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Root growth: homogenization in domains with time dependent partial perforations

Published online by Cambridge University Press:  14 October 2011

Yves Capdeboscq  and
Mariya Ptashnyk
Yves Capdeboscq
Affiliation:
Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, UK. [email protected]
Mariya Ptashnyk
Affiliation:
Department of Mathematics I, RWTH Aachen University, Wüllnerstr. 5b, 52056 Aachen, Germany; [email protected]
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Abstract

In this article we derive a macroscopic model for the time evolution of root density, starting from a discrete mesh of roots, using homogenization techniques. In the microscopic model each root grows vertically according to an ordinary differential equation. The roots growth rates depend on the spatial distribution of nutrient in the soil, which also evolves in time, leading to a fully coupled non-linear problem. We derive an effective partial differential equation for the root tip surface and for the nutrient density.

Keywords

Homogenizationroot growthtime dependent domains
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 856 - 876
Copyright
© EDP Sciences, SMAI, 2011

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References

Acerbi, E., Chiado Piat, V., Dal Maso, G. and Percivale, D., An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18 (1992) 481–496. Google Scholar
Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. Google Scholar
G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, edited by A. Bourgeat et al., World Scientific Pub., Singapore (1996) 15–25.
Bastian, P., Chavarría-Krauser, A., Engwer, Ch., Jäger, W., Marnach, S. and Ptashnyk, M., Modelling in vitro growth of dense root networks. J. Theor. Biol. 254 (2008) 99–109. Google ScholarPubMed
Caloin, M. and Yu, O., An extension of the logistic model of plant growth. Ann. Bot. 49 (1982) 599–607. Google Scholar
Chavarria-Krauser, A. and Schurr, U., A cellular growth model for root tips. J. Theor. Biol. 230 (2004) 21–32. Google ScholarPubMed
S. Chuai-Aree, W. Jäger, H.G. Bock and S. Siripant, Modeling, simulation and visualization of plant root growth and diffusion processes in soil volume, 4th International Workshop on Functional-Structural Plant Models, edited by C. Godin et al. Montpellier, France (2004) 289–293.
D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer (1999).
Dupuy, L., Fourcaud, T., Stokes, A. and Danjon, F., A density-based approach for the modelling of root architecture : application to Maritime pine (Pinus pinaster Ait.) root systems. J. Theor Biol. 236 (2005) 323–334. Google ScholarPubMed
Dupuy, L., Gregory, P.J. and Bengough, A.G., Root growth models : towards a new generation of continuous approaches. J. Exp. Bot. 61 (2010) 2131–2143. Google ScholarPubMed
Erickson, R.O., Modeling of plant growth. Ann. Rev. Plant Physiol. 27 (1976) 407–434. Google Scholar
Grabarnik, P., Pagès, L. and Bengough, A.G., Geometrical properties of simulated maize root systems : consequences for length density and intersection density. Plant Soil 200 (1998) 157–167. Google Scholar
Marciniak-Czochra, A. and Ptashnyk, M., Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Anal. 40 (2008) 215–237. Google Scholar
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. Google Scholar
Pagès, L., How to include organ interactions in models of the root system architecture? The concept of endogenous environment. Ann. For. Sci. 57 (2000) 535–541. Google Scholar
Pagès, L., Picard, M.O. Jordan and D., A simulation-model of the three-dimensional architecture of the maize root-system. Plant Soil 1989 (1989) 147–154. CrossRefGoogle Scholar
Pagès, L., Vercambre, G., Drouet, J.-L., Lecompte, F., Collet, C. and Le Bot, J., Root Typ : a generic model to depict and analyse the root system architecture. Plant Soil 258 (2004) 103–119. Google Scholar
Prusinkiewicz, P., Modeling plant growth and development. Current Opinion in Plant Biol. 7 (2004) 79–83. Google ScholarPubMed
P. Prusinkiewicz and A. Lindenmayer, The algorithmic beauty of plants. Springer-Vergal, New York, USA (1990).
Wilderotter, O., An adaptive numerical method for the Richards equation with root growth. Plant Soil 251 (2003) 255–267.Google Scholar