Existence of positive periodic solutions for a generalized Nicholson’s blowflies model

In this paper, by using the Krasnoselskii cone fixed point theorem, we obtain a sufficient condition as well as a necessary condition for the existence of positive periodic solutions of the following generalized Nicholson’s blowflies model: x ′ ( t ) = − δ ( t ) x ( t ) + ∑ i = 1 m p i ( t ) x ( t − τ i ( t ) ) e − q i ( t ) x ( t − τ i ( t ) ) , t ≥ 0 . In the degenerate case, i.e., where the coefficients and delays of the above equation are all constants, a sufficient and necessary condition for the existence of positive periodic solutions is obtained. Our results are completely new, and generalize and improve some results from the literature.