Accelerated Fronts in a Two-Stage Invasion Process

We study wavespeed selection in a staged invasion process. That is, we study a model in which an unstable homogeneous state is replaced via an invading front with a secondary state. This secondary state is also unstable and, in turn, replaced by a stable homogeneous state via a secondary invasion front. In particular, we are interested in the selected wavespeed of the secondary front and under what conditions does the influence of the primary front increase this speed. We find three regimes: a locked regime where both fronts travel at the same speed, a pulled regime where the secondary front travels at the linear spreading speed associated to the intermediate state, and an accelerated regime where the selected speed is between these two speeds. We show that the transition to locked fronts can be described by the crossing of a resonance pole in the linearization about the primary front. In addition, we use properties of this resonance pole to derive an estimate for the selected wavespeed in the accelerated case and to determine the transition between accelerated and pulled fronts. MSC numbers: 35C07, 35K57, 34A26