Turing Instability in Reaction-Diffusion Systems with a Single Diffuser: Characterization Based on Root Locus

TL;DR

This paper develops a unified framework to analyze Turing instability in reaction-diffusion systems with a single diffuser, using root locus techniques to classify and derive conditions for biologically plausible patterns.

Contribution

It introduces a novel root locus-based classification and analytic conditions for Turing instability in single-diffuser reaction-diffusion systems, aiding biological network design.

Findings

Classified Turing instabilities as biologically plausible or not

Derived analytic conditions for plausible Turing patterns

Validated results on an extended Gray-Scott model

Abstract

Cooperative behaviors arising from bacterial cell-to-cell communication can be modeled by reaction-diffusion equations having only a single diffusible component. This paper presents the following three contributions for the systematic analysis of Turing instability in such reaction-diffusion systems. (i) We first introduce a unified framework to formulate the reaction-diffusion system as an interconnected multi-agent dynamical system. (ii) Then, we mathematically classify biologically plausible and implausible Turing instabilities and characterize them by the root locus of each agent's dynamics, or the local reaction dynamics. (iii) Using this characterization, we derive analytic conditions for biologically plausible Turing instability, which provide useful guidance for the design and the analysis of biological networks. These results are demonstrated on an extended Gray-Scott model with a single diffuser.