Effect of cross-diffusion on pattern formation - a nonlinear analysis
type: Item
instance of: scholarly article
title: Effect of cross-diffusion on pattern formation - a nonlinear analysis
zbMATH Open document ID: 0904.92011
DOI: 10.1023/A:1005764514684
author: P. K. Tapaswi
author: Joydev Chattopadhyay
published in: Acta Applicandae Mathematicae
publication date: 1999-01-24T00:00:00Z
review text: The authors consider a model for the switching behaviour (determination) of a cell proposed by \textit{H. Meinhardt} [Models of biological pattern formation (1982)] and traced the main characteristics of a reactional system in the presence of diffusion, in order to maintain inhomogeneities leading to spatial and spatio-temporal ordering and find out the criteria for global stability of these spatio-temporal ordered structures. The most important observation in this paper is that the introduction of cross-diffusion terms in this model system are necessary for the emergence of a spatially ordered structure in the Turing sense. This case agrees well with the realistic situation of the coexistence of self-diffusion with a contribution due to cross-effects. Self-diffusion implies passive diffusion where the diffusing substance moves along its concentration gradient. Cross-diffusion implies counter-transport. The cross-diffusion coefficient may be positive or negative. When it is positive, the substance moves along the concentration gradient of another substance (passive counter transport) and when it is negative, the substance moves against the concentration gradient of the other substance (active counter-transport). Examples of cross-diffusion in chemical and biological processes are discussed in the last section. We consider both the cross-diffusion coefficients to be positive.
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 1057376
zbMATH Keywords: Bendixson-Dulac criteria
zbMATH Keywords: Lyapunov functional method
zbMATH Keywords: global stability
zbMATH Keywords: switching behaviour
zbMATH Keywords: spatio-temporal ordered structure
zbMATH Keywords: cross-diffusion terms
MaRDI profile type: Publication
full work available at URL: a:1005764514684
OpenAlex ID: W1518340778
description: scientific article; zbMATH DE number 1057376