Alessandra Jannelli†, Maria Paola Speciale‡
Abstract: In this paper, we introduce a space-fractional mathematical model to describe the dynamics and interactions between vegetation and water in arid and semi-arid environments, both on flat and sloped terrains. The fractional model links two processes, such as water flows over in- clined surfaces and water spreads on flat terrain, enabling the study of pattern formation with different slopes of the domain. The first process is usually described by the Klausmeier model, and the second one by the Klausmeier-Gray-Scott model, which describes water diffusion. In the proposed fractional model, the fractional Caputo derivative term allows for modeling an anomalous transport phenomenon and the non-locality of the water advection process. Oscillatory dynamics and vegetation patterns are demonstrated through numerical simulations, using a migration speed derived from a stability analysis and resulting as a function of the fractional parameter. The computational results demonstrate the robustness and effectiveness of the fractional model, which captures ecological behaviors and anomalous transport mechanisms in different terrains.
Keywords: pattern dynamics, Klausmeier and Klausmeier-Gray-Scott models, space-fractional Caputo derivative, Hopf bifurcation of the migration speed, explicit rectangular method.
MSC: 93C20, 26A33, 34C23,65C20.
DOI 10.56082/annalsarscimath.2025.3.227
† ajannelli@unime.it, Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina
‡ mpspeciale@unime.it, Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 17 no 3, 2025
