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

We will consider the following nonlinear impulsive delay differential equationN′(t)=r(t)N(t)((K(t)−N(t−mw))/(K(t)+λ(t)N(t−mw))), a.e.t>0,t≠tk,N(tk+)=(1+bk)N(tk),K=1,2,…, wheremis a positive integer,r(t),K(t),λ(t)are positive periodic functions of periodicω. In the nondelay case(m=0), we show that the above equation has a unique positive periodic solutionN*(t)which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity ofN*(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.