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

Slower recovery in space before collapse of connected populations

Slower recovery from perturbations near a tipping point and its indirect signatures in fluctuation patterns have been suggested to foreshadow catastrophes in a wide variety of systems. Recent studies of populations in the field and in the laboratory have used time-series data to confirm some of the theoretically predicted early warning indicators, such as an increase in recovery time or in the size and timescale of fluctuations. However, the predictive power of temporal warning signals is limited by the demand for long-term observations. Large-scale spatial data are more accessible, but the performance of warning signals in spatially extended systems needs to be examined empirically. Here we use spatially extended yeast populations, an experimental system with a fold bifurcation (tipping point), to evaluate early warning signals based on spatio-temporal fluctuations and to identify a novel spatial warning indicator. We found that two leading indicators based on fluctuations increased before collapse of connected populations; however, the magnitudes of the increases were smaller than those observed in isolated populations, possibly because local variation is reduced by dispersal. Furthermore, we propose a generic indicator based on deterministic spatial patterns, which we call ‘recovery length’. As the spatial counterpart of recovery time, recovery length is the distance necessary for connected populations to recover from spatial perturbations. In our experiments, recovery length increased substantially before population collapse, suggesting that the spatial scale of recovery can provide a superior warning signal before tipping points in spatially extended systems.United States. National Institutes of Health (NIH R00 GM085279-02)United States. National Institutes of Health (NIH DP2)Alfred P. Sloan FoundationNational Science Foundation (U.S.

Recovery and resilience of tropical forests after disturbance

The time taken for forested tropical ecosystems to re-establish post-disturbance is of widespread interest. Yet to date there has been no comparative study across tropical biomes to determine rates of forest re-growth, and how they vary through space and time. Here we present results from a meta-analysis of palaeoecological records that use fossil pollen as a proxy for vegetation change over the past 20,000 years. A total of 283 forest disturbance and recovery events, reported in 71 studies, are identified across four tropical regions. Results indicate that forests in Central America and Africa generally recover faster from past disturbances than those in South America and Asia, as do forests exposed to natural large infrequent disturbances compared with post-climatic and human impacts. Results also demonstrate that increasing frequency of disturbance events at a site through time elevates recovery rates, indicating a degree of resilience in forests exposed to recurrent past disturbance

Avoiding, diagnosing and treating well leg compartment syndrome after pelvic surgery

Background Patients undergoing prolonged pelvic surgery may develop compartment syndrome of one or both lower limbs in the absence of direct trauma or pre‐existing vascular disease (well leg compartment syndrome). This condition may have devastating consequences for postoperative recovery, including loss of life or limb, and irreversible disability. Methods These guidelines represent the collaboration of a multidisciplinary group of colorectal, vascular and orthopaedic surgeons, acting on behalf of their specialty associations in the UK and Ireland. A systematic analysis of the available peer‐reviewed literature was undertaken to provide an evidence base from which these guidelines were developed. Results These guidelines encompass the risk factors (both patient‐ and procedure‐related), diagnosis and management of the condition. Key recommendations for the adoption of perioperative strategies to facilitate prevention and effective treatment of well leg compartment syndrome are presented. Conclusion All surgeons who carry out abdominopelvic surgical procedures should be aware of well leg compartment syndrome, and instigate policies within their own institution to reduce the risk of this potentially life‐changing complication

Evidence for 'critical slowing down' in seagrass:a stress gradient experiment at the southern limit of its range

The theory of critical slowing down, i.e. the increasing recovery times of complex systems close to tipping points, has been proposed as an early warning signal for collapse. Empirical evidence for the reality of such warning signals is still rare in ecology. We studied this on Zostera noltii intertidal seagrass meadows at their southern range limit, the Banc d'Arguin, Mauritania. We analyse the environmental covariates of recovery rates using structural equation modelling (SEM), based on an experiment in which we assessed whether recovery after disturbances (i.e. seagrass & infauna removal) depends on stress intensity (increasing with elevation) and disturbance patch size (1 m(2) vs. 9 m(2)). The SEM analyses revealed that higher biofilm density and sediment accretion best explained seagrass recovery rates. Experimental disturbances were followed by slow rates of recovery, regrowth occurring mainly in the coolest months of the year. Macrofauna recolonisation lagged behind seagrass recovery. Overall, the recovery rate was six times slower in the high intertidal zone than in the low zone. The large disturbances in the low zone recovered faster than the small ones in the high zone. This provides empirical evidence for critical slowing down with increasing desiccation stress in an intertidal seagrass system

Detecting the Collapse of Cooperation in Evolving Networks

The sustainability of biological, social, economic and ecological communities is often determined by the outcome of social conflicts between cooperative and selfish individuals (cheaters). Cheaters avoid the cost of contributing to the community and can occasionally spread in the population leading to the complete collapse of cooperation. Although such collapse often unfolds unexpectedly, it is unclear whether one can detect the risk of cheater’s invasions and loss of cooperation in an evolving community. Here, we combine dynamical networks and evolutionary game theory to study the abrupt loss of cooperation with tools for studying critical transitions. We estimate the risk of cooperation collapse following the introduction of a single cheater under gradually changing conditions. We observe an increase in the average time it takes for cheaters to be eliminated from the community as the risk of collapse increases. We argue that such slow system response resembles slowing down in recovery rates prior to a critical transition. In addition, we show how changes in community structure reflect the risk of cooperation collapse. We find that these changes strongly depend on the mechanism that governs how cheaters evolve in the community. Our results highlight novel directions for detecting abrupt transitions in evolving networks

Building connectomes using diffusion MRI: why, how and but

Why has diffusion MRI become a principal modality for mapping connectomes in vivo? How do different image acquisition parameters, fiber tracking algorithms and other methodological choices affect connectome estimation? What are the main factors that dictate the success and failure of connectome reconstruction? These are some of the key questions that we aim to address in this review. We provide an overview of the key methods that can be used to estimate the nodes and edges of macroscale connectomes, and we discuss open problems and inherent limitations. We argue that diffusion MRI-based connectome mapping methods are still in their infancy and caution against blind application of deep white matter tractography due to the challenges inherent to connectome reconstruction. We review a number of studies that provide evidence of useful microstructural and network properties that can be extracted in various independent and biologically-relevant contexts. Finally, we highlight some of the key deficiencies of current macroscale connectome mapping methodologies and motivate future developments

Assessing ecological resilience to human induced environmental change in shallow lakes

Sudden unpredictable changes in ecosystems are an increasing source of concern because of their inherent unpredictability and the difficulties involved in restoration. Our understanding of the changes that occur across different trophic levels and the form of this change is lacking. This is especially true of large shallow lakes, where characteristics such as fetch and depth are close to theoretical boundary values for hysteretic behaviour. The development of reliable indicators capable of predicting these changes has been the focus of much research in recent years. The success of these early warning indicators (EWIs) has so far been mixed. There remain many unknowns about how they perform under a wide variety of conditions and parameters. Future climate change is predicted to have a wide range of impacts through the interaction of combined pressures, making the understanding of EWIs and the in-lake processes that occur during regime shifts imperative. Loch Leven, Scotland, UK, is a large shallow lake with a history of eutrophication, research and management and as such is an ideal study site to better understand resilience and regime shifts under a range of interacting stressors. The objectives of this research are to: (1) analyse long term data to identify the occurrence of common tipping points within the chemical (water column nutrient concentrations) and biological (macrophytes, phytoplankton, zooplankton) components of the loch, then test these tipping points using five statistical early warning indicators (EWIs) across multiple rolling window sizes; and (2) quantify the changes in lake ecology using a before/after analysis and testing for non-linearity, combined with modelling using the aquatic ecosystem process model PCLake to determine the level of resilience following a regime shift during recovery from eutrophication; (3) using PCLake, examine the sensitivity of Loch Leven to regime shifts in the face of predicted environmental change (e.g. climate change, nutrient pollution). Statistical analysis identified tipping points across all trophic levels included, from physical and chemical variables through to apex predators. The success of EWIs in predicting the tipping points was highly dependent on the number of EWIs used, with window size having a smaller impact. The 45% window size had the highest overall accuracy across all EWIs but only detected 16.5% more tipping points than the window size with the lowest overall accuracy. Differences between individual EWI performance and usage of them as a group was substantial with a 29.7% increase between the two. In both individual and group use of EWIs, false positives (early warning without a tipping point) were more common than true positives (tipping point preceded by EWI), creating significant doubts about their reliability as management tools. Significant change was seen across multiple variables and trophic levels in the before/after analysis following sudden recovery from eutrophication, with most variables also showing evidence of non-linear change. Modelling of responses to nutrient loading for chlorophyll, zooplankton and macrophytes, under states from before and after the shift, indicate hysteresis and thus the presence of feedback mechanisms. The modelling of responses to nutrient loading and predicted climate change in temperature and precipitation demonstrated that increases in temperature and decreases in summer precipitation individually had large impacts on chlorophyll and zooplankton at medium to high phosphorus (P) loads. However, modelling of the combined effects of these changes resulted in the highest lake chlorophyll concentrations of all tested scenarios. At low P loads higher temperatures and increased winter precipitation had the greatest impact on system resilience with a lower Critical Nutrient Load (CNL). The difference between chlorophyll and zooplankton as opposed to macrophytes was in the presence of a lower CNL for the increased winter precipitation-only scenarios which was not seen in the macrophytes. This highlights the potential role of high winter inputs potentially loaded with particulate matter in reducing resilience at lower P loads. This research has highlighted the vulnerability and low resilience of Loch Leven to environmental change. The presence of multiple tipping points and high levels of EWI activity show a high level of flexibility in the system. Coupled with the occurrence of widespread trophic change during a sudden recovery and a small level of hysteresis and high levels of sensitivity to climate change, the low levels of resilience become clear. The impact of lake-specific characteristics such as moderate depth, large fetch and a heterogeneous bed morphology is particularly evident in the limitations on macrophyte cover and the reliance on zooplankton to determine the hysteresis offset (amount of phosphorus (P) loading between the two CNL). The presence of these characteristics can be used to identify other lakes vulnerable to change. Improving the predictive capabilities of resilience indicators such as EWIs, and better understanding of the ecological changes that occur during non-linear change in response to recovery and climate change, can help target relevant ecosystem components for preventative management. These actions may become necessary under even the most conservative estimates of environmental change