Persistence and global stability in discrete models of Lotka–Volterra type

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka–Volterra type: ⎧⎪ ⎪⎨⎪ ⎪⎩ Ni(p +1) = Ni(p) exp ci −aiNi(p)− n j=1 aijNj (p − kij ) , p 0, 1 i n, Ni(p) = Nip 0, p 0, and Ni0 > 0, 1 i n, where each Nip for p 0, each ci , ai and aij are finite and ai > 0, ai +aii > 0, 1 i n, and kij 0, 1 i, j n. Applying the former results [Y. Muroya, Persistence and global stability for discrete models of nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 273 (2002) 492–511] on sufficient conditions for the persistence of nonautonomous discrete Lotka–Volterra systems, we first obtain conditions for the persistence of the above autonomous system, and extending a similar technique to use a nonnegative Lyapunovlike function offered by Y. Saito, T. Hara and W. Ma [Y. Saito, T. Hara, W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka–Volterra system with two delays, J. Math. Anal. Appl. 236 (1999) 534–556] for n = 2 to the above systemfor n 2, we establish new conditions for global asymptotic stability of the positive equilibrium. In some special cases that kij = kjj, 1 i, j n, and n j=1 ajiajk = 0, i = k, these conditions become ai > n j=1 a2 ji, 1 i n, and improve the well-known stability conditions ai > n j=1 |aji |, 1 i n, obtained by K. Gopalsamy [K. Gopalsamy, Global asymptotic stability in Volterra’s population systems, J. Math. Biol. 19 (1984) 157–168]. © 2006 Elsevier Inc. All rights reserved