Turing diffusive instabilities

We finally mention that wave propagation arises from diffusion, but that the excitable behavior can give rise to interesting spatiotem-poral phenomena even without diffusion. For example, Matthews and Brindley (1997) show for model (4.32)-(4.33) with D = 0 that if there is a spatially localized forcing (a point source of nutrients, a local change in temperature, etc.), a localized excitation and recovery plankton pulse will occur, which may be a mechanism for localized (i.e. non-propagating) plankton blooms. 

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- 

Now a number of chemical and biological systems (Epstein and Pojman, 1998 Murray, 1993) are known to display Turing patterns. The mechanism has also been invoked as possible source of plankton patchiness (Levin and Segel, 1976). Although diffusion coefficients of planktonic organisms have roughly the same value for both predator 

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Diffusive instability can appear in simple predator-prey models. Bartumeus et al. (2001) used the linear stability analysis and demonstrated that a simple reaction-diffusion predator-prey model with a ratio-dependent functional response for the predator can lead to Turing structures due to diffusion-driven instabilities. 

Some well-known autocatalytic reaction systems are periodic in time and sometimes are called biological clocks. Turing s instabilities, published in 1952, suggested that the morphogens transported by diffusion and other mechanisms from a production site may cause the pa... 

Diffusive instabilities usually require a special relationship between dififiision coefficients. Ordinarily, for Turing instability the diffusion coefficient of the inhibitor, Anh, should be significantly larger than that of the activator. Act- This theoretical relation applies, however, only in the absence of cross-diffusion [66],... 

I would like to make some further remarks on the topic of spatial and temporal structures in chemical instabilities. There are three topics I wish to discuss briefly, topics of possible interest to this conference. First, let me say that Professor Prigogine introduced me to the field of instabilities in reaction-diffusion systems more than ten years ago by discussing with me Turing s pioneering work and the plans for his incisive research. This is an appropriate occasion to thank him for that and many helpful talks on the subject since then. 

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. 

Classic Turing Models. The most well studied models of spontaneous pattern formation are based on Turing s original idea that symmetry breaking in biomorphogenesis occurs via an instability in a reaction-diffusion system. In order for this to be operative the time scales for reaction and diffusion must be comparable. Since cellular systems are in general quite small... 

We consider the stability of a general two-variable system, first in the absence of diffusion and then with diffusion terms. Because the Turing bifurcation is a diffusion-induced instability, we will first show that the system is stable... 

Point B is much farther from the Turing bifurcation, and we anticipate the pattern formation to be more complex. Following the same procedure as described above, we find the following range of wavenumbers over which the diffusion-induced instability occurs... 

Over the past several years there have been many experimental and theoretical studies aimed at developing a better understanding of pattern formation in reaction-diffusion systems. The focus of recent studies has been on more complex behavior away from the onset of instability. For some parameter values, spatiotemporal chaos may occur near the boundary between the Turing region and the region of homogeneous oscillations (Figure 12). 

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... 

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... 

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