowflies model with patch structure.By using the fluctuation lemma and some differential inequality technique,delay-dependent criteria for the global attractivity of the addressed system are obtained.Our results relax the relevant restrictions in the existing literature and further reveal that time delay can affect the biological characteristics of the equilibrium state of the population model.In the last section of this chapter,some numerical examples are given to illustrate the feasibility of the theoretical results.In Chapter 4,by constructing an invariant set and using Lyapunov functional and inequality technique we first establish a criterion on exponential stability of positive almost periodic solutions for a non-autonomous SIS epidemic model with time-delay.Next,new criteria for the global asymptotic stability and global exponential stability of a delayed HIV infection model with a nonlinear incidence rate are established respectively.It is especially rewarding to note that the required conditions for the global exponential stability axe simple and easy to verify,and our estimation of the convergence rate of the equilibrium point is completely new.Finally,we present some examples with simulations to support the theoretical results,respectively.In Chapter 5,by constructing new differential inequality technique,we estab-lish some sufficient conditions for the existence and exponential stability of pseudo almost periodic solutions for a multidirectional associative memory neural network with oscillating leakage coefficients and distributed delays,which extend and im prove the corresponding results in some existing literatures.At the same time,combining with the definition of anti-periodic functions,we construct a suitable nonlinear operator and obtain completely new criteria on the existence and global exponential st ability of anti-periodic solutions of a class of shunting inhibitory cel lular neural networks model with oscillating coefficients in leakage terms,and some examples with simulations are also given to verify its rationality.Chapter 6 gives a summary of the thesis work and prospects to the further research fields.