AI 中文总结
该研究证明趋化性驱动的失稳无需反应性，扩展了反应-扩散系统的面向矩阵数值框架，通过示例验证了趋化性可单独诱导模式形成，揭示了两类空间自组织机制的差异。
AI 中文摘要
Neubert、Caswell和Murray的经典结论指出，空间均匀平衡态的反应性是扩散驱动（图灵）失稳的必要条件。本研究探究在存在趋化性的情况下该结论是否仍然成立。我们考虑耦合趋化通量的一般反应-扩散系统，建立渐近失稳的必要条件，证明当趋化贡献足够强时，可放宽经典的反应性要求：反应性仍是图灵失稳的必要条件，但并非趋化性驱动失稳的必要条件。从计算角度，我们将为反应-扩散系统开发的面向矩阵的公式扩展至更通用的反应-扩散-趋化模型类，趋化输运项以与扩散算子的面向矩阵近似兼容的形式离散化，形成用于模拟趋化性驱动模式形成的高效数值框架。我们给出失稳区域的几何解释，凸显扩散与趋化的不同作用。通过两个代表性示例说明理论与数值进展：趋化性扩展的Schnakenberg模型，展示趋化性如何修改经典图灵模式；捕食者-猎物模型，证明在无反应性和扩散驱动失稳的情况下，仅趋化性即可诱导模式形成。这些结果揭示了扩散驱动与趋化性驱动空间自组织机制的根本差异，为非对称输运过程在生物模式形成中的作用提供新的理论与计算见解。
英文摘要
A classical result by Neubert, Caswell and Murray states that reactivity of a spatially homogeneous equilibrium is a necessary condition for diffusion-driven (Turing) instability. In this work, we investigate whether the same conclusion remains valid in the presence of chemotaxis. We consider a general reaction--diffusion system coupled with a chemotactic flux and establish necessary conditions for asymptotic instability. We show that the classical requirement of reactivity can be relaxed when the chemotactic contribution is sufficiently strong. In particular, while reactivity remains necessary for Turing instability, it is not a necessary condition for chemotaxis-driven instability. From a computational viewpoint, we extend the matrix-oriented formulation developed for reaction--diffusion systems to the more general class of reaction--diffusion--chemotaxis models. The chemotactic transport term is discretized in a form compatible with the matrix-oriented approximation of the diffusion operator, yielding an efficient numerical framework for the simulation of chemotaxis-driven pattern formation. A geometric interpretation of the instability region is presented, highlighting the distinct roles played by diffusion and chemotaxis. The theoretical and numerical developments are illustrated through two representative examples: a chemotaxis-extended Schnakenberg model, showing how chemotaxis modifies classical Turing patterns, and a predator--prey model, demonstrating that chemotaxis alone can induce pattern formation in the absence of both reactivity and diffusion-driven instability. These results reveal a fundamental difference between diffusion-driven and chemotaxis-driven mechanisms of spatial self-organization and provide new theoretical and computational insights into the role of non-symmetric transport processes in biological pattern formation.
