In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize a uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize a uniform equilibrium which...

(1) ∂tu = a∆u− f0(u, t) + g0(x, t), u|∂Ω = 0 (or ∂u/∂ν|∂Ω = 0) where u = u(x, t) = (u, . . . , u ), x ∈ Ω b R, t ≥ 0, f0(v, s) = (f 0 , . . . , f 0 ), (v, s) ∈ R × R+, g0(x, s) = (g 0 , . . . , g 0 ), x ∈ Ω, s ≥ 0. We assume that the matrix a and the functions f0, g0 satisfy some general conditions (see Section 2). These conditions provide the existence of a solution u of the Cauchy problem for...

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are pre...

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Reaction-diffusion systems contribute to various morphogenetic processes, and can also be used as computation models in real and artificial chemistries. Evolving reaction-diffusion solutions automatically is interesting because it is otherwise difficult to engineer them to achieve a target pattern or to perform a desired task. However most of the existing work focuses on the optimization of par...

In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally s...

Vision systems require fundamental algorithms of image processing and vision computing. Algorithms of edge detection, grouping and stereo disparity detection are typical examples. Marr and his collaborators proposed several effective algorithms of edge detection and stereo disparity detection, in particular, in a computational approach (Marr, 1982). Marr and Hildreth had previously proposed an ...

In this paper a simple reaction-diffusion system, namely a binary fluid mixture with an association-dissociation reaction between the two components, is considered. Fluctuations at hydrodynamic spatiotemporal scales when a temperature gradient is present in this chemically reacting system are studied. First, fluctuating hydrodynamics when the system is in global equilibrium (isothermal) is revi...

A random channel approach is developed for reaction-diffusion processes in disordered systems. Although the starting point of our research is the kinetic study of the decay and preservation of marine organic carbon, our approach can be used for describing other disordered kinetic catalytic processes with random pathways. We consider a generic catalytic mechanism with two species: a) a catalyst,...

Reversible reaction-diffusion systems display anomalous dynamics characterized by a power-law relaxation toward stationarity. In this paper we study in the aging regime the nonequilibrium dynamical properties of some model systems with reversible reactions. Starting from the exact Langevin equations describing these models, we derive expressions for two-time correlation and autoresponse functio...