Identifying, discerning, and managing bifurcation-induced transitions in complex systems

Abstract

Bifurcation-induced transitions from one stable steady state to another alternative state occur in a broad range of dynamical systems including ecosystems, f inancial markets, and climate systems. Such transitions can be catastrophic or critical (a large and sudden change in the state of a system) or non-catastrophic/ non-critical (a gradual change in the state of a system), but are mostly unwanted and are typically di cult to reverse. Hence, understanding critical transitions or tipping points via mathematical modelling and empirical data, and their early predictions are crucial to managing catastrophes. This has led researchers to develop statistical methods which attempt to predict these transitions-so called Early Warning Signals (EWSs)- which have had mixed success in predicting whether asystemisapproachingatransition, andcannotdiscernwhetheranapproaching transition is catastrophic (and therefore should be avoided if at all possible) or not. This thesis aims to better understand sudden transitions in complex systems, develop novel indicators to detect and distinguish a critical transition from other bifurcation-induced transitions, and further avert or mitigate it. In complex systems, fluctuations are characterized by time-varying frequencies and large variances with correlations that can extend over short- or long-range. We derive the approximate Fokker-Planck equation from the Langevin equation of a spatially extended system perturbed by Ornstein-Uhlenbeck (OU) noise using the unified color noise approximation. Further, we theoretically interpret the role of OU noise on the resilience of a bistable spatial system. We also propose spatial mutual information as an EWS after examining its e↵ectiveness in simulated and real data. Next, we take a new approach to predicting transitions in complex systems while also discerning the types of transitions, using time series data generated by nine mathematical models to train a machine learning (ML) based EWS indicator (EWSNet), and then applying this to real world data sets. We have also developed a novel detectionmethod trainedusingmathematicalmodels to detect critical transitions in spatial systems-the Early Warning Signal Network (SEWSNet). However, ML methods rely largely on the data that is fed to the model as input and accordingly can be an asset or nuisance. Having this in mind, we have taken a sampling approach that forces the ML model towards learning features that can aid in distinguishing catastrophic and non-catastrophic transitions. Further, we mitigate critical transition in mutualistic networks by modeling it under the framework of the human-environment system using opinion dynamics from evolutionary game theory. We demonstrate that applying social norms at pollinator nodes e↵ectively prevents sudden community collapse in networks with various topologies. In this study, we introduce an optimal conservation strategy (OCS) based on network structure to identify the optimal nodes for norm implementation, thereby averting community collapse. We find that networks with intermediate levels of nestedness require conservation at a minimal number of nodes to prevent collapse. We validate the robustness of the OCS through simulations and empirical networks of diverse complexities, across a wide range of system parameters. Finally, this thesis contributes to the study of mutualistic networks by quantifying and minimizing uncertainty in tipping thresholds for this significant class of ecological networks. Through Bayesian inference, we demonstrate how uncertainty in tipping points can be narrowed, providing estimates of tipping bounds. This thesis seeks to enhance our understanding of critical transitions and develop novel indicators to distinguish critical transitions from other bifurcation-induced transitions, a gap that was yet to be addressed. Moving forward, the work focuses on mitigating critical transitions in higher-dimensional networks by creating optimal conservation strategies that minimize costs by conserving only the minimum number of nodes necessary via OCS. Finally, the thesis explores the impact of uncertainty on critical transitions in networks, opening up avenues for future research, such as developing more improved methods to detect and reduce uncertainty in systems experiencing tipping points.