Distributed delays stabilize negative feedback loops

TL;DR

This paper demonstrates that in linear differential equations with distributed delays, stability of negative feedback loops depends only on the mean delay, and such delays tend to stabilize the system regardless of the specific delay distribution shape.

Contribution

It establishes that distributed delays stabilize negative feedback loops by showing stability depends solely on the mean delay, independent of the distribution shape.

Findings

Distributed delays stabilize negative feedback loops.

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

Distributed delays can enhance system stability.

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 oscillation 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 the linear equation with distributed delays is asymptotically stable if the associated differential equation with a discrete delay of the same mean is asymptotically stable. Therefore, distributed delays stabilize negative feedback loops.