The Brusselator is a generic reaction-diffusion model for a tri-molecular chemical reaction. We consider the case when the input and output reactions are slow. In this limit, we show the existence of K-periodic, spatially bi-stable structures, mesas, and study their stability. Using singular perturbation techniques, we find a threshold for the stability of K mesas. This threshold occurs in the ...

Various patterns of standing waves are found beyond the onset of the short-wave instability in a model reactiondiffusion system. These include plain and modulated stripes, squares, and rhombi in systems with square and rectangular geometry and patterns with rotational symmetry in systems with circular geometry. We also find standing waves consisting of periodic time sequences of stripes and rho...

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction–diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry of the patterns generated by the nonli...

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to ...

We study the Turing instability to stationary spatial patterns in reaction-transport systems with inertia. After a brief discussion of hyperbolic reaction-diffusion equations and reaction-Cattaneo equations we focus on reaction randomwalks, which are themost natural generalization of reactiondiffusion equations. We analyze the effect of inertia in the transport on spatial instabilities of the h...

In this paper we establish a general theoretical framework for Turing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lyapunov exponents of the evolution family associated with the linearised system (obtained by linear...

We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentia...

Although the Turing structures, or stationary reaction-diffusion patterns, have received increasing attention in biology and chemistry, making such unusual patterns on inorganic solids is fundamentally challenging. We report a simple cation exchange approach to produce Turing-type Ag2Se CoSe2 nanobelts relied diffusion-driven instability. The resultant Ag2Se-CoSe2 material highly effective cata...