Global Dynamics Of Several High-dimensional Biological Systems With Delays

There are a number of practical problems that can be characterized by delay differential equations in many areas of the objective world,such as nuclear physics,ecosystems,epidemiology,economic mathematics,and automatic control systems,etc.In particular,most of the dynamic behaviors of biological population models and neural network models are significantly affected by delay,so it is more and more important to study the delay population models and the delay neural network models.In this dissertation,the basic theory of delay differential equations,inequality techniques,fluctuation lemma and the lyapunov function method are applied to qualitatively study several kinds of delay population models and a class of high-order inertial neural network model.The influence of delay on its dynamic behavior is analyzed,including the attractivity of the equilibrium point,the existence and stability of the almost periodic solution,and the existence and stability of the anti-periodic solution.The conclusion complements and improves the related results of the existed literatures,and is verified by numerical experiments.This paper consists of six chapters.In chapter 1,the historical background and development status of the research problems are summarized,and the work to be carried out is briefly stated.Finally,the basic notations,concepts and lemmas commonly used in this paper are listed.In chapter 2,a class of patch structure Nicholson's blowflies model with multiple pairs of different time-varying delays is explored.Both the global generalized exponential convergence and the local stability on the zero equilibrium point of the addressed system are established by employing some novel inequality techniques.In addition,two numerical examples with simulations are given to illustrate the effectiveness and feasibility of the results.In chapter 3,two classes of patch structure Nicholson's blowflies systems involving nonlinear density-dependentmortality terms and multiple pairs of time-varying delays are studyed.By utilizing differential inequality techniques and the fluctuation lemma,some sufficient conditions for the global asymptotical stability of the model are established under the condition that the maturity delay and the feedback delay are allowed to be different.The effectiveness of the obtained result is illustrated by some numerical simulations.In chapter 4,a delayed Nicholson-type system involving patch structure is investigated.Applying differential inequality techniques and the fluctuation lemma,the global convergence conditions of the positive asymptotically almost periodic solution of the system are established in the asymptotically almost periodic environment.By constructing proper Lyapunov function,the new criteria for the existence and global exponential stability of the system are given in the almost periodic environment.Finally,the effectiveness and feasibility of the obtain results is confirmed by some numerical examples.In chapter 5,a class of high order inertia Hopfield neural network with time-varying delay and anti-periodic environment is studied.By developing an approach based on differential inequality techniques coupled with Lyapunov function method,some assertions are demonstrated to guarantee the existence and global exponential stability of anti-periodic solutions for the addressed networks.Finally,Numerical examples and simulations are given to illustrate these analytical conclusions.In chapter 6,we summarize the work of this paper and prospect the next step.