A user's guide to PDE models for chemotaxis
A user's guide to PDE models for chemotaxis
复制标题
DOI：
10.1007/s00285-008-0201-3
复制
发表时间：
2009-01-01
影响因子：
1.9
通讯作者：
Painter, K. J.
中科院分区：
文献类型：
--
作者：
Hillen, T.;Painter, K. J.
趋化性（生物细胞或生物体响应化学梯度的运动）的数学建模已发展成为一门大型且多样化的学科，其方面包括其机械基础、特定系统的建模以及基础方程的数学行为。趋化性的 Keller-Segel 模型（Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971）为这项工作的大部分提供了基石，其成功归功于其直观的简单性、分析的易处理性和复制趋化群体关键行为的能力。其中一个特性，即显示“自动聚集”的能力，使其成为生物系统自组织机制的重要组成部分。这种现象已被证明会在模型的某些公式下导致有限时间爆炸，并且大量工作致力于确定何时发生爆炸或是否存在全局现有解决方案。在本文中，我们详细探讨了原始 Keller-Segel 模型的许多变体。我们从生物学角度审查它们的配方，对比它们的图案特性，总结其分析特性的关键结果并对它们的溶液形式进行分类。最后，我们进行了简短的讨论，并扩展了这项工作所揭示的一些悬而未决的问题。
Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.