Turing Instabilities in HRDEs and Reaction-Cattaneo Systems

Note that the Brusselator does not have a Takens-Bogdanov point. Theorems 10.2 and 10.3 are not applicable in this case, and the Brusselator displays somewhat pathological behavior as decreases toward unity. Equation (10.51) implies that a becomes arbitrarily large as Infinite a corresponds to an infinite Hopf 

2 Turing Instabilities in Reaction-Transport Systems with Inertia 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. 

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