Reaction-Subdiffusion Equations

As shown above, the standard diffusion equation (2.1) has a fractional diffusion equation (2.59) as its analog in the subdiffusive case. As in the case of reaction-transport equation with inertia, see Sect. 2.2, the question arises how to combine reactions and subdiffusion in the activation-controlled regime. (For a discussion of the subdiffusion-limited case, which is outside the scope of this monograph as mentioned on page 34, see for example [491-493, 369, 391, 392, 389, 409, 410, 390, 411, 203, 187].) In some schemes, [188, 189, 186, 187], reactions terms are simply added to the fractional diffusion equation, in a manner similar to the ad hoc HRDEs (2.16), assuming at the outset that the effects of subdiffusion and reactions are separable as in the standard reaction-diffusion (2.11). However, it is easy to 

The memory kernel in (2.59), recall that v] represents a nonlocal-in-time integral operator, is a clear indication that subdiffusive transport is non-Markovian. Incorporating kinetic terms into a non-Markovian transport equation requires great care and is best carried out at the mesoscopic level. We show in Sect. 3.4 how to proceed directly at the level of the mesoscopic balance equations for non-Markovian CTRWs. Here we pursue a different approach. As stated above, if all processes are Markovian, then contributions from different processes are indeed separable and simply additive. As is well known, processes often become Markovian if a sufficiently large and appropriate state space is chosen. For the case of reactions and subdiffusion, the goal of a Markovian description can be achieved by taking the age structure of the system explicitly into account as done by Vlad and Ross [460,461]. This approach is equivalent to Model B, see Sect. 3.4. 

In terms of chemical and related systems, reactions typically create and destroy particles. In terms of ecological systems and populations dynamics, individuals are bom and die. In other words, kinetic events affect the waiting times of particles. We assume that new particles are created with zero age. The same holds for newborn individuals. Our assumption implies that all processes resulting in the arrival of a particle or individual at a given site x are treated equally. We do not distinguish between arrival via a jump to x from another site x or arrival by a reactive or birth event at x. Any arrival event sets the waiting time t at x equal to zero. We assume that locally the reactions obey classical kinetic laws, as is the case in porous media for instance, and that the local kinetics of particles or individuals can be written in production-loss form. Flip) = / (/ )-F (/)),see(1.3). As discussed in Sect. 1.1, F (p) 0 as /Oj - 0. To ensure the nonnegativity of the age-dependent densities 

This boundary condition implies that entities with zero age at a particular position are either created there with a rate F p x, t )) or arrive there from other positions, as discussed above. 

In the following, we consider the usual case of spatially homogeneous CTRWs with independent jump and waiting time PDFs, i.e., j(x x, x) = 

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Remark 2.4 In the derivation of the generalized reaction-diffusion equation (2.82) we do not explicitly refer to the particular form of the waiting time PDF. Equation (2.82) is valid for arbitrary waiting time PDFs < (t) and has much wider applicability than subdiffusive transport. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

In this section, we use Model B, see Sect. 3.4.2, to explore the effects of subdiffusion on the Turing instability. We consider the two-variable generalized reaction-diffusion equation [484],... 

These equations describe subdiffusion-limited chemical reactions. As stated in Chap. 2, such reactions are outside the scope of this monograph. For results on fronts and Turing instabilities in systems with subdiffusion-controlled chemical reactions, see for example [490,240, 147, 148, 235, 318,191]. 

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