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For a family of $n$-dimensional periodic delay differential equations whichencompasses a broad set of models used in structured population dynamics, theexistence of a positive periodic solution is obtained under very mildconditions. The proof uses the Schauder fixed point theorem and relies on thepermanence of the system. A general criterion for the existence of a positiveperiodic solution for Nicholson's blowflies periodic systems (with bothdistributed and discrete time-varying delays) is derived as a simpleapplication of our main result, generalizing the few existing resultsconcerning multi-dimensional Nicholson models.  In the case of a Nicholson system with discrete delays all multiples of theperiod, the global attractivity of the positive periodic solution is furtheranalyzed, improving results in recent literature.

Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems