The Fisher equation in the presence of an active boundary

The Fisher equation (or Fisher-Kolmogorov equation) is the prototypical equation for non-linear propagating wavefronts. As such it plays an important role in many biological, chemical and mechanical systems. Associated with the Fisher equation is a well-defined and easily calculated linear spreading velocity, i.e. the velocity by which a perturbation about the unstable state will propagate through the system. A driving force can be introduced into the system by adding an advection term to the equation. This advective term can be transformed away resulting only in a shift in the linear spreading velocity. However if introducing both advection and an active boundary, the advective velocity cannot simply be transformed away. As I will show, one has to take the stationary profile induced by the active boundary into account to fully understand the wave propagation.