Numerical Study of a Fractional Vegetation–Water Model describing Anomalous Advection Processes with Variable Rainfall Alessandra Jannelli and Maria Paola Speciale Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina Viale F. Stagno d’Alcontres 31, I–98166 Messina, Italy Email: ajannelli@unime.it, mpspeciale@unime.it ABSTRACT In this paper, we present a numerical study of a new fractional vegetation-water model describing the effects of the anomalous water advection process and a variable rainfall rate on the vegetation pattern formation, in semi-arid and arid environments. Assuming the fractional order linked to the slope of the domains, the model, involving the fractional Caputo operator, describes a connection between the Klausmeier model and the generalized Klausmeier-Gray-Scott model [1]. The variable rainfall is assumed as a periodic function of time, describing the change between high and low season rains [2]. Numerical solutions show the efficiency of the method and the reliability of the new proposed model that ensures periodic solutions and vegetation patterns [3]. REFERENCES [1] Jannelli A. and Speciale M.P. Fractional Vegetation-Water Model in Arid and Semi-Arid Environments: Pattern Formation and Numerical Simulations. (2025) Submitted. [2] Jannelli A. and Speciale M.P. Variable Rainfall Rate Effects on Fractional Vegetation–WaterModel: Numerical Solution and Pattern Formation. (2025) Preprint. [3] Jannelli A. Numerical Study of a New Fractional Vegetation-Water Model. (2025) Preprint.