The evolution of traveling waves in a KPP reaction–diffusion model with cut‐off reaction rate. II. Evolution of traveling waves

Alex D. O. Tisbury, David J. Needham, Alexandra Tzella

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Abstract

In Part II of this series of papers, we consider an initial‐boundary value problem for the Kolmogorov–Petrovskii–Piscounov (KPP)‐type equation with a discontinuous cut‐off in the reaction function at concentration urn:x-wiley:00222526:media:sapm12352:sapm12352-math-0001. For fixed cut‐off value urn:x-wiley:00222526:media:sapm12352:sapm12352-math-0002, we apply the method of matched asymptotic coordinate expansions to obtain the complete large‐time asymptotic form of the solution, which exhibits the formation of a permanent form traveling wave (PTW) structure. In particular, this approach allows the correction to the wave speed and the rate of convergence of the solution onto the PTW to be determined via a detailed analysis of the asymptotic structures in small time and, subsequently, in large space. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut‐off Fisher reaction function.
Original languageEnglish
Pages (from-to)330-370
JournalStudies in Applied Mathematics
Volume146
Issue number2
Early online date16 Dec 2020
DOIs
Publication statusPublished - 1 Feb 2021

Keywords

  • matched asymptotic expansions
  • permanent form traveling waves
  • reaction–diffusion equations

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