Reaction-Transport Fronts Propagating into Unstable States

Reaction-Transport Fronts Propagating into Unstable States 

Mendez et al., Reaction-Transport Systems. Springer Series in Synergetics, DOI 10.1007/978-3-642-11443-4 5, Springer-Verlag Berlin Heidelberg 2010 

This case corresponds to the random walk model where the jumps Z, can only take two values, -a or a, with equal probability 1/2, see Sect. 3.1.1. Then (5.1) takes the form 

---

Reaction-Transport Fronts Propagating into Unstable States A, In rcosh(flp) - ll + r... 

Fedotov, S. Front propagation into an unstable state of reaction-transport systems. Phys. Rev. Lett. 86(5), 926-929 (2001). http //dx.doi.org/10.1103/PhysRevLett.86. 926... 

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. 

---