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Effect of cross-diffusion on pattern formation - a nonlinear analysis. (English) Zbl 0904.92011
The authors consider a model for the switching behaviour (determination) of a cell proposed by 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.

MSC:
92C15 Developmental biology, pattern formation
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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