Direction-independent reaction walk

As discussed in Sect. 2.2, persistent random walks provide a mesoscopic description of reaction-transport systems with inertia. This approach provides another opportunity to explore the effects of a finite velocity in the transport mechanism on propagating fronts. We consider two cases. The first corresponds to reaction walks where the kinetic terms do not depend on the direction of the particles. This corresponds to choosing /f = 1/2 in (2.38), and persistent random walks with such kinetics are called direction-independent reaction walks (DIRW). The second case corresponds to walks where reactions occur only between particles with opposite velocities. We call such systems direction-dependent reaction walks (DDRWs). 

If we replace Brownian motion by its simplest generalization, the persistent random walk, we obtain direction-independent reaction walks as the simplest generalization of reaction-diffusion equations. Both describe chemical reactions in the reaction-limited or activation-controlled regime. However, the activation barrier is only implicitly taken into account it is incorporated into the kinetic coefficients... 

Turing Instabilities in Direction-Independent Reaction Walks... 

Horsthemke, W. Spatial instabilities in reaction random walks with direction-independent kinetics. Phys. Rev. E60(3),2651-2663 (1999). http //dx.doi.org/10.1103/PhysRevE. 60.2651... 

An interesting question arises when the dispersal is biased in a direction away from the region occupied by the unstable state [286]. What are the conditions on the reaction rate and bias that will result in a stalled front Or phrased differently, what is the critical (minimal) value of the reaction rate to sustain front propagation when the underlying random walk has a bias in the opposite direction The goal of this section is to show the following (i) The standard diffusion approximation of the transport process always provides an inaccurate value for the critical reaction rate, (ii) If the reaction rate exceeds the jump frequency of the random walk, then the front cannot stall and will always propagate into the unstable state, independently of the values of the other statistical parameters of the random walk. 

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