Pulled fronts are not (just) pulled

TL;DR

This paper challenges the traditional view of pulled fronts by showing that in some cases, the invasion dynamics are influenced by factors beyond the linear leading edge, resulting in more complex front behaviors.

Contribution

It introduces examples where front propagation involves multiple invasion scenarios, demonstrating that not all fronts are solely determined by linear leading edge effects.

Findings

Leading edge behavior predicts only two invasion scenarios.

Some fronts exhibit three or more invasion fronts.

The resulting state is influenced by factors beyond the linear tail.

Abstract

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.