Noise-induced degradation in a coupled human-forest system: Tipping time and probability as well as risk assessments

In this letter, we revisit the stochastic model coupling forest growth with human activities developed in one of our previous papers Yu et al. (2025). It was shown there that environmental noise may trigger critical transitions from the high forest cover state to the low cover state. However, the likelihood and time for the occurrence of such tipping events remain not explicitly quantified, which are essential for anticipating ecological degradation and designing sustainable management strategies. This letter addresses this issue by examining the relationship between stochasticity and the probability or time of tipping to the degradation state (the collapse state or the extinction state). Our results indicate that the time of tipping to the degradation state falls as the noise intensity increases; but the probabilities of tipping to the collapse state and the extinction state are different: the tipping probability to the extinction state increases monotonically with noise intensity, whereas the tipping probability to the collapse state exhibits a nonmonotonic dependence on noise intensity. Moreover, we establish a risk classification for different initial conditions, finding that stochasticity exerts a dual effect on human–environment interactions: it may promote forest persistence in regions that would otherwise degrade deterministically, while increasing the degradation risk in otherwise persistent areas. Also, forest-related noise inhibits forest degradation, whereas human-related noise hastens it. These results offer valuable theoretical insights for protecting forest ecosystems, suggesting that precautionary management and early warning strategies can be formulated to avert catastrophic shifts.