Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics

In this paper we shall consider the following nonlinear impulsive delay differential equation x′(t)+αV(t)x(t)xn(t−mω)θn+xn(t−mω)=λ(t), a.e. t>0, t≠tk,x(tk+)=1(1+bk) x(tk), k=1,2,…,where m and n are positive integers, V(t) and λ(t) are positive periodic continuous functions with period ω>0. In the nondelay case (m=0), we show that the above equation has a unique positive periodic solution x∗(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x∗(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation shown above preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.