Skip to main content
Log in

On Travelling Waves for the Stochastic Fisher–Kolmogorov–Petrovsky–Piscunov Equation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

This paper is concerned with properties of the wave speed for the stochastically perturbed Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation. It was shown in the classical 1937 paper by Kolmogorov, Petrovsky and Piscunov that the large time behavior of the solution to the FKPP equation with Heaviside initial data is a travelling wave. In a seminal 1995 paper Mueller and Sowers proved that this also holds for a stochastically perturbed FKPP equation. The wave speed depends on the strength σ of the noise. In this paper bounds on the asymptotic behavior of the wave speed c(σ) as σ→0 and σ→∞ are obtained.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, in Partial Differential Equations and Related Topics, J. Goldstein, ed. (Springer, New York, 1975), Lecture Notes in Mathematics no. 446, pp. 5–49.

  2. M. Bramson (1983) ArticleTitleConvergence of solutions of the Kolmogorov equation to travelling waves Mem. Amer. Math. Soc. 44 285

    Google Scholar 

  3. E. Brunet B. Derrida (1997) ArticleTitleShift in the velocity of a front due to a cutoff Phys. Rev. E. 56 2597–2604 Occurrence Handle10.1103/PhysRevE.56.2597

    Article  Google Scholar 

  4. C.R. Doering (1987) ArticleTitleA stochastic partial differential equation with multiplicative noise Phys. Lett. A 122 133–139 Occurrence Handle10.1016/0375-9601(87)90791-2

    Article  Google Scholar 

  5. C.R. Doering C. Mueller P. Smereka (2003) ArticleTitleInteracting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation and duality. Physica A 325 243–259

    Google Scholar 

  6. U. Ebert W. Saarloos Particlevan (2000) ArticleTitleFront propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts Physica D 146 1–99

    Google Scholar 

  7. P. Fife J.B. McLeod (1977) ArticleTitleThe approach of solutions of nonlinear diffusion equations to travelling wave front solutions Arch. Rat. Mech. Anal. 65 335–362 Occurrence Handle10.1007/BF00250432

    Article  Google Scholar 

  8. R.A. Fisher (1937) ArticleTitleThe wave of advance of advantageous genes Ann. Eugen. 7 355–369

    Google Scholar 

  9. D.A. Kessler Z. Ner L.M. Sander (1998) ArticleTitleFront propagation: Precursors, cutoffs, and structural stability Phys. Rev. E 58 107–114 Occurrence Handle10.1103/PhysRevE.58.107

    Article  Google Scholar 

  10. A. Kolmogorov I. Petrovsky N. Piscunov (1937) ArticleTitleÈtude de l'\'{equation de la diffusion avec croissance de la quantit\'{eacute; de matière et son application à un problème biologique Moscou Universitet Bull. Math. 1 1–25

    Google Scholar 

  11. T. Liggett (1985) Interacting Particles Systems Springer-Verlag Berlin

    Google Scholar 

  12. H. McKean (1975) ArticleTitleApplication of Brownian motion to the equation of Kolmogorov-Petrowskii- Piskunov Comm. Pure Appl. Math. 28 323–331

    Google Scholar 

  13. C. Mueller R. Sowers (1995) ArticleTitleRandom Travelling Waves for the KPP Equation with Noise J. Functional Analysis 128 439–498 Occurrence Handle10.1006/jfan.1995.1038

    Article  Google Scholar 

  14. C. Mueller R. Tribe (1995) ArticleTitleStochastic p.d.e.’s arising from the long range contact and long range voter processes Probab. Theory Relat. Fields 102 519–545 Occurrence Handle10.1007/BF01198848

    Article  Google Scholar 

  15. L. Pechenik H. Levine (1999) ArticleTitleInterfacial velocity corrections due to multiplicative noise Phys. Rev E 59 3893–3900 Occurrence Handle10.1103/PhysRevE.59.3893

    Article  Google Scholar 

  16. E. Perkins (2004) ArticleTitleSuper-Brownian motion and critical spatial stochastic systems Canad. Math. Bull. 47 280–297

    Google Scholar 

  17. V. Privman C. Doering H. Frisch (1993) ArticleTitleCrossover from rate-equation to diffusion- controlled kinetics in two-particle coagulation Phys. Rev. E. 48 846–851 Occurrence Handle10.1103/PhysRevE.48.846

    Article  Google Scholar 

  18. T. Shiga (1994) ArticleTitleTwo contrasting properties of solutions for one-dimensional stochastic partial differential equations Can. J. Math. 46 415–437

    Google Scholar 

  19. T. Shiga K. Uchiyama (1986) ArticleTitleStationary states and the stability of the stepping stone model involving mutation and selection. Probab Theory Relat. Fields 73 87–117 Occurrence Handle10.1007/BF01845994

    Article  Google Scholar 

  20. R. Tribe (1995) ArticleTitleLarge time behavior of interface solutions to the heat equation with Fisher-Wright white noise Probab. Theory Relat. Fields 102 289–311 Occurrence Handle10.1007/BF01192463

    Article  Google Scholar 

  21. J. Xin (2000) ArticleTitleFront propagation in heterogeneous media SIAM Review 42 161–230 Occurrence Handle10.1137/S0036144599364296

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joseph G. Conlon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Conlon, J.G., Doering, C.R. On Travelling Waves for the Stochastic Fisher–Kolmogorov–Petrovsky–Piscunov Equation. J Stat Phys 120, 421–477 (2005). https://doi.org/10.1007/s10955-005-5960-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-005-5960-2

Key words

Navigation