Abstract
Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. The difficulty is rooted in the lack of mathematical tools to analyze systems with high dimensionality, nonlinearity, and stochastic effects. Here we show that nucleation theory can be employed to advance resilience restoration in spatially-embedded ecological systems. We find that systems may exhibit single-cluster or multi-cluster phases depending on their sizes and noise strengths. We also discover a scaling law governing the restoration time for arbitrary system sizes and noise strengths in two-dimensional systems. This approach is not limited to ecosystems and has applications in various dynamical systems, from biology to infrastructural systems.
Highlights
Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples
Environmental conditions should be recovered to the critical point where the undesired state is destabilized and resilience restoration would occur
It has been shown that noise can induce transitions between alternative stable states and the required time to transition has been established by computing the mean first passage time (MFPT)[2,19,20,21]
Summary
Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. Hindered by the high-dimensionality of interaction topology and the nonlinear evolution dynamics, few analyses of critical transitions and resilience restoration had been done directly on high-dimensional systems consisting of a great number of participants until the effective reduction theory was recently developed by Gao et al.[3] This mean-field theory can be used to effectively reduce a multi-dimensional complex system to a one-dimensional system by capturing the average activities of the original system. Our study discovers that noise eventually eliminates the deterministic critical threshold, and the recovery of the entire system from the dysfunctional state is possible in the presence of perturbations as long as noise can trigger the transition for just one component This scenario is likely to occur when the system is close to the deterministic threshold where the undesired state loses stability. The farther away the system is from this point, the more difficult it is for noise to induce the transition
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