On the Theoretical and Numerical Analysis of a Vegetation Model with Cross-Diffusion

We consider a nonlinear parabolic coupled system of partial differential equations describing the dynamic of a living species (vegetation) interacting with one resource system (soil water). In addition to the natural diffusion of water and plants, the system incorporates a cross-diffusion term given by the hydraulic diffusivity due to the suction of water by the roots. The model also considers a monotonously decreasing vegetation death rate, capturing the infiltration feedback between the plant and the groundwater. We first establish the existence and uniqueness of global solutions in a large class of initial data, allowing non-singular ones, and then, we propose a fully discrete numerical scheme, based on a semi-implicit Euler discretization in time and finite element discretization (with “mass-lumping”) in space, for approximating the solutions of the continuous model. The proposed numerical scheme is well-posed and verifies some qualitative properties of the discrete solutions including, non-negativity, uniform estimates, convergence towards strong solutions and optimal error estimates. Finally, we present some numerical experiments in order to verify the good behavior of the numerical scheme including the capture of Turing patterns, as well as validate the convergence order in the error estimates.