2020/01/02 by Andreas Buttenschön, Buttenschön, Andreas, Thomas Hillen +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Cell Behavior (q-bio.CB) #Cellular Mechanics and Interactions #FOS: Biological sciences #FOS: Mathematics #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.2001.00286
openalex publication_date 2020/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.