On The Dynamical Behaviors In Discrete-time Population Models With Feedback Controls And Delays

In recent years,a lot of experiment show that it may realie the permanence of thespecies by feedback controls.These will have very important theoretical and practicalmeaning to keep the population diversity and to sustainable development of ecosys-tem.In this paper,we will investigate the permanence and extinction of the general nonau-tonomous discrete N-species competitive system with feedback controls,and the perma-nence of discrete n-species food chain model with feedback controls.The main contents inthis paper can be summarized as follows:The first section is introduction.We first introduce the dynamical behaviors in discrete-time model of species research background,purpose and significance.The second,we intro-duce discrete-time model of species with feedback control present research results.Finally,wewill give this paper research contents and the organizational structure.The second section is preliminaries,in which we introduced some hypotheses andlemmas to prove the problem,and obtained the better results.In section 3, we study extinction in nonautonomous discrete Lotka-Volterra competi-tive system with pure-delays and feedback controls. A new sufficient conditions for whicha part of the N-species is driven to extinction are established by using the method of mul-tiple discrete Lyapunov functionals,there is no question of such research findings.Finally,some numerical examples and simulations are presented to show the feasibility and effec-tiveness of the proposed methods.On the base of section 3,in section 4,we study permanence in nonautonomous discreteLotka-Volterra N-species competitive systems with pure-delays and feedback controls.Newsu?cient conditions for which the surplus species remains permanent are established byusing the method of multiple discrete Liapunov functionals,there is no question of suchresearch findings.Finally, some numerical examples and simulations are presented to showthe feasibility and effectiveness of the proposed methods. In Section 5,we study permanence for a discrete time Lotka-Volterra type food-chainmodel with feedback controls and delays.By applying the comparison theorem of differenceequation,sufficient conditions are obtained of the permanence of the system.Finally,somenumerical simulations are presented to verify the obtained results.