Transport-Reaction Systems: Higher-Dimensional Domains and a Qualitative Dichotomy

TL;DR

This paper analyzes linear transport-reaction systems on high-dimensional domains, establishing spectral properties, classifying transport-driven instabilities, and deriving conditions for Turing pattern formation.

Contribution

It extends the understanding of transport-reaction systems to higher dimensions, introduces hyperbolic instabilities, and provides a new algebraic criterion for Turing patterns.

Findings

Established a weak spectral mapping theorem for these systems.

Identified hyperbolic instabilities as a framework for transport-driven patterns.

Derived a new algebraic condition for Turing pattern existence.

Abstract

We study general linear transport-reaction systems on an arbitrary dimensional hypercube with periodic boundary conditions. Transport-reaction systems are often used to model the finite speed movement and interaction of particles, bacteria or animals. We first show a weak spectral mapping theorem and demonstrate its' application. Secondly, we introduce a certain class of so-called hyperbolic instabilities, which provide a natural framework for transport-driven instabilities on one-dimensional domains: They are either Turing patterns or increasingly oscillating hyperbolic instabilities. A new algebraic condition for the existence of Turing patterns is obtained as a side-product.