On the growth of filamentary structures in planar media

Authors

  • D. Andreucci,

    1. Dipartimento di Metodi e Modelli Matematici, Università di Roma La Sapienza, via Scarpa 16, 00161 Roma, Italy
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  • M. A. Herrero,

    Corresponding author
    1. Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
    • Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Cindad Universitaria s/n 28040 Madrid, Spain
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  • J. J. L. Velázquez

    1. Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
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Abstract

We analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a growth factor), and including a fourth cell variable characterising irreversible incorporation to a filament. Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, in the case when the system has parameters very large and very small, thereby allowing the onset of different time and space scales. In particular, we describe the motion of the tip of a filament, and the changes in the relevant chemical species nearby. Copyright © 2004 John Wiley & Sons, Ltd.

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