SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 382-401, January 2020.
In 2006 Armstrong, Painter, and Sherratt formulated a nonlocal differential equation model for cell-cell adhesion. For the one-dimensional case on a bounded domain we derive various types of biological boundary conditions, describing adhesive, repulsive, and neutral boundaries. We prove local and global existence and uniqueness for the resulting integrodifferential equations. In numerical simulations we consider adhesive, repulsive, and neutral boundary conditions, and we show that the solutions mimic known behavior of fluid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion.

中文翻译：

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有界域上微生物的非局部黏附模型

SIAM应用数学杂志，第80卷，第1期，第382-401页，2020
年1月。2006年，Armstrong，Painter和Sherratt为细胞-细胞粘附建立了非局部微分方程模型。对于有界域上的一维情况，我们导出了各种类型的生物边界条件，描述了粘附，排斥和中性边界。我们证明了积分微分方程的局部和全局存在性和唯一性。在数值模拟中，我们考虑了粘合，排斥和中性边界条件，并且我们证明了这些解决方案模仿了流体对边界的粘合行为。另外，我们观察到由于细胞间粘附而形成的内部图案。