Reaction-Biased Random Walks. Propagation Failure

7 Reaction-Biased Random Walks. Propagation Failure 

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. 

We choose the initial conditions to be /o(x, 0) = 1 for jc 0 and /o(Jt, 0) = 0 for X 0. This initial condition describes, for example, a territory divided into an invaded zone, x 0, and a noninvaded zone, x 0, separated by a frontier at X = 0. If particles disperse according to an isotropic random walk with KPP kinetics, this initial condition turns into a front propagating from left to right, i.e., the invasion starts. Since the particle jumps are isotropic, the reaction is responsible for the motion of the front from left to right. It is the reaction process that starts and maintains a successful invasion. A bias to the left in the random walk will hinder the invasion. Therefore we expect that the critical reaction rate is given by a balance between the factor favoring the invasion, the reaction process, and the factor opposing the invasion, the bias in the transport process. 

We use Model C, given by (5.27), for the mean-field equation for p x,t) with z) = (p(t)w z). The standard diffusion approximation of (5.27), i.e., taking the limit of small jump lengths and small waiting times, yields the reaction-diffusion-advection equation 

The propagation velocity is obtained from (4.46), involving H and p. The critical condition, i.e., a stalled front, v = 0, is realized if H(p ) = 0 with 0. Writing 

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Reaction-Biased Random Walks. Propagation Failure (x- ) = lim... 

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