Turing instability

In a reaction-diffusion (R-D) system with two intermediates X and Y where X is an activator and Y is an inhibitor, criterion for Turing symmetry breaking instability is that 

For developing a theoretical formalism, under such circumstances, the modified form of reaction-diffusion equation would be as follows  

Primary function of diffusion in the Turing instability is thus to spatially disengage the counteracting species. 

In the case of lOj-arsenite system [35] convection enhances the velocity of wave front [33]. The density is decreased on account of exothermicity of reaction as well as positive V. This effect is similar to the observed effect. 

More complex behaviour during wave propagation has also been observed. Thus, Bazsa and Epstein [37] in the case of Fe-HN03 system have found anisotropic wave velocity in the system as follows  

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Murray, J. D. (1982). Parameter space for Turing instability in reaction diffusion mechanisms a comparison of models. J. Theor. Biol., 98, 143-63. 

Pearson, J. E. and Horsthemke, W. (1989). Turing instabilities with nearly equal diffusion coefficients. J. Chem. Phys., 90, 1588-99. 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

The Turing instability was proposed to describe qualitatively natural phenomena such as animal skin patterns (Murray, 1993), but it took close to 40 years to be reproduced experimentally in a well con-... 

There are a number of excellent treatments of the Turing instability. A classic and comprehensive account can be found in the book by Murray,which includes fascinating treatments of pattern formation in animal-shaped domains (to address the question of animal coat patterns ). A less advanced but highly recommended presentation can be found in the book by Edelstein-Keshet, which contains a wealth of information on mathematical treatments of biological systems. The following description is adapted from Ref. 34, which draws on both of these sources. For more advanced discussions, the reader may wish to consult the thermodynamics oriented treatment of Nicolis and Prigogine. ... 

We now further explore the Turing instability by examining the parameter dependence of the sign of det(B). First, we define the function H k ) ... 

The Turing instability arises from the combination of short-range activation and long-range inhibition. This can be seen by subtracting the first inequality in Eq. [60] from the first inequality in Eq. [66] to give... 

The first corresponds to the classical activator-inhibitor system, where the elements fy<0 and g > 0 represent, respectively, Y (the inhibitor) inhibiting the formation of X (the activator), and X promoting the formation of Y. The second, with the opposite sign pattern for these off-diagonal elements, corresponds to a positive-feedback system such as the Gray-Scott model, where X is the autocatalyst and Y is typically a consumable reactant. In this case, both the autocatalyst and the reactant promote the formation of the autocatalyst, and, in turn, both species participate in the consumption of the reactant. In either case, a Turing instability can exist. 

Figure 12 Constraint diagram showing parameter values of Turing bifurcation locus (left line) and Hopf bifurcation locus (right line). The Turing instability occurs for parameter values in the middle region. (Reprinted with permission from Ref. 34.)...

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. 

Turing Instabilities in Standard Reaction-Diffusion Systems... 

To obtain the threshold criterion for the onset of a Turing instability, we set k = 0 and analyze the determinant... 

Theorem 10.4 The critical wavenumber kj of the Turing instability near a doublezero point is given by... 

In this section we study the onset of Turing instabilities in more detail for two-variable systems. We will focus on systems with Neumann boundary conditions, which are most relevant for experimental systems. 

In other words, for a Turing instability to occur, the activator must diffuse slower than the inhibitor. This is known as the principle of short-range activation and long-range inhibition. It is also known as local autocatalysis with lateral inhibition or local auto-activation-lateral inhibition (LALI), see for example [332, 319], local self-activation and lateral inhibition [280], or self-enhancement and lateral inhibition (SELI) [315] and has been applied to mechanisms other than reaction-diffusion. 

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

If we consider b to be the control parameter, then the threshold for the Turing instability is given by... 

Turing Instabilities in HRDEs and Reaction-Cattaneo Systems... 

As discussed in Sect. 2.2 the diffusion equation has the well-known unrealistic feature that localized disturbances spread infinitely fast, though with heavy attenuation, through the system. In that section we described three approaches to address the unphysical behavior of the diffusion equation and reaction-diffusion equation. Since the Turing instability is a diffusion-driven instability, it is of particular interest to explore how this bifurcation depends on the characteristics of the transport process. In this section, we address the effects of inertia in the dispersal of particles or individuals on the Turing instability. Does the finite speed of propagation of perturbations in such systems affect Turing instabilities We determine the stability properties of the uniform steady state for the three approaches presented in Sect. 2.2. 

Turing Instabilities in Hyperbolic Reaction-Diffusion Equations... 

We first use hyperbolic reaction-diffusion equations, see Sect. 2.2.1, to study the effect of inertia on Turing instabilities [206]. Specifically, we consider two-variable HRDEs,... 

The Turing condition C4 = 0 for hyperbolic reaction-diffusion equations leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40). In other words, the Turing condition is independent of and Ty. If inertia in the transport is modeled by HRDEs, then the inertia has no effect whatsoever on the Turing instability to stationary patterns. 

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