Study On Periodicity And Permanence Of Impulsive Delayed Systems

In the real world,there exit many phenomena in nature with the characteristics of sudden change of state at some time,which can be described by impulsive system.Impulsive differential equation theory provides a more accurate and natural framework for mathematical modeling in many real world phenomena,such as population dynamic system,neural network model,infectious disease model and so on.At the same time,in real life,the time delays are ubiquitous.In recent years,many important achievements have been made in the study of impulsive delay differential equations.The study of periodic solution and almost periodic solution of differential dynamical system is an important branch of differential dynamical system theory,which is widely used in biological model and neural network.Thus,this paper is concerned with the existence of almost periodic solutions for impulsive time-delay systems,pulse vaccination control of an SIR epidemic model with distributed delays and dynamical analysis for the hybrid network model of delayed predator-prey Gompertz systems with impulsive diffusion between two patches.The main contents are given as follows:In the first part,the exponential dichotomy of linear differential equations firstly is extended to the exponential dichotomy of impulsive systems.Then,the existence of almost periodic solutions of impulsive systems is proved by using the exponential dichotomy of impulsive systems.Finally,the existence of almost periodic solutions of impulsive delay differential equations is studied by using the fixed point theorem of contractive mapping and differential inequality techniques.In the second part,firstly,a new SIR epidemic model with impulsive vaccination and distributed delay is established,which is described by impulsive delay equation,then,using the impulsive comparison theory and analysis technique,we obtain the sufficient conditions on existence and global asymptotic stability of disease-free periodic solution.Furthermore,we obtain sufficient conditions for the permanence of the system by using the properties of linear distributed delay differential equations,the method of constructing function and the method of disprovement.In the third part,we consider a hybrid network model of delayed predator-prey Gompertz systems with impulsive diffusion between two patches.Using the discrete dynamical system determined by the stroboscopic map which has a globally stable positive fixed point,we obtain the global attractive condition of predator-extinction periodic solution for the network system.Furthermore,by employing the theory of delay functional and impulsive differential equation,we obtain sufficient condition with time delay for the permanence of the network.