Root Growth: Homogenization in Domains with Time Dependent Partial Perforations

In this article we consider derive a macroscopic model for root length density evolution, starting from a discrete mesh of roots, using homogenization. In the microscopic model each roots grows vertically according to an ordinary di erential equation. The roots growth rates depend on the spatial repartition of nutrient in the soil, which also evolves in time, leading to a fully coupled nonlinear problem. We derive an e ective partial di erential equation for the root tip surface and for the nutrient density.