Cross-Diffusion in Reaction-Diffusion Models: Analysis, Numerics, and Applications

Abstract

Cross-diffusion terms are nowadays widely used in reaction-diffusion equations encountered in models from mathematical biology and in various engineering applications. In this contribution we review the basic model equations of such systems, give an overview of their mathematical analysis, with an emphasis on pattern formation and positivity preservation, and finally we present numerical simulations that highlight special features of reaction-cross-diffusion models.

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

“…This has become a popular continuum description of cell adhesion, with a number of variations proposed to study tumour invasion and cell movement across the ECM [15,22,25,37,43,79,100], development and cell-sorting dynamics [8,23,43,80]. Attention has also been given over the years to variations of the nonlocal modelling of adhesion by considering di↵erent ways to enforce limits on the cell density and furthermore to include nonlinear cross di↵usion, that is substituting linear di↵usion modelling random movement with nonlinear di↵usion modelling movement of cells down a density-dependent pressure gradient, into the overall model, see [21,23,63,72]. The principal e↵ect of these modifications is that they lead to sharper interfaces between the densities of di↵erent cell types or the ECM or can even lead to strict segregation of them.…”