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

Hemolymph osmolality and cation concentrations in Litopenaeus vannamei during exposure to artificial sea salt or a mixed-ion solution: Relationship to potassium flux

Interest in culturing the Pacific white shrimp Litopenaeus vannamei in low-salinity and brackish-well waters has led to questions about the ability of this species to osmo- and ionoregulate in environments containing low concentrations of ions and in environments with ionic ratios that differ from those found in sea water. After seven days, hemolymph osmolality and potassium, sodium and calcium values were all significantly affected by salinity (as artificial sea salt) with values decreasing with decreasing salinity. These decreases were small, however, relative to decreases in salinity, indicating iono- and osmoregulation with adjustment for gradients. The hemolymph osmolality and sodium and calcium concentrations in shrimp exposed to either 2 g/L artificial sea salt or 2 g/L mixed-ion solution (a mixture of sodium, potassium, calcium, and magnesium chlorides that approximate the concentrations and ratios of these cations found in 2 g/L dilute seawater) did not differ significantly. However, hemolymph potassium levels were significantly lower in shrimp held in the mixed-ion environment. Potassium influx rates were similar in shrimp held in either artificial sea salt or mixed ions. The results of this study indicate that salinity affects hemolymph-cation concentrations and osmolality. Further, differential potassium-influx rates do not appear to be the basis for low hemolymph potassium levels observed in shrimp held in mixed-ion environments

Oxygen consumption of Litopenaeus vannamei juveniles in heterotrophic medium with zero water exchange Consumo de oxigênio de juvenis de Litopenaeus vannamei em meio heterotrófico sem renovação de água

This work aimed at determining the dissolved oxygen consumption rate of Litopenaeus vannamei juveniles maintained in a microbial biofloc raceway system at high density with no aeration. Three 4 L bottles were filled for each treatment, sealed hermetically, and placed in an enclosed greenhouse raceway system. Four shrimp (13.2±1.42 g) were assigned to two sets of the bottles, which underwent the following treatments: light conditions with no shrimp; dark conditions with no shrimp; light conditions with shrimp; and dark conditions with shrimp. Dissolved oxygen content was measured every 10 min for 30 min. A quadratic behavior was observed in dissolved oxygen concentration over time. Significant differences for oxigen consumption were observed only at 10 and 20 min between shrimp maintained in the dark and those under light conditions. At 10 min, a higher value was observed in shrimp maintained under light, and at 20 min, in the dark. Significant differences between 10 and 20 min and between 10 and 30 min were observed when oxygen consumption was analyzed over time in the presence of light. Under dark conditions there were significant differences only between 20 and 30 min. Lethal oxygen concentration (0.65 mg L-1) would be reached in less than one hour either under light or dark conditions with no aeration.<br>O objetivo deste trabalho foi determinar o consumo de oxigênio dissolvido (OD) de juvenis de Litopenaeus vannamei mantidos em sistema de cultivo de bioflocs bacterianos em alta densidade e ausência de aeração. Três garrafas de 4 L foram preenchidas para cada tratamento, fechadas hermeticamente e colocadas em sistema de cultivo fechado. Quatro camarões (13,2±1,42 g) foram colocados em dois dos conjuntos de garrafas. Os tratamentos aplicados foram: luminosidade, sem camarões; escuro, sem camarões; luminosidade, com camarões; escuro, com camarões. A concentração de oxigênio dissolvido foi determinada a cada 10 min durante 30 min. Foi observado um comportamento quadrático na concentração de OD ao longo do tempo. Diferenças significativas para consumo de oxigênio foram observadas apenas aos 10 e 20 min entre camarões mantidos no escuro e camarões em luminosidade. Aos 10 min, foi observada maior concentração no sistema mantido em luminosidade e, aos 20 min, no sistema mantido no escuro. Na análise do consumo de oxigênio ao longo do tempo e com luminosidade, foram constatadas diferenças significativas entre 10 e 20 min e entre 10 e 30 min. Em condições de escuridão, houve diferença significativa apenas entre 20 e 30 min. Concluiu-se que, sem aeração, a condição anóxica pode ser alcançada em menos de uma hora, tanto ao dia quanto à noite