Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach

We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction-diffusion-advection system. The model describes the activator-inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction-diffusion-advection contexts is also discussed.