ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2012 - Volume 18, Issue 03  

Research Article

Root growth: homogenization in domains with time dependent partial perforations

Yves Capdeboscqa1 and Mariya Ptashnyka2

a1 Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, UK. capdeboscq@maths.ox.ac.uk

a2 Department of Mathematics I, RWTH Aachen University, Wüllnerstr. 5b, 52056 Aachen, Germany; ptashnyk@math1.rwth-aachen.de

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.

(Received October 28 2010)

(Revised June 27 2011)

(Online publication October 14 2011)

Key Words:

  • Homogenization;
  • root growth;
  • time dependent domains

Mathematics Subject Classification:

  • 35B27;
  • 35K55;
  • 92C99s