Abstract
AbstractWe consider Kolmogorov‐Petrovskii‐Piscounov (KPP) type models in the presence of a discontinuous cut‐off in reaction rate at concentration . In Part I, we examine permanent form traveling wave solutions (a companion paper, Part II, is devoted to their evolution in the large time limit). For each fixed cut‐off value , we prove the existence of a unique permanent form traveling wave with a continuous and monotone decreasing propagation speed . We extend previous asymptotic results in the limit of small and present new asymptotic results in the limit of large which are, respectively, obtained via the systematic use of matched and regular asymptotic expansions. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut‐off Fisher reaction function.
Highlights
Traveling waves arise in a wide range of applications in mathematical chemistry and biology
For each fixed cut-off value 0 < uc < 1, we prove the existence of a unique permanent form traveling wave with a continuous and monotone decreasing propagation speed v∗(uc)
We have considered a canonical evolution problem for a reaction-diffusion process when the reaction function is of standard KPP-type, but experiences a cut-off in the reaction rate below the normalized cut-off concentration uc ∈ (0, 1)
Summary
Traveling waves arise in a wide range of applications in mathematical chemistry and biology (for example, in combustion[1] and in ecology, epidemiology, and genetics[2,3]). A more rigorous approach was employed by Dumortier et al[20] who used geometric desingularization, to prove the existence and uniqueness of a permanent form traveling wave with v∗ (uc ) All these results have restricted validity to the small uc limit with specific choices of cut-off KPPtype reaction function (8d), the most common based on f(u) given by (3b)[1]. The analysis is carried out on the direct problem (rather than the phase plane) It highlights that higher-order corrections are controlled by two global constants A∞ and B∞ associated with the leading behaviour of Um, the permanent form traveling wave solution to the non cut-off problem (1) with minimal speed v = vm (see equation (7a)).
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