Skip to main content
Springer Nature Link
Log in

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

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

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 \(\partial _{\nu \nu }d,\partial _\lambda d\ne 0\).

References

  • Alhasanat, A., Ou, C.: Minimal-speed selection of traveling waves to the Lotka–Volterra competition model. J. Differ. Equ. 266(11), 7357–7378 (2019a)

    Article  MathSciNet  MATH  Google Scholar 

  • Alhasanat, A., Ou, C.: On a conjecture raised by Yuzo Hosono. J. Dynam. Differ. Equ. 31(1), 287–304 (2019b)

    Article  MathSciNet  MATH  Google Scholar 

  • An, J., Henderson, C., Ryzhik, L.: Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation. J. Eur. Math. Soc. (JEMS) (to appear)

  • Avery, M.: Front selection in reaction–diffusion equations via diffusive normal forms. Preprint (2022)

  • Avery, M., Garénaux, L.: Spectral stability of the critical front in the extended Fisher-KPP equation. Z. Angew. Math. Phys. 74, 71 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  • Avery, M., Scheel, A.: Universal selection of pulled fronts. Commun. Am. Math. Soc. 2, 172–231 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Avery, M., Dedina, C., Smith, A., Scheel, A.: Instability in large bounded domains–branched versus unbranched resonances. Nonlinearity 34(11), 7916–7937 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Benguria, R., Depassier, M.: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Commun. Math. Phys. 175, 221–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Bers, A., Rosenbluth, M., Sagdeev, R.: Handbook of plasma physics. MN Rosenbluth and RZ Sagdeev eds, 1(3.2) (1983)

  • Beyn, W.-J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10(3), 379–405 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Beyn, W.-J., Thümmler, V.: Freezing solutions of equivariant evolution equations. SIAM J. Appl. Dyn. Syst. 3(2), 85–116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31(5), 531–581 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M.: Convergence of solutions of the Kolmogorov equation to traveling waves. Mem. Am. Math. Soc. (1983)

  • Champneys, A.R., Kuznetsov, Y.A., Sandstede, B.: A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bifur. Chaos Appl. Sci. Eng. 6(5), 867–887 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, K., Deiman, Z., Goh, R., Jankovic, S., Scheel, A.: Strain and defects in oblique stripe growth. Multiscale Model. Simul. 19(3), 1236–1260 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Collet, P., Eckmann, J.-P.: Instabilities and Fronts in Extended Systems. Princeton Series in Physics, Princeton University Press, Princeton, NJ (1990)

    Book  MATH  Google Scholar 

  • Ebert, U., van Saarloos, W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Phys. D 146, 1–99 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Faye, G., Holzer, M., Scheel, A.: Linear spreading speeds from nonlinear resonant interaction. Nonlinearity 30(6), 2403–2442 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Faye, G., Holzer, M., Scheel, A., Siemer, L.: Invasion into remnant instability: a case study of front dynamics. Indiana Univ. Math. J. 71, 1819–1896 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Focant, S., Gallay, T.: Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system. Phys. D 120(3–4), 346–368 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Gohberg, I., Lancaster, P., Rodman, L.: Invariant Subspaces of Matrices with Applications. SIAM (2006)

    Book  MATH  Google Scholar 

  • Hadeler, K.-P., Rothe, F.: Traveling fronts in nonlinear diffusion equations. J. Math. Biol. 2(1), 251–263 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Hamel, F., Nolen, J., Roquejoffre, J.-M., Ryzhik, L.: A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Netw. Heterog. Media 8(1), 275–289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Henderson, C.: Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis. J. Math. Pures. Appl. 167, 175–203 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Holzer, M.: Anomalous spreading in a system of coupled Fisher-KPP equations. Phys. D 270, 1–10 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Holzer, M., Scheel, A.: A slow pushed front in a Lotka–Volterra competition model. Nonlinearity 25(7), 2151–2179 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Holzer, M., Scheel, A.: Criteria for pointwise growth and their role in invasion processes. J. Nonlinear Sci. 24(1), 661–709 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Hosono, Y.: The minimal speed of traveling fronts for a diffusive Lotka–Volterra competition model. Bull. Math. Biol. 60(3), 435–448 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Huang, W.: Problem on minimum wave speed for a Lotka–Volterra reaction–diffusion competition model. J. Dynam. Differ. Equ. 22(2), 285–297 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Kolmogorov, A., Petrovskii, I., Piskunov, N.: Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sect. A 1, 1–26 (1937)

    Google Scholar 

  • Lau, K.-S.: On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J. Differ. Equ. 59(1), 44–70 (1985)

    Article  MathSciNet  Google Scholar 

  • Lewis, M.A., Li, B., Weinberger, H.F.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45(3), 219–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Lloyd, D.J.B., Scheel, A.: Continuation and bifurcation of grain boundaries in the Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 16(1), 252–293 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Lord, G.J., Peterhof, D., Sandstede, B., Scheel, A.: Numerical computation of solitary waves in infinite cylindrical domains. SIAM J. Numer. Anal. 37(5), 1420–1454 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Pogan, A., Scheel, A.: Instability of spikes in the presence of conservation laws. Z. Angew. Math. Phys. 61(6), 979–998 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Sandstede, B., Scheel, A.: Absolute and convective instabilities of waves on unbounded and large bounded domains. Phys. D 145(3), 233–277 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Sandstede, B., Scheel, A.: On the structure of spectra of modulated travelling waves. Math. Nachr. 232, 39–93 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Sandstede, B., Scheel, A.: Defects in oscillatory media: toward a classification. SIAM J. Appl. Dyn. Syst. 3(1), 1–68 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Sandstede, B., Scheel, A.: Relative Morse indices, Fredholm indices, and group velocities. Discrete Contin. Dyn. Syst. 20(1), 139–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Sattinger, D.: Weighted norms for the stability of traveling waves. J. Differe. Equ. 25(1), 130–144 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Stegemerten, F., Gurevich, S.V., Thiele, U.: Bifurcations of front motion in passive and active Allen-Cahn-type equations. Chaos 30(5), 053136, 12 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)

    Article  MATH  Google Scholar 

Download references

Author information

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
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Matt Holzer
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Arnd Scheel
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

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

Corresponding author

Correspondence to Montie Avery.

Ethics declarations

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.

Additional information

Communicated by Paul Newton.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-023-09957-3

Keywords

Mathematics Subject Classification