Convective Instability and Boundary Driven Oscillations in a Reaction-Diffusion-Advection Model

TL;DR

This paper investigates how boundary conditions induce local oscillations in a reaction-diffusion-advection system with convective instability, combining analytical and numerical methods to characterize this new instability.

Contribution

It introduces a novel boundary-induced instability mechanism in reaction-diffusion-advection models, supported by analytical analysis and numerical simulations.

Findings

Boundary-induced instability acts as a continuous wave source.

High advection speeds suppress the instability.

The wave packet behavior matches analytical predictions.

Abstract

In a reaction-diffusion-advection system, with a convectively unstable regime, a perturbation creates a wave train that is advected downstream and eventually leaves the system. We show that the convective instability coexists with a local absolute instability when a fixed boundary condition upstream is imposed. This boundary induced instability acts as a continuous wave source, creating a local periodic excitation near the boundary, which initiates waves traveling both up and downstream. To confirm this, we performed analytical analysis and numerical simulations of a modified Martiel-Goldbeter reaction-diffusion model with the addition of an advection term. We provide a quantitative description of the wave packet appearing in the convectively unstable regime, which we found to be in excellent agreement with the numerical simulations. We characterize this new instability and show that in the limit of high advection speed, it is suppressed. This type of instability can be expected for reaction-diffusion systems that present both a convective instability and an excitable regime. In particular, it can be relevant to understand the signaling mechanism of the social amoeba Dictyostelium discoideum that may experience fluid flows in its natural habitat.