The evolution of traveling waves in a KPP reaction‐diffusion model with cut‐off reaction rate. I. Permanent form traveling waves
Alex D. O. Tisbury, D. J. Needham, Alexandra Tzella
Abstract
Abstract We 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.