Persistence and global stability in discrete models of pure-delay nonautonomous Lotka–Volterra type ✩

Consider the persistence and the global asymptotic stability of the following discrete model of pure-delay nonautonomous Lotka–Volterra type:    Ni(p + 1) = Ni(p) exp ci(p) − n j=1 m l=0 al ij (p)Nj (p −kl ) , p= 0, 1, 2, . . . , 1 i n, Ni(p) = Ni,p 0, p 0, and Ni,0 > 0, 1 i n, where each ci(p) and al ij (p) are bounded for p 0 and inf p 0 m l=0 al ii(p) > 0, al ij (p) 0, i j n, 1 i n, and kl 0, 1 l m. In this paper, for the above discrete system of pure-delay type, by improving the former work [J.Math. Anal. Appl. 273 (2002) 492–511] which extended the averaged condition offered by S. Ahmad and A.C. Lazer [Nonlinear Anal. 40 (2000) 37–49], we offer conditions of persistence, and considering a Lyapunov-like discrete function to the above discrete system, we establish sufficient conditions of global asymptotic stability.  2004 Elsevier Inc. All rights reserved