Asymptotic behaviour of general nonautonomous Nicholson equations with mixed monotonicities

TL;DR

This paper analyzes the long-term behavior of a broad class of nonautonomous Nicholson equations with delays and mixed monotonic nonlinearities, providing conditions for stability, boundedness, and existence of globally attractive solutions.

Contribution

It introduces new sufficient conditions for permanence and global attractivity in general nonautonomous Nicholson equations with multiple delays and mixed monotonicities.

Findings

All positive solutions are bounded and persistent under certain conditions.

A criterion for the existence of a globally attractive positive solution in periodic cases.

The results improve existing literature with sharper criteria and broader applicability.

Abstract

A general nonautonomous Nicholson equation with multiple pairs of delays in {\it mixed monotone} nonlinear terms is studied. Sufficient conditions for permanence are given, with explicit lower and upper uniform bounds for all positive solutions. Imposing an additional condition on the size of some of the delays, and by using an adequate difference equation of the form $x_{n+1}=h(x_n)$, we show that all positive solutions are globally attractive. In the case of a periodic equation, a criterion for existence of a globally attractive positive solution is provided. The results here constitute a significant improvement of recent literature, in view of the generality of the equation under study and of sharper criteria obtained for situations covered in recent works. Several examples illustrate the results.