Global Dynamics Analyses On Two Classes Of Delayed Systems

With the progress of science and technology,the theory of functional differential equations has been widely used to characterize practical problems in many fields of natural science and social science,such as mechanics,ecology,control theory,epidemiology and economics.In recent decades,the descriptions of cellular neural network models and population ecology models by delay differential equations have attracted extensive attention.At present,the dynamic behavior of delay differential equations on high-order cellular neural networks with D operator and Nicholson's blowflies equation with multiple different developmental or maturation delays need more attention and urgent research.Therefore,in this thesis,the differential inequality techniques,fluctuation lemma,fixed point theorem and the Lyapunov functional method are applied to study qualitative and stability related the dynamic properties for a class of high-order proportional delayed cellular neural networks with D operator and Nicholson's blowflies equation with multiple different developmental or maturation delays,including the existence and global exponential stability of positive periodic solutions and positive pseudo almost periodic solutions.The full text consists of four chapters.In Chapter 1,the research background,significance and development of the research issues in this thesis are introduced.The main research contents of this paper are addressed as well.In Chapter 2,a class of high-order proportional delayed cellular neural networks with D operator is explored.First of all,we have developed some new techniques and methods to overcome the difficulties brought by neutral operators,and prove the boundedness and global existence of positive solutions.Secondly,by exploiting differential inequality techniques and Lyapunov function method,some testable conditions are obtained to assure the positiveness and global exponential stability of pseudo almost periodic solutions for the proposed models,which improve and complement the corresponding works of some existing literature.In addition,the validity of the theoretical results is verified by some numerical simulations.In Chapter 3,the dynamic behavior of a class of Nicholson's blowflies equation with multiple different developmental or maturation delays is analyzed on the premise of abandoning the restriction condition that the maximum reproductive rate function is greater than or equal to 1 or less than or equal to 1.Above all,by using differential inequality techniques and fluctuation lemma,we obtain the results of positive,global existence and persistence of the solution of the system.Then,we establish a new criterion to ensure the global exponential stability of the positive periodic solutions for the addressed equation by using differential inequality techniques and the Lyapunov functional.The obtained results are completely new because its abandon the limitation of the existing literature on the maximum reproduction rate function.Finally,the correctness of the theoretical results are illustrated by a numerical simulation.In Chapter 4,we summarize the research and look forward to the future research.