Analyzing vegetation pattern formation through a time-ordered fractional vegetation-sand model: A spatiotemporal dynamic approach

Yimamu Maimaiti; Zunyou Lv; Ahmadjan Muhammadhaji; Wang Zhang; Yimamu Maimaiti; Zunyou Lv; Ahmadjan Muhammadhaji; Wang Zhang

doi:10.3934/nhm.2024055

Networks and Heterogeneous Media

2024, Volume 19, Issue 3: 1286-1308. doi: 10.3934/nhm.2024055

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Analyzing vegetation pattern formation through a time-ordered fractional vegetation-sand model: A spatiotemporal dynamic approach

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China
2.
Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China

Received: 07 October 2024 Revised: 03 November 2024 Accepted: 05 November 2024 Published: 12 November 2024

This paper contributes to the field by developing a fractional-order vegetation-sand model that incorporates memory effects into the traditional integer-order framework. By studying the spatiotemporal dynamics of a time-order fractional vegetation-sand model, the research aimed to deepen our understanding of the complex interactions between vegetation and sand environments, providing insights for effective management and conservation strategies in arid and semi-arid regions. First, using the linear stability theory of fractional differential equations, we conducted a stability analysis of the spatially homogeneous fractional-order vegetation-sand model and provided the parametric conditions for stability and instability. Next, we performed a stability analysis of the spatiotemporal model, utilizing Turing instability to reveal the effects of diffusion and fractional order on vegetation distribution. Through numerical simulations, we demonstrated the spatiotemporal evolution patterns of the model under different environmental conditions and discussed the implications of these dynamic changes for ecological restoration and land management.
- sand-vegetation model,
- fractional-order derivative,
- bifurcation,
- vegetation patterns
Citation: Yimamu Maimaiti, Zunyou Lv, Ahmadjan Muhammadhaji, Wang Zhang. Analyzing vegetation pattern formation through a time-ordered fractional vegetation-sand model: A spatiotemporal dynamic approach[J]. Networks and Heterogeneous Media, 2024, 19(3): 1286-1308. doi: 10.3934/nhm.2024055

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Abstract

This paper contributes to the field by developing a fractional-order vegetation-sand model that incorporates memory effects into the traditional integer-order framework. By studying the spatiotemporal dynamics of a time-order fractional vegetation-sand model, the research aimed to deepen our understanding of the complex interactions between vegetation and sand environments, providing insights for effective management and conservation strategies in arid and semi-arid regions. First, using the linear stability theory of fractional differential equations, we conducted a stability analysis of the spatially homogeneous fractional-order vegetation-sand model and provided the parametric conditions for stability and instability. Next, we performed a stability analysis of the spatiotemporal model, utilizing Turing instability to reveal the effects of diffusion and fractional order on vegetation distribution. Through numerical simulations, we demonstrated the spatiotemporal evolution patterns of the model under different environmental conditions and discussed the implications of these dynamic changes for ecological restoration and land management.

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