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

TL;DR

This paper proves the existence of positive periodic solutions for a broad class of delayed differential equations used in population dynamics, including Nicholson systems, using fixed point theorems and system permanence.

Contribution

It introduces a general criterion for positive periodic solutions in delayed systems, extending previous results to multi-dimensional Nicholson models with distributed and discrete delays.

Findings

Existence of positive periodic solutions under mild conditions

A general criterion for Nicholson's blowflies systems

Global attractivity of solutions in certain Nicholson systems

Abstract

For a family of $n$-dimensional periodic delay differential equations which encompasses a broad set of models used in structured population dynamics, the existence of a positive periodic solution is obtained under very mild conditions. The proof uses the Schauder fixed point theorem and relies on the permanence of the system. A general criterion for the existence of a positive periodic solution for Nicholson's blowflies periodic systems (with both distributed and discrete time-varying delays) is derived as a simple application of our main result, generalizing the few existing results concerning multi-dimensional Nicholson models. In the case of a Nicholson system with discrete delays all multiples of the period, the global attractivity of the positive periodic solution is further analyzed, improving results in recent literature.