Absolute and Delay-Dependent Stability of Equations with a Distributed Delay: a Bridge from Nonlinear Differential to Difference Equations

TL;DR

This paper investigates the stability of nonlinear equations with distributed delays, establishing a connection to difference equations to determine stability conditions, especially focusing on delay-independent stability in models relevant to population dynamics.

Contribution

It introduces a method linking stability of distributed delay equations to associated difference equations, clarifying stability conditions for finite and infinite memory systems.

Findings

Difference equation stability implies stability of distributed delay equations.

Distributed delay equations are stable for small enough delays.

Delay-independent stability is characterized for models with finite memory.

Abstract

We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent attracting set and also demonstrate that the equation with a distributed delay is stable for small enough delays.