Dynamic Analyses Of Nicholson's Blowflies Systems Involving Time-varying Delays

Due to the functional differential equation(FDE)can fully consider the influence of the historical state of the system on the present situation,the description of the objective world is often more accurate than ordinary differential equation(ODE).It has been widely studied and applied in mechanics,ecology,epidemiology and many other fields.With the improvement of the theory of functional differential equations,the theoretical study of biological population dynamics has also developed rapidly,among them,the research on the dynamics of FDE,which describes the law of population development,has received extensive attention.In particular,the research on dynamics of delayed Nicholson's blowflies system has always been a hot issue.By utilizing Lyapunov functional method,differential inequality technique and dynamic system method,three kinds of Nicholson's blowflies systems with time delay are investigated in this paper.And the conclusions obtained supplement and improve the relevant result of the existing literature.The paper consists of the following four parts:In Chapter 1,the background,development and research significance of the three systems are summarized,and the research content of this paper is briefly stated.In Chapter 2,we investigate the almost periodic stability of a Nicholson's blowflies system with patch structure and nonlinear mortality term.By using Lyapunov functional method and differential inequality technique,we propose some new assertions to guarantee the existence and exponential stability of positive almost periodic solutions of the model,and improve the corresponding results in some literatures.Last but not least,the validity of the results in this chapter is verified by numerical simulation.In Chapter 3,the stability of a scalar Nicholson's blowflies system with distinctive time-varying maturation and self-limitation delays is studied.Firstly,the positive,global existence and uniqueness of the solution of the system are proved by using the differential inequality analysis technique.Secondly,by using the expansion and contraction of the differential inequality and the fluctuation lemma.,we establish two criteria of global exponential stability and global asymptotic stability for the zero equilibrium of the model.The obtained result improve and perfect some existing result and the feasibility of the theoretical result is verified by numerical simulations.In Chapter 4,we consider the stability of a Nicholson's blowflies system with distinctive time-varying maturation and self-limitation delays.Firstly,some omissions in recent references are pointed out.Then,without assuming the uniform positiveness of the death rate and the boundedness of coefficients,we establish three novel criteria to check the global convergence,generalized exponential convergence and asymptotical stability on the zero equilibrium point of the addressed system,respectively.Our theoretical proof corrects the omissions in the existing literatures,and improves and perfects all the results in these literatures.Furthermore,a numerical example is given to verify the feasibility of the theoretical results.Finally,we summarize the research work and look forward to the future research direction.