The exponential ordering for non-autonomous delay systems with applications to compartmental Nycholson systems

TL;DR

This paper introduces the exponential ordering for non-autonomous delay systems, providing new insights into their long-term dynamics, persistence, and attractors, with applications to Nicholson systems ensuring unique positive solutions.

Contribution

It develops the exponential ordering framework for non-autonomous delay systems, relaxing conditions for monotonicity and applying it to Nicholson systems to establish existence and attraction of positive solutions.

Findings

Monotone skew-product semiflows under exponential ordering

Conditions for uniform persistence and global attractors

Existence of unique almost periodic positive solutions in Nicholson systems

Abstract

The exponential ordering is exploited in the context of non-auto\-no\-mous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.