生物入侵与衰退的移动边界模型的斑图形成与锋面稳定性
Pattern formation and front stability for a moving-boundary model of biological invasion and recession
摘要：
本文研究了二维Fisher-Stefan模型中的斑图形成，该模型涉及到在紧支区域上求解Fisher-KPP方程。Fisher-Stefan模型将Fisher-KPP方程与经典Stefan理论相结合，克服了Fisher-KPP方程对生物种群的两个局限性。在这项工作中，我们调查是否二维Fisher-Stefan模型预测斑图形成，通过分析线性稳定性的平面行波解正弦横向扰动。Fisher-KPP方程的平面波前是线性稳定的。同样，我们证明了入侵平面锋（$c > 0$）的Fisher-Stefan模型是线性稳定的扰动的所有波数。然而，我们的分析表明，后退平面锋（$c < 0$）的Fisher-Stefan模型是线性不稳定的所有波数。这类似于经典Stefan问题中平面凝固的不稳定解。引入一个表面张力正则化稳定后退前短波长扰动，引起一系列的不稳定模式和最不稳定的波数。我们补充线性稳定性分析水平集数值解，证实理论结果。总的来说，前面的不稳定性的Fisher-Stefan模型提出了一个新的机制，斑图形成在后退的生物种群。
英文摘要：
We investigate pattern formation in a two-dimensional (2D) Fisher--Stefan model, which involves solving the Fisher--KPP equation on a compactly-supported region with a moving boundary. By combining the Fisher--KPP and classical Stefan theory, the Fisher--Stefan model alleviates two limitations of the Fisher--KPP equation for biological populations. In this work, we investigate whether the 2D Fisher--Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher--KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts ($c > 0$) of the Fisher--Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts ($c < 0$) of the Fisher--Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher--Stefan model suggests a new mechanism for pattern formation in receding biological populations.
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