An efficient, nonlinear stability analysis for detecting pattern formation in reaction diffusion systems

TL;DR

This paper introduces a simple, efficient nonlinear stability analysis method for reaction diffusion systems, enabling detection of pattern formation regimes, especially in complex biological models with disparate diffusion rates.

Contribution

The paper presents a novel nonlinear stability technique that simplifies analysis of reaction diffusion systems, applicable to complex biological models with significant diffusion rate differences.

Findings

Identified previously undetected non-linear patterning regimes in simple models.

Validated the method's predictions against numerical simulations and bifurcation analyses.

Demonstrated applicability to complex biological networks like chemotaxis.

Abstract

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, non-linear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and non-linear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have non-linear patterning regimes not previously detected. Analysis of a protein regulatory network related to chemotaxis shows its application in a more complex setting where other non-linear methods become intractable. Predictions of this method are verified against results of numerical simulation, linear stability, and full PDE bifurcation analyses.