含种群压力与退化迁移率的单组分非局部粘附模型中的分岔结构与台地模式形成
Bifurcation structure and mesa pattern formation in a one-component nonlocal adhesion model with population pressure and degenerate mobility
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中文总结 AI 辅助
该研究分析含种群压力与退化迁移率的单组分非局部粘附模型的模式形成，通过线性与弱非线性分析推导分岔特性，结合数值模拟确认台地模式形成，将其与容量受限结构关联。
中文摘要 AI 辅助
我们分析了含种群压力与退化迁移率的单组分非局部粘附模型中，从均匀定态出发的模式形成。首先，通过线性稳定性分析，我们推导了失稳阈值与最快增长模的选择规则，阐明了随平均密度增加，所选波数向低波数偏移的机制。接着，我们在临界粘附强度附近开展弱非线性分析，推导了Stuart--Landau方程中Landau系数的显式表达式。该表达式表明，临界分岔根据平均密度、非线性指数及核的二次谐波响应比，可分为超临界或亚临界。此外，我们发现大的非线性指数会促进向亚临界的转变，并通过数值分岔分析与随时间演化的模拟，确认了带有折叠点的分岔结构及台地模式的形成。最后，通过m→∞时的能量极限，我们将观测到的台地剖面与容量受限的极限结构关联起来。
英文摘要
We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart--Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.
