Oscillations in a scalar differential equation coupled to a diffusive field

TL;DR

This paper investigates how periodic oscillations emerge in a scalar diffusion equation with dynamic boundary conditions, highlighting the role of delay effects and spectral properties in biological systems.

Contribution

It demonstrates the occurrence of Hopf bifurcations in a diffusive system with boundary dynamics, accounting for essential spectrum challenges and biological relevance.

Findings

Hopf bifurcation occurs in the presence of essential spectrum.

Delay effects influence boundary oscillations in biological models.

The analysis extends understanding of oscillations beyond classical center-manifold methods.

Abstract

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.