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<p>Vegetation pattern transitions in arid and semi-arid regions are widely regarded as an important pathway for reversing ecosystem desertification, and the time scale required for intervention plays a critical role in determining the feasibility and implementation cost of ecological restoration. However, quantitative prediction and optimization of this time scale remain relatively limited. Motivated by this, we adopt the Klausmeier-Gray-Scott vegetation-water model and introduce a direct human intervention term into the vegetation dynamics. Using vegetation patterns as the control medium, we formulate a time-optimal control problem with a free terminal time, in which the objective function simultaneously accounts for the pattern-matching error, control cost, and terminal-time penalty. Furthermore, the concept of spatial zonal management in ecological restoration is incorporated to endow the control variable with practical interpretability. We first establish the existence of optimal solutions for the proposed control problem. Subsequently, by applying Pontryagin’s minimum principle and Karush–Kuhn–Tucker (KKT) theory, the first-order necessary optimality conditions are systematically derived, and numerical solutions are obtained via a joint gradient descent method. The results show that, during the transition from low- to high-density vegetation patterns, a minimum feasible control time exists and can be predicted; the terminal time penalty significantly shortens the control duration while maintaining pattern matching accuracy, and the trade-offs among control error, intervention intensity, and control time are further revealed. By incorporating Google Maps satellite imagery, illustrative studies of real vegetation patterns in regions such as Niger, Sudan, and Australia are carried out to validate the effectiveness of the proposed method and its ability to predict intervention durations. The results show that, under optimal intervention conditions, vegetation pattern transitions that would normally require several years can be accomplished within a matter of months. These findings should be interpreted as model-based theoretical estimates rather than empirically validated field predictions. This study provides a quantitative modeling framework for evaluating time scales in rapid vegetation pattern restoration and ecological management.</p>

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From undesirable to desirable vegetation patterns: How long does it take under optimal intervention?

  • 期刊论文

  • 发表日期：

    2026年6月14日

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  • Na Zheng,
  • Zhen Wang,
  • Li Li

摘要

Vegetation pattern transitions in arid and semi-arid regions are widely regarded as an important pathway for reversing ecosystem desertification, and the time scale required for intervention plays a critical role in determining the feasibility and implementation cost of ecological restoration. However, quantitative prediction and optimization of this time scale remain relatively limited. Motivated by this, we adopt the Klausmeier-Gray-Scott vegetation-water model and introduce a direct human intervention term into the vegetation dynamics. Using vegetation patterns as the control medium, we formulate a time-optimal control problem with a free terminal time, in which the objective function simultaneously accounts for the pattern-matching error, control cost, and terminal-time penalty. Furthermore, the concept of spatial zonal management in ecological restoration is incorporated to endow the control variable with practical interpretability. We first establish the existence of optimal solutions for the proposed control problem. Subsequently, by applying Pontryagin’s minimum principle and Karush–Kuhn–Tucker (KKT) theory, the first-order necessary optimality conditions are systematically derived, and numerical solutions are obtained via a joint gradient descent method. The results show that, during the transition from low- to high-density vegetation patterns, a minimum feasible control time exists and can be predicted; the terminal time penalty significantly shortens the control duration while maintaining pattern matching accuracy, and the trade-offs among control error, intervention intensity, and control time are further revealed. By incorporating Google Maps satellite imagery, illustrative studies of real vegetation patterns in regions such as Niger, Sudan, and Australia are carried out to validate the effectiveness of the proposed method and its ability to predict intervention durations. The results show that, under optimal intervention conditions, vegetation pattern transitions that would normally require several years can be accomplished within a matter of months. These findings should be interpreted as model-based theoretical estimates rather than empirically validated field predictions. This study provides a quantitative modeling framework for evaluating time scales in rapid vegetation pattern restoration and ecological management.