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

We will consider the following nonlinear impulsive delaydifferential equation N ′ ( t ) = r ( t ) N ( t ) ( ( K ( t ) − N ( t − m w ) ) / ( K ( t ) + λ ( t ) N ( t − m w ) ) ) , a.e. t > 0 , t ≠ t k , N ( t k + ) = ( 1 + b k ) N ( t k ) , 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 sufficientconditions 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 equation. In particular, our work extends and improves some known results.