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The evolution of traveling waves in a KPP reaction–diffusion model with cut‐off reaction rate. II. Evolution of traveling waves
Author(s) -
Tisbury Alex D. O.,
Needham David J.,
Tzella Alexandra
Publication year - 2021
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12352
Subject(s) - traveling wave , reaction–diffusion system , mathematics , mathematical analysis , initial value problem , boundary value problem , function (biology) , convergence (economics) , asymptotic expansion , cut off , diffusion , rate of convergence , asymptotic analysis , physics , thermodynamics , channel (broadcasting) , engineering , evolutionary biology , electrical engineering , economics , biology , economic growth , power (physics)
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 u = u c . For fixed cut‐off valueu c ∈ ( 0 , 1 ) , 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.