From tipping point to tipping set: Extending the concept of regime shift to uncertain dynamics for real-world applications

The concept of regime shift generally refers to a deterministic dynamical system switching from an attraction basin to another after an isolated perturbation. However, when the dynamics is constantly submitted to random perturbations, the system can go back and forth between attraction basins and the usual concept seems inadequate. To address this issue, we consider a stochastic dynamical system and we assume that its functioning is satisfactory in a subset of its state space, called the satisfaction set. We define regimes with respect to the propensity of the system to become and to remain satisfactory. We investigate two indicators available in the literature: (i) the first-exit time from the satisfaction set and (ii) the sojourn time in the satisfaction set. Using statistics on these indicators, we define regimes of durable or resilient satisfaction. Applying the same definitions to the dissatisfaction set, we define durable or resilient dissatisfaction. Our results show the emergence of a tipping set, equivalent to tipping point in the deterministic case. We illustrate our approach using three different types of dynamics: two theoretical models based on the exploitation of natural resources and predator–prey dynamics, as well as the eutrophication of the French Lake Bourget.