A mathematical framework for critical transitions: normal forms, variance and applications

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 early-warning 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.Comment: minor corrections to previous versio

The importance of diastolic velocities in the detection of celiac and mesenteric artery disease by duplex ultrasound.

AbstractPurpose: To assess the predictive value of ultrasound duplex scanning in the detection of superior mesenteric artery (SMA) and celiac artery (CA) occlusive disease.Methods: Thirty-nine patients in whom mesenteric ischemia was suspected were examined with duplex scan and arteriography.Results: Duplex scan visualized both vessels in all examined patients. An absent Doppler signal from an adequately visualized SMA or CA was a pathognomonic sign for total occlusion of the arteries. Elevated diastolic velocities (peak diastolic velocity >70 cm/sec for SMA and >100 cm/sec for CA) were accurate predictors of arteriographically confirmed stenoses ≥50%. Increased blood velocity was occasionally observed in a thyrotoxic patient with malabsorption-induced weight loss and abdominal pain but arteriographically-normal SMA. Further examinations of hyperthyroid patients showed systematic elevation of peak systolic velocity in the SMA greater than 300 cm/sec but normal diastolic velocities and resistivity indexes.Conclusion: Duplex ultrasound is a highly selective and specific method for detection of SMA and CA occlusive disease. The most accurate predictors of the disease were increased diastolic velocities, which in contrast to peak systolic velocity also correctly classified thyrotoxic patients. The method can be used before arteriographic evaluation as a screening examination of patients in whom mesenteric ischemia is suspected. (J Vasc Surg 1997 26:288-93.