Analyzing diffusive vegetation-sand model: Instability, bifurcation, and pattern formation

Gaihui Guo; Xinyue Zhang; Jichun Li; Tingting Wei; Gaihui Guo; Xinyue Zhang; Jichun Li; Tingting Wei

doi:10.3934/era.2025243

Electronic Research Archive

2025, Volume 33, Issue 9: 5426-5456. doi: 10.3934/era.2025243

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Research article

Analyzing diffusive vegetation-sand model: Instability, bifurcation, and pattern formation

1.
School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an, Shaanxi 710021, China
2.
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710126, China

Received: 26 June 2025 Revised: 07 August 2025 Accepted: 14 August 2025 Published: 15 September 2025

In this study, we explored a diffusive vegetation-sand model with Neumann boundary conditions, investigating the role of Turing instability in vegetation pattern formation. A priori estimates for steady-state solutions were established using the maximum principle and Poincaré inequality. Bifurcation analysis was performed for simple and double eigenvalue cases. By employing bifurcation theory, a local bifurcation was extended globally, and the direction of bifurcation was characterized. Double eigenvalue cases were analyzed through spatial decomposition and the implicit function theorem. Finally, numerical simulations validated and complemented the theoretical results.
- vegetation-sand model,
- turing instability,
- steady-state bifurcation,
- vegetation pattern,
- double eigenvalue
Citation: Gaihui Guo, Xinyue Zhang, Jichun Li, Tingting Wei. Analyzing diffusive vegetation-sand model: Instability, bifurcation, and pattern formation[J]. Electronic Research Archive, 2025, 33(9): 5426-5456. doi: 10.3934/era.2025243

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Abstract

In this study, we explored a diffusive vegetation-sand model with Neumann boundary conditions, investigating the role of Turing instability in vegetation pattern formation. A priori estimates for steady-state solutions were established using the maximum principle and Poincaré inequality. Bifurcation analysis was performed for simple and double eigenvalue cases. By employing bifurcation theory, a local bifurcation was extended globally, and the direction of bifurcation was characterized. Double eigenvalue cases were analyzed through spatial decomposition and the implicit function theorem. Finally, numerical simulations validated and complemented the theoretical results.

References

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