Upper limits on the robustness of Turing models and other multiparametric dynamical systems

TL;DR

This paper introduces an efficient $O(n)$ method using Gershgorin's theorem to determine upper bounds on Turing instability regions, significantly reducing computational complexity in analyzing multiparametric dynamical systems.

Contribution

The authors develop a novel $O(n)$ approach for bounding Turing instabilities, improving upon traditional $O(n^3)$ methods for stability analysis in complex systems.

Findings

The method effectively identifies parameter regions where Turing patterns cannot form.

It simplifies the exploration of phase diagrams in high-dimensional models.

Applicable to systems biology and other multiparametric models.

Abstract

Traditional linear stability analysis based on matrix diagonalization is a computationally intensive $O(n^3)$ process for $n$-dimensional systems of differential equations, posing substantial limitations for the exploration of Turing systems of pattern formation where an additional wave-number parameter needs to be investigated. In this study, we introduce an efficient $O(n)$ technique that leverages Gershgorin's theorem to determine upper limits on regions of parameter space and the wave number beyond which Turing instabilities cannot occur. This method offers a streamlined avenue for exploring the phase diagrams of other complex multiparametric models, such as those found in systems biology.