Formation and Persistence of Spatiotemporal Turing Patterns

TL;DR

This paper investigates the stability and long-term behavior of Turing patterns in reaction-diffusion systems, providing rigorous analysis and applying it to biological models like Schnakenberg and Gierer-Meinhardt equations.

Contribution

It offers a rigorous analysis of the stability and asymptotic behavior of Turing patterns in reaction-diffusion systems, with applications to biological morphogenesis models.

Findings

Stability conditions for bifurcated solutions are derived.

Long-time asymptotic behavior of patterns is rigorously established.

Application to Schnakenberg and Gierer-Meinhardt models demonstrates practical relevance.

Abstract

This article is concerned with the stability and long-time dynamics of structures arising from a structureless state. The paradigm is suggested by developmental biology, where morphogenesis is thought to result from a competition between chemical reactions and spatial diffusion. A system of two reaction-diffusion equations for the concentrations of two morphogens is reduced to a finite system of ordinary differential equations. The stability of bifurcated solutions of this system is analyzed, and the long-time asymptotic behavior of the bifurcated solutions is established rigorously. The Schnakenberg and Gierer-Meinhardt equations are discussed as examples.