论文标题
模式形成和前稳定性，用于生物学入侵和衰退的移动模型
Pattern formation and front stability for a moving-boundary model of biological invasion and recession
论文作者
论文摘要
我们研究了二维（2D）Fisher-Stefan模型中的模式形成，该模型涉及在紧凑型支撑区域上求解Fisher-kpp方程，并具有移动的边界。通过将Fisher（KPP和古典Stefan理论）结合在一起，Fisher-Stefan模型减轻了Fisher的两个局限性 - KPP方程生物种群。在这项工作中，我们通过分析平面运动波溶液对正弦横向扰动的线性稳定性来研究2D Fisher-Stefan模型是否可以预测模式形成。 Fisher-KPP方程的平面前部线性稳定。同样，我们证明了Fisher-Stefan模型的入侵平面前线（$ c> 0 $）是线性稳定的，可与所有波数的扰动。但是，我们的分析表明，Fisher-Stefan模型的后退平面前线（$ C <0 $）对于所有波数都是线性不稳定的。这类似于经典Stefan问题中平面固化的不稳定解决方案。引入表面张力正则化稳定了后退的前部，以实现短波长度扰动，从而产生一系列不稳定的模式和最不稳定的波浪数。我们使用级别的数值解决方案来补充线性稳定性分析，以证实理论结果。总体而言，Fisher-stefan模型中的前部不稳定性提出了一种新的机制，用于恢复生物学种群的模式形成。
We investigate pattern formation in a two-dimensional (2D) Fisher--Stefan model, which involves solving the Fisher--KPP equation on a compactly-supported region with a moving boundary. By combining the Fisher--KPP and classical Stefan theory, the Fisher--Stefan model alleviates two limitations of the Fisher--KPP equation for biological populations. In this work, we investigate whether the 2D Fisher--Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher--KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts ($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.
