On the global attractivity for a logistic equation with piecewise constant arguments

In this paper, we consider the following logistic equation with piecewise constant arguments: dN(t) dt = rN(t){1− m j=0 ajN([t − j])}, t 0, m 1, N(0) = N0 > 0, N(−j) = N−j 0, j= 1, 2, . . .,m, where r > 0, a0,a1, . . . , am 0, m j=0 aj > 0, and [x] means the maximal integer not greater than x. The sequence {Nn}∞n=0, where Nn = N(n), n = 0, 1, 2, . . . , satisfies the difference equation Nn+1 = Nn exp r 1 − m j=0 ajNn−j , n= 0, 1, 2, . . . . Under the condition that the first term a0 dominates the other m coefficients ai, 1 i m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N∗ = 1/( m j=0 aj ).  2004 Elsevier Inc. All rights reserved