On the effects of advection and network structure in spiky patterns for the Gierer-Meinhardt model

Abstract

In this paper, we consider the existence of spike stationary solutions for the Gierer-Meinhardt model with advection term on the Y-shaped metric graph. By using Liapunov-Schmidt reduction method, we give the rigorous proof of the existence theorems. We are interested in the effects of the advection velocity and geometry of the graph on the location and amplitude of a spike. In particular, by considering the Neumann boundary condition and the Robin boundary condition, we reveal that how the direction of the shift of a spike is decided by the choice of the boundary conditions. We also emphasize that the effects of the advection and geometry of the graph are represented by the associated Green's function.