Numerical Simulation of a Fractional Vegetation–Water Model Describing Anomalous Processes and Pattern Formation in Arid and Semi-Arid Environments

In this paper, we present a new fractional mathematical model to describe the dynamics and the interaction between plants and water in arid and semi-arid environments with and without slope. By the Caputo fractional operator, the model allows for simulating the phenomena related to the vegetation migration, which occur in domains with different slopes. The new fractional model represents a connection between the Klausmeier model, where water advection occurs, to the Klausmeier–Gray–Scott model, where water diffuses. The proposed model describes an anomalous physical phenomenon that changes as the fractional parameter changes, modeling an anomalous water advection. We obtain the Hopf bifurcation of migration speed as function of the fractional parameter. The oscillatory solutions and the vegetation pattern formation, obtained by an explicit first-order numerical method, validate the analytical results and confirm the reliability and efficiency of the fractional formulation of the considered model.