Fronts in Direction-Independent Reaction Walks

We consider RRWs with KPP kinetics [204]. Specifically we choose the branching-coalescence kinetic scheme, see Sect. 1.4.1. A DIRW with such kinetics obeys the following nondimensionalized evolution equations  

The prime denotes the derivative with respect to the argument of the function and u = + u. The fixed points of (5.64) are (0,0) and (1/2, 1 /2), and a front corresponds to a nonnegative heteroclinic orbit in the phase plane connecting 

For V y, the vector field of (5.64a) and (5.64c) points away from the nullcline N(u ), and a stable heteroclinic orbit does not exist. For u y, the vector field points toward the nullcline, and for front velocities close to y, i.e., 0 y - n = e 1, the density is a fast variable and follows the evolution of m adiabatically. The system (5.64a) and (5.64c) is then well approximated by (5.65) and 

This approximation becomes exact in the limit v y. The trajectory in the phase plane is then given by the nullcline N(u ). Substituting (5.65) into (5.66), we obtain 

The nonnegative heteroclinic orbit will persist as v decreases, as long as no bifurcation occurs in the vector field of (5.64a) and (5.64c) and (1/2, 1 /2) remains a saddle poinf and (0, 0) a sfable node, whose eigenvectors lie strictly within the positive quadrant. For w y, the eigenvalues for the fixed point (0, 0) are 

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