The effect of growth and curvature on pattern formation
Ramón G. Plaza, Faustino Sanchez-Garduno, Pablo Padilla, Rafael A. Barrio,and Philip K. Maini
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Submission date: 28. Sep. 2004 (revised version: October 2004)
published in: Journal of dynamics and differential equations, 16 (2004) 4, p. 1093-1121
DOI number (of the published article): 10.1007/s10884-004-7834-8
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Based on first principles, we derive a general model to describe the spatio-temporal dynamics of two morphogens. The diffusive part of the model incorporates the dynamics, growth and curvature of one and two-dimensional domains embedded in . Our generalized diffusion process includes spatio-temporal varying diffusion coefficients, advection and dilution terms. We present specific examples by analyzing a third order activator-inhibitor mechanism for the kinetic part. We carry out illustrative numerical simulations on two-dimensional growing domains having different geometries. Comparisons with former results on fixed domains show the crucial role of growth and curvature on pattern selection. Evidence is given that both effects might be biologically relevant in explaining the selection of some observed patterns and in changing or enhancing their stability.