Pulled fronts are not (just) pulled

Front propagation into unstable states is often determined by the linearization; that is, propagation speeds agree with predictions from the linearized equation at the unstable state. The leading edge behavior is then a Gaussian tail propagating with the linear spreading speed. Fronts following this leading edge are commonly referred to as pulled fronts, alluding to the idea that they are “pulled” by this leading edge Gaussian tail. We describe here a class of examples that exhibits how these leading order effects do not completely describe the dynamics in the wake of the front. In fact, leading edge behavior predicts at most two possible invasion scenarios, associated with positive and negative amplitudes of the Gaussian tail, but our examples exhibit three or more invasion fronts with different states in the wake. The resulting invasion process therefore leaves behind a state that is not solely determined by the leading edge and thus not just pulled by the Gaussian tail.