种群生物学中无限陡峭行波的前沿稳定性
Front stability of infinitely steep travelling waves in population biology
AI总结：
本文研究了Fadai提出的具有无限陡峭前沿的生物入侵行波模型的两个关键扩展，建立了行波解的存在性，并通过横向扰动分析和2D数值模拟揭示了入侵行波稳定而衰退行波不稳定的特性。
摘要：
反应扩散模型常被用于描述生物入侵，其中经历随机运动和增殖的个体种群会导致移动前沿。许多生物入侵模型是Fisher-KPP模型的扩展，该模型描述了一维种群密度作为线性扩散和logistic增长结果的演化。2020年Fadai引入了一种新的生物入侵模型，该模型被表述为具有非线性退化扩散通量的移动边界问题。Fadai的模型导致在移动边界处具有无限陡峭、明确定义的前沿的行波解，并且该模型具有在某些参数极限下易于分析的数学优势。我们旨在通过首先提出两个关键扩展来提供一般性见解，考虑：(i)具有通量的广义非线性退化扩散；以及(ii)描述生物入侵和生物衰退的解。我们建立了这两个扩展的行波解的存在性，然后通过引入行波前沿的横向扰动来考虑行波解的稳定性。完整的2D时间相关水平集数值解表明，入侵行波对小横向扰动是稳定的，而衰退行波是不稳定的。这些初步的数值观察通过线性稳定性分析得到了证实，该分析为波前沿扰动幅度的短时增长/衰减提供了更正式的见解。
英文摘要：
Reaction-diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many models of biological invasion are extensions of the Fisher-KPP model that describes the evolution of a 1D population density as a result of linear diffusion and logistic growth. In 2020 Fadai introduced a new model of biological invasion that was formulated as a moving boundary problem with a nonlinear degenerate diffusive flux. Fadai's model leads to travelling wave solutions with infinitely steep, well-defined fronts at the moving boundary, and the model has the mathematical advantage of being analytically tractable in certain parameter limits. We aim to provide general insight by first presenting two key extensions by considering: (i) generalised nonlinear degenerate diffusion with flux; and, (ii) solutions describing both biological invasion, and biological recession. We establish the existence of travelling wave solutions for these two extensions, and then consider stability of the travelling wave solutions by introducing a lateral perturbation of the travelling wavefront. Full 2D time-dependent level-set numerical solutions indicate that invasive travelling waves are stable to small lateral perturbations, whereas receding travelling waves are unstable. These preliminary numerical observations are corroborated through a linear stability analysis that gives more formal insight into short time growth/decay of wavefront perturbation amplitude.
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