In this paper we develop a mathematical model of a root-soil system, and also accurate and efficient finite element and finite difference algorithms for approximating this model. The goal of our work is to develop an understanding of the properties of root systems, which can be modified by using genetic engineering techniques, in order to improve the performance of plants when water availability is limited. The results of some numerical simulations are presented, which demonstrate the effectiveness of genetic and physical changes to the root-soil system.