On The Dynamical Behaviors In Nonautonomous Lotka-Volterra Competitive System With Pure-Delays And Feedback Controls

Dynamical behaviors of mathematical ecological system is an important conceptionwith rich connotation, which mainly includes extinction, permanence, local or global at-tractive, periodicity and oscillation and so on. In the population dynamics, people maymake full use of nature and remold nature by means of study of these properties. Thesewill have very important theoretical and practical meaning to protect and save valuableand rare species which is on the verge of becoming extinction, to keep the populationdiversity and sustainable development of ecosystem. In the paper, we investigate thedynamical behaviors, which include extinction, permanence and global attractive, in gen-eral n-species nonautonomous Lotka-Volterra competitive system with pure-delays andfeedback controls.By improve and extend the methods in [1] and [2], firstly, the boundedness of thesystem is discussed using the di?erential inequalities method and comparison theorem.Secondly, based on the boundedness of the system, new su?cient conditions of which apart species xr+1,xr+2,···,xn of the n species is driven to extinction are established byusing the method of multiple Lyapunov functionals, analysis technique and induction.Then, new su?cient conditions of which the rest part x1,x2,···,xr of the n species isdriven to permanence are established by constructing an auxiliary function and math-ematical analysis. Finally, based on the above conclusions, new su?cient conditions ofwhich all solution of the system is global attractive are established by constructing Li-apunov functionals and using di?erential inequality method, the theory of stability andBarbalat's lemma. And some examples to illustrate the conclusions obtained in this paperare given as the application of theorems.