Pushed-to-Pulled Front Transitions: Continuation, Speed Scalings, and Hidden Monotonicty

Montie Avery, Matt Holzer, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We analyze the transition between pulled and pushed fronts both analytically and numerically from a model-independent perspective. Based on minimal conceptual assumptions, we show that pushed fronts bifurcate from a branch of pulled fronts with an effective speed correction that scales quadratically in the bifurcation parameter. Strikingly, we find that in this general context without assumptions on comparison principles, the pulled front loses stability and gives way to a pushed front when monotonicity in the leading edge is lost. Our methods rely on far-field core decompositions that identify explicitly asymptotics in the leading edge of the front. We show how the theoretical construction can be directly implemented to yield effective algorithms that determine spreading speeds and bifurcation points with exponentially small error in the domain size. Example applications considered here include an extended Fisher-KPP equation, a Fisher–Burgers equation, negative taxis in combination with logistic population growth, an autocatalytic reaction, and a Lotka-Volterra model.

Original languageEnglish (US)
Article number102
JournalJournal of Nonlinear Science
Issue number6
StatePublished - Dec 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Front propagation
  • Invasion into unstable states
  • Pulled fronts
  • Pushed fronts
  • Traveling waves


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