Global stability of Wright-type equations with negative Schwarzian

TL;DR

This paper extends Wright's classical global stability criterion to a broader class of Wright-type delay differential equations with negative Schwarzian, using qualitative analysis and interval methods for validation.

Contribution

It generalizes the 37/24 stability condition to Wright-type equations with decreasing nonlinearities and negative Schwarzian, providing a rigorous analytical framework.

Findings

Extended stability criterion to new class of equations

Validated results using interval analysis at finite points

Provided qualitative properties of bounding relations

Abstract

Simplicity of the $37/24$-global stability criterion announced by E.M. Wright in 1955 and rigorously proved by B. B\'{a}nhelyi et al in 2014 for the delayed logistic equation raised the question of its possible extension for other population models. In our study, we answer this question by extending the $37/24$-stability condition for the Wright-type equations with decreasing smooth nonlinearity $f$ which has a negative Schwarzian and satisfies the standard negative feedback and boundedness assumptions. The proof contains the construction and careful analysis of qualitative properties of certain bounding relations. To validate our conclusions, these relations are evaluated at finite sets of points; for this purpose, we systematically use interval analysis.