A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications

Abstract

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find earlywarning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.

Critical slowing down as early warning for the onset of collapse in mutualistic communities

“…S8. Previous studies have shown that CSD indicators change smoothly before bifurcation points (8,41). We confirmed smooth changes in CV and AR1 in our mutualistic communities under a gradual decline of mutualistic strength (Fig.…”

“…This phenomenon of "critical slowing down" appears to be generic for a wide class of local bifurcations (8), at which the current equilibrium state of a system loses stability before being replaced by another equilibrium state. Critical slowing down may be captured by two simple statistical signals in the dynamics of complex systems (6): increasing variance and rising correlation.…”

Linear stability theory as an early warning sign for transitions in high dimensional complex systems

“…It is expected to be of particular relevance when a slow change in some external parameter which drives the system towards the bifurcation point. In this case transitions are called critical transitions and the mathematical framework used comes from dynamical systems theory [4,5,6]. This method is pertinent to systems that are dynamically effectively low dimensional in which the transition takes the form of a bifurcation captured by a robust macroscopic variable, which emerges from the micro dynamics.…”

Effects of allochthonous inputs in the control of infectious disease of prey

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