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

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Abstract

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.

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Data Availability

Matlab code to perform continuation for scalar PDEs is available at GitHub/mholzergmu/scalar-pushed-pulled-continuation. The datasets generated during and analyzed during the current study are available from the corresponding author (msavery@umn.edu) on reasonable request.

Notes

  1. The terminology “simple double root” is motivated by the fact that \((0,\nu _0)\) is “simple” in a degree counting sense as a solution to the double root equation \(d=\partial _\nu d=0\) assuming that \(\partial _{\nu \nu }d,\partial _\lambda d\ne 0\).

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, 55455, USA

    Montie Avery & Arnd Scheel

  2. Department of Mathematics and Statistics, Boston University, 665 Commonwealth Ave, Boston, MA, 02215, USA

    Montie Avery

  3. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Matt Holzer

Authors
  1. Montie Avery
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  2. Matt Holzer
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  3. Arnd Scheel
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Contributions

All authors contributed equally to the proofs and implementation of algorithms.

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Correspondence to Montie Avery.

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Conflict of interest

The material here is based on work supported by the National Science Foundation, through through the GRFP-00074041 (MA), NSF-DMS-2202714 (MA), NSF-DMS-2007759 (MH) and NSF-DMS-1907391 (AS). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors have no other competing interests that are relevant to the content of this article.

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Communicated by Paul Newton.

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Avery, M., Holzer, M. & Scheel, A. Pushed-to-Pulled Front Transitions: Continuation, Speed Scalings, and Hidden Monotonicty. J Nonlinear Sci 33, 102 (2023). https://doi.org/10.1007/s00332-023-09957-3

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