On the permanence in non-autonomous Lotka–Volterra competitive system with pure-delays and feedback controls

Abstract

This paper,which is a continuation of [H. Hu, Z. Teng, H. Jiang, On the extinction in nonautonomous Lotka–Volterra competitive system with pure-delays and feedback controls, Appl. Math. Comput. (2007) (submitted for publication)], considers general n -species non-autonomous Lotka–Volterra competitive systems with pure-delays and feedback controls. New sufficient conditions for which a part of the n species remains permanent, are established by applying the method of multiple Lyapunov functionals and introducing a new analysis technique. In contrast to previous works (see, for example, references [F. Chen, The permanence and global attractivity of Lotka–Volterra competition system with feedback controls, Nonlinear Anal. RWA 7 (2006) 133–143; F. Chen, Global stability of a single species model with feedback control and distributed time delay, Appl. Math. Comput. 178 (2006) 474–479; F. Chen, Permamence in nonautonomous multi-species predator–prey system with feedback controls, Appl. Math. Comput. 173 (2006) 694–709; F. Chen, Z. Li, Y. Huang, Note on the permamence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA 8 (2007) 680–687]), we introduce a new research method and obtain feedback controls that can not influence the permanence of species.