Quantifying the global stability and transition dynamics of a coupled human-environment system via a landscape-flux approach.

Human and environmental systems should not be viewed in isolation from each other but as a complex integrated system since humans not only influence ecosystem services and functions but also respond to changes in the ecosystem. Additionally, stochastic perturbations play a crucial role in natural systems, and stochastic factors associated with social and ecological systems can significantly affect the dynamics of coupled models, such as noise-induced tipping. In this paper, we propose a coupled human-environment model with noisy disturbances that includes the dynamics of forest conservation opinions within a population and the natural expansion and harvesting of forest ecosystems. We investigate how stochasticity triggers critical transitions between high and low forest cover states (or a stable oscillatory state) using social and ecological fitting parameters from old-growth forests in Oregon. Based on landscape-flow theory from non-equilibrium statistical mechanics, we quantify the global stability and robustness of equilibria and limit cycles using barrier height and average flux. We find that the stability of the high forest cover state weakens, and the low forest cover state becomes increasingly stable as noise intensity increases. Conversely, an increase in the intensity of injunctive social norms favors the global stability of the high forest cover state. Moreover, only a sufficiently small forest protection cost will allow forest cover to be maintained at a high level. Finally, a sensitivity analysis of the parameters of the coupled system is conducted, revealing the key factors affecting the global stability and critical transitions of high and low forest cover states.