Solutions for a Flux-Dependent Diffusion Model

We study a one-dimensional continuous analogue of a system proposed by Mitchison to model vein formation in meristematic tissues of young leaves. The signal concentration satisfies a diffusion equation where the diffusion coefficient changes according to a differential equation which is flux depende...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 1982-09, Vol.13 (5), p.758-769
Main Authors: Bell, Jonathan, Cosner, Chris, Bertiger, Willy
Format: Article
Language:English
Subjects:
Applied mathematics
Boundary conditions
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Summary:We study a one-dimensional continuous analogue of a system proposed by Mitchison to model vein formation in meristematic tissues of young leaves. The signal concentration satisfies a diffusion equation where the diffusion coefficient changes according to a differential equation which is flux dependent. We show that the system possesses a unique, global solution. We then examine the stability of the steady state solution which depends on a source strength parameter $\psi > 0$. For $\psi $ sufficiently small, the steady state is linearly and $L^2 $ stable. But as $\psi $ passes through a critical value, the stability changes and a Hopf bifurcation takes place.
ISSN:0036-1410
1095-7154
DOI:10.1137/0513052