Numerical bifurcation analysis and pattern formation in a minimal reaction–diffusion model for vegetation

• Published in: Journal of theoretical biology 2022-03, Vol.536, p.110997-110997, Article 110997
• Main Authors: Kabir, M. Humayun, Gani, M. Osman
• Format: Article
• Language: English
• Subjects: Biomass; Desert Climate; Diffusion; Ecosystem; Klausmeier model; Minimal model; Models, Biological; Numerical bifurcation analysis; Reaction–diffusion system; Vegetation patterns
• ISSN: 0022-5193; 1095-8541
• DOI: 10.1016/j.jtbi.2021.110997

•Extension of the classical Klausmeier model focusing the water flow in flat terrains.•Local dynamics of the diffusive Klausmeier model considering the rainfall rate as a free parameter.•Numerical bifurcation analysis of the diffusive Klausmeier model reveals that the occurrence of spatially inhomogeneous stationary solutions.•Occurrence of vegetation patterns due to the Turing instability as found in field observation.•Minimality of the diffusive Klausmeier model from the viewpoint of a two component reaction-diffusion system. Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier in 1999 (Klausmeier, 1999) that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction–diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction–diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction–diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction–diffusion system.