Abstract :
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].
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