Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions

Fronts between two different stable homogeneous domains are considered in the framework of two-component reaction-diffusion (rd) systems with FitzHugh-Nagumo type dynamics. The dominance principle - ensuring that for a system with homogeneous parameters the direction of front propagation is uniquely determined by the relative strengths of the two domains - fails when the inhibitor's (controller's) time scale is slowed down sufficiently. This leads to a regime of bidirectional propagation, where for a given set of parameters either of the two domains may win the competition. Bi-directional propagation may be viewed as being organized by a bifurcation of the stationary front - obtained for equally strong domains - which is destabilized when a critical value of the relative time constant is crossed. Although, generically, one would expect a transcritical scenario, this bifurcation turns out to be of supercritical pitchfork type for arbitrary nonlinearity, i.e., the bifurcation is independent of an accidental inflection symmetry of the dynamics. For inhomogeneous parameter distributions the behavior becomes richer, leading to stationary and oscillating pinning as well as to reflections of incoming fronts. This situation is analyzed performing a multiple scale perturbation expansion up to third order in the vicinity of the above bifurcation leading to a two-dimensional reduced dynamics, globally valid in space. The two degrees of freedom relevant for the description are the position of the front and a characteristic feature of its shape. The latter is reminiscent of the momentum of classical particle physics. Results are compared with full system numerics.