Pattern dynamics of vegetation in ecologically fragile areas based on sparse optimal control theory

In recent years, due to the impacts of climate change and human activities, the vegetation ecosystem in ecologically fragile areas has manifested a significant phenomenon of degradation, accompanied by rich vegetation pattern structures during the degradation process. The dynamics of vegetation patterns enable both quantitative characterization and qualitative analysis of how vegetation forms and is distributed. For this purpose, this paper establishes a dynamic model of vegetation and water that incorporates cross-diffusion, investigates the conditions for Turing pattern formation through mathematical analysis, provides robustness indicators for vegetation patterns. By incorporating human activities as a control function, this study investigates the sparse optimal control issue associated with the model, revealing how to enhance ecosystem robustness through human activities from the perspective of control-related vegetation pattern formation. Then the presence of an optimal solution is investigated. Numerical simulations demonstrate the efficacy of the control strategy in terms of control effectiveness, error reduction and cost efficiency. This research provides a theoretical foundation for combating desertification in vegetation ecosystems within ecologically fragile regions.