Global attractivity of positive periodic solution to periodic Lotka–Volterra competition systems with pure delay

Abstract

We consider a periodic Lotka–Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems) (*) x ˙ i ( t ) = x i ( t ) [ r i ( t ) − ∑ j = 1 n a i j ( t ) x j ( t − τ i j ( t ) ) ] , i = 1 , 2 , … , n . We establish some 3/2-type criteria for global attractivity of a positive periodic solution of the system, which generalize the well-known Wright's 3/2 criteria for the autonomous delay logistic equation, and thereby, address the open problem proposed by both Kuang [Y. Kuang, Global stability in delayed nonautonomous Lotka–Volterra type systems without saturated equilibria, Differential Integral Equations 9 (1996) 557–567] and Teng [Z. Teng, Nonautonomous Lotka–Volterra systems with delays, J. Differential Equations 179 (2002) 538–561].