Turing Instabilities in Standard Reaction-Diffusion Systems

1 Iliring Instabilities in Standard Reaction-Difrusion Systems 

We begin the discussion of Turing instabilities by presenting some general results for n-variable systems. We focus particularly on properties of the diffusion matrix D that facilitate or hinder the onset of spatial instabilities. 

Mendez et al., Reaction-Transport Systems, Springer Series in Synergetics, DOI 10.1007/978-3-642-11443-4 10, Springer-Verlag Berlin Heidelberg 2010 

---

Turing Instabilities in Standard Reaction-Diffusion Systems... 

Part III focuses on spatial instabilities and patterns. We examine the simplest type of spatial pattern in standard reaction-diffusion systems in Chap. 9, namely patterns in a finite domain where the density vanishes at the boundaries. We discuss methods to determine the smallest domain size that supports a nontrivial steady state, known as the critical patch size in ecology. In Chap. 10, we provide first an overview of the Turing instability in standard reaction-diffusion systems. Then we explore how deviations from standard diffusion, namely transport with inertia and anomalous diffusion, affect the Turing instability. Chapter 11 deals with the effects of temporally or spatially varying diffusivities on the Turing instability in reaction-diffusion systems. We present applications of Turing systems to chemical reactions and biological systems in Chap. 12. Chapter 13 deals with spatial instabilities and patterns in spatially discrete systems, such as diffusively and photochemically coupled reactors. 

To be able to compare directly a reaction-subdiffusing system with a standard reaction-diffusion system, we proceed in the following way. As shown in Sect. 10.1.2, a Turing instability can occur in a standard reaction-diffusion system only... 

Turing Instability in the Standard Brusselator Reaction-Diffusion System... 

Front propagation and Turing instabilities are two emblematic phenomena displayed by nonlinear reaction-diffusion systems. In the former, a stable state invades an unstable or less stable state. In the latter, diffusion couples with the local nonlinear transformations and drives the uniform steady state of the system unstable. We focus on the effect of deviations from standard diffusion and spatial homogeneity of the medium on these two signature phenomena. 

---