Global attractivity of positive periodic solutions for an impulsive delay periodic “food limited” population model

We will consider the following nonlinear impulsive delay differential equation N (t) = r(t)N(t)((K(t)−N(t −mw))/(K(t) + λ(t)N(t −mw))), a.e. t> 0, t = tk, N(t+k) = (1 + bk)N(tk), K = 1,2,..., where m is a positive integer, r(t), K(t), λ(t) are positive periodic functions of periodic ω. In the nondelay case (m = 0), we show that the above equation has a unique positive periodic solution N∗(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of N∗(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation preserves the original periodic property of the nonimpulsive delay equa-tion. In particular, our work extends and improves some known results. Copyright © 2006 Jian Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited. 1