Asymptotic behaviour for a class of non-monotone delay differential systems with applications

TL;DR

This paper investigates the long-term behavior of a broad class of non-monotone delay differential systems, providing conditions for population extinction or permanence, with applications to structured population models like Nicholson systems.

Contribution

It introduces new sufficient conditions for extinction and permanence in non-monotone delay differential equations, extending existing criteria and applying to complex population models.

Findings

Established criteria for population extinction and permanence.

Improved existing stability conditions for autonomous systems.

Analyzed a non-autonomous Nicholson system with delays.

Abstract

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.