Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholsonʹs blowflies model

In this paper we shall consider the following nonlinear impulsive delay population model:(0.1) x ′ ( t ) = - δ ( t ) x ( t ) + p ( t ) x ( t - m ω ) e - α ( t ) x ( t - m ω ) a.e. t > 0 , t ≠ t k , x ( t k + ) = ( 1 + b k ) x ( t k ) , k = 1 , 2 , … , where m is a positive integer, δ ( t ) , α ( t ) and p ( t ) are positive periodic continuous functions with period ω > 0 . In the nondelay case ( m = 0 ), we show that (0.1) 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 linear periodic impulsive perturbations, the impulsive delay equation (0.1) preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.