AI 中文总结
本文针对含两种斑块的周期环境，研究KPP-双稳反应扩散方程的传播动力学，分析持续生存、脉动行波存在性及物种灭绝条件，丰富了异质环境下的反应扩散传播理论。
AI 中文摘要
本文首先研究由两种不同斑块类型构成的空间周期环境中，一维反应扩散方程柯西问题解的传播动力学。本研究的创新之处在于系统分析了KPP-双稳异质框架。在该框架下，各斑块长度、零解与正周期稳态的线性稳定性、初始数据的大小对长时间动力学起关键作用。我们首先建立物种的持续生存性质：当相关周期斑块模型的零稳态不稳定时，一致持续成立；在额外合适条件下，得到局部持续生存。利用动力学系统方法，我们进一步建立传播性质，并根据平凡解不稳定或稳定的两种不同情况，证明了脉动行波的存在性。最后，我们给出两组表征物种灭绝的充分条件。
英文摘要
This paper first investigates the propagation dynamics of solutions to the Cauchy problem for a one-dimensional reaction-diffusion equation in a spatially periodic environment consisting of two distinct patch types. The novelty of this work lies in the systematic analysis of a KPP-bistable heterogeneous framework. In this setting, the respective patch lengths, the linear stability of the zero solution and the positive periodic steady state, and the magnitude of the initial data play crucial roles in the long-time dynamics. We first establish persistence properties of the species, showing that uniform persistence holds when the zero steady state of the associated periodic patch model is unstable, while local persistence is obtained under additional suitable conditions. Using a dynamical systems approach, we further establish spreading properties and demonstrate the existence of pulsating traveling waves in two different cases, depending on whether the trivial solution is unstable or stable. Finally, we present two sets of sufficient conditions characterizing species extinction.
