Optimal linear stability condition for scalar differential equations with distributed delay

TL;DR

This paper establishes a stability criterion for scalar differential equations with distributed delays, showing that stability depends only on the mean delay and not on the delay distribution shape, with applications to biological models.

Contribution

It provides a shape-independent stability condition for equations with distributed delays based solely on the mean delay, advancing understanding in biological and physical systems.

Findings

Stability depends only on the mean delay, not the distribution shape.

Discrete delay equations' stability implies distributed delay equations' stability.

Applied criterion to a biological hematopoietic cell model.

Abstract

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability.