论文标题
非局部细胞粘附模型：稳态和分叉
Non-Local Cell Adhesion Models: Steady States and Bifurcations
论文作者
论文摘要
在本手稿中，我们考虑了细胞粘连的建模，这是生物细胞之间的关键相互作用。扩散 - 调节反应类型的连续模型长期以来一直在组织建模中使用。 2006年，阿姆斯特朗（Armstrong），画家和谢拉特（Sherratt）提出了一个扩展，以考虑粘附效应。最终的方程是非本地对流扩散方程。尽管在应用方面取得了巨大成功，但缺乏与稳态形成有关的数学理论的发展。 非本地粘附模型的数学分析具有挑战性。在这一专着中，我们为稳态及其分叉结构的分析做出了贡献。稳态的重要性在于，这些是自然界和组织中观察到的模式（例如细胞分类实验）。在周期性边界条件的情况下，我们结合了由Rabinowitz，Eopiriast分叉理论和非局部项的数学特性（最大原理）结合了全局分叉的结果，以获得非琐事解决方案分支的全局分叉结果。
In this manuscript, we consider the modelling of cellular adhesions, which is a key interaction between biological cells. Continuum models of the diffusion-advection-reaction type have long been used in tissue modelling. In 2006, Armstrong, Painter, and Sherratt proposed an extension to take adhesion effects into account. The resulting equation is a non-local advection-diffusion equation. While immensely successful in applications, the development of mathematical theory pertaining to steady states and pattern formation is lacking. The mathematical analysis of the non-local adhesion model is challenging. In this monograph, we contribute to the analysis of steady states and their bifurcation structure. The importance of steady-states is that these are the patterns observed in nature and tissues (e.g. cell-sorting experiments). In the case of periodic boundary conditions, we combine global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties (maximum principle) of the non-local term to obtain a global bifurcation result for the branches of non-trivial solutions.
