Reaction-Cattaneo system

Reaction-Cattaneo Systems and Reaction-Telegraph Equations... 

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Eu and Al-Ghoul have derived such systems from generalized hydrodynamic theory [9, 7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversible thermodynamics [223], see for example [282]. If we differentiate (2.17) with respect to t and (2.18) with respect to x and eliminate mixed second derivatives, we obtain the so-called reaction-telegraph equation. 

This relation between the relaxation time t of the flux and the time scale l/F (p) of the reaction appears to be a purely mathematical requirement. The following mesoscopic approach will shed light on the foundational problems of the reaction-Cattaneo system (2.17) and (2.18) and the reaction-telegraph equation (2.19) hinted at by points (i) and (ii). 

For n-variable systems, the reaction-Cattaneo systems and reaction-telegraph equations read, i = 1,... 

This choice is based on the assumption that F(p) is a source term for the particles, that the reaction does not depend on the direction of motion, and that new particles choose either direction with equal probability. With (2.35) we obtain from (2.34a) and (2.34b) the reaction-Cattaneo system... 

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

As a second approach, we use reaction-Cattaneo equations, see Sect. 2.2.2, to study the effect of inertia on Turing instabilities [206]. The uniform steady state of the two-species reaction-Cattaneo system. 

The appropriate spatial modes for the fluxes 8J and 8Jy are sin jr), and we obtain the characteristic equation for the reaction-Cattaneo system, det J = 0, where... 

As for HRDEs, the Turing condition C4 = 0 for the reaction-Cattaneo system leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40) the Turing condition is independent of r and ty for reaction-Cattaneo equations. As for HRDEs, the spatial Hopf bifurcation cannot occur in the regime of small inertia, if (10.23) is satisfied. 

Remark 10.2 These equations do not constitute a reaction-Cattaneo system because of the contribution of the intrinsic death rates f (p , Py), / = 1, 2, to the decay rate of the flows. Further, no reaction-telegraph equations can be derived for the total densities, unless the death rates /i (p , Py) are constants. 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

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