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.

中文翻译：

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具有截止反应速率的KPP反应扩散模型中行波的演化。一，永久形式的行波

我们考虑在浓度下反应速率不连续地截止的情况下考虑的Kolmogorov-Petrovskii-Piscounov（KPP）型模型。在第一部分中，我们研究了永久形式的行波解（第二部分的辅助论文专门讨论了它们在较长时间内的演化）。对于每个固定的截止值，我们证明了存在唯一的，连续且单调递减的传播速度的永久形式行波。我们在小范围内扩展以前的渐近结果，在大范围内扩展新的渐近结果它们分别通过系统地使用匹配的和规则的渐近展开而获得。渐近结果与截断Fisher反应函数的特殊情况下获得的数值结果得到了确认。