Periodicity in delayed self-regulation is a predator-prey process

TL;DR

This paper unifies delayed self-regulation and predator-prey feedback mechanisms in delay differential equations, revealing the structure of periodic solutions as delay varies, with implications for understanding complex biological and ecological oscillations.

Contribution

It demonstrates that periodic orbits form annuli characterized by predator-prey dynamics within delay differential equations, providing a complete characterization of solutions.

Findings

Periodic orbits form annuli with global coordinates.

Variables satisfy predator-prey relations on each annulus.

Complete characterization of periodic solutions via two time maps.

Abstract

We unify two different periodicity mechanisms: delayed self-regulation and planar predator-prey feedback. We consider scalar delay differential equations $\dot x(t) = rf(x(t), x(t - 1))$ where $f$ is monotone in the delayed component. Due to a Poincar\'{e}--Bendixson theorem for monotone delayed feedback systems, the typical global dynamics present periodic orbits as the delay parameter $r$ increases. In this article, we show that, as we vary the delay, each connected component of periodic orbits is an annulus with global coordinates given by the time and the amplitude of the corresponding periodic solutions. On each annulus, the variables $x(t)$ and $x(t - 1)$ solve an integrable ordinary differential equation that satisfies a predator-prey feedback relation. Moreover, we completely characterize the set of periodic solutions of the delay differential equation in terms of two time maps generated by the underlying predator-prey system.