Spatiotemporal complexity of vegetation dynamics in view of optimal control

The investigation of vegetation pattern transitions in arid and semi-arid areas plays a pivotal role in evaluating ecosystem health and averting ecosystem degradation. Nonetheless, extant research predominantly concentrates on transitions triggered by natural phenomena such as precipitation, with scant attention given to reversing pattern structures via intervention strategies. In this study, we leverage optimal control theory and incorporate human activities, characterized by significant controllability, as control variables within the Rietkerk model, to conduct an in-depth analysis of pattern transitions. Our findings reveal that under challenging natural conditions, it is feasible to induce pattern transitions by devising appropriate spatiotemporal distributions of human activities. Additionally, our research indicates that enhancing sparsity within the anticipated error margin can substantially reduce control expenditures without detracting from the efficacy of the pattern transitions. In essence, this paper introduces a novel methodological approach for examining pattern transitions from a control standpoint, offering fresh perspectives for the development of strategies aimed at desertification mitigation and control in arid landscapes.