Pattern formation in reaction-diffusion systems with piece-wise kinetic modulation: an example study of heterogeneous kinetics

TL;DR

This paper investigates how spatially discontinuous reaction kinetics influence pattern formation in reaction-diffusion systems, providing a local stability analysis approach for heterogeneous kinetic parameters.

Contribution

It introduces a stability analysis method for reaction-diffusion systems with piece-wise constant kinetics, highlighting local properties of Turing instability conditions.

Findings

Stability or instability is a local property in piece-wise kinetic models.

Local assessment of parameters predicts large-region pattern formation.

Discontinuities in kinetics can be analyzed to understand pattern emergence.

Abstract

The study of pattern emergence together with exploration of the exemplar Turing model is enjoying a renaissance both from theoretical and experimental perspective. Here, we implement a stability analysis of spatially dependent reaction kinetics by exploring the effect of a jump discontinuity within piece-wise constant kinetic parameters, using various methods to identify and confirm the diffusion-driven instability conditions. Essentially, the presence of stability or instability in Turing models is a local property for piece-wise constant kinetic parameters and, as such, may be analysed locally. In particular, a local assessment of whether parameters are within the Turing space provides a strong indication that for a large enough region with these parameters, an instability can be excited.