Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

Network based early warning indicators of vegetation changes in a land–atmosphere model

Numerous model studies demonstrate that ecosystems might not shift smoothly with a gradual change in resource concentration. At specific points, vegetation can suddenly shift from one stable state to another. To predict such undesirable shifts, statistical indicators are proposed for early warning prediction. These so-called classical indicators can address whether vegetation state is moving towards the tipping point of an abrupt transition, however when the transition will occur is hard to predict. Recent studies suggest that complex network based indicators can improve early warning signals of abrupt transitions in complex dynamic systems. In this study, both classical and network based indicators are tested in a coupled land–atmosphere ecological model in which a scale-dependent hydrology-infiltration feedback and a large scale vegetation–precipitation feedback are represented. Multiple biomass equilibria are found in the model and abrupt transitions can occur when rainfall efficiency is decreased. Interaction network based indicators of these transitions are compared with classical indicators, such as the lag-1 autocorrelation and Moran's coefficient, with particular focus on the transition associated with desertification. Two criteria are used to evaluate the quality of these early warning indicators and several high quality network based indicators are identified.     

Resilience and tipping points of an exploited fish population over six decades

Complex natural systems with eroded resilience, such as populations, ecosystems and socio‐ecological systems, respond to small perturbations with abrupt, discontinuous state shifts, or critical transitions. Theory of critical transitions suggests that such systems exhibit fold bifurcations featuring folded response curves, tipping points and alternate attractors. However, there is little empirical evidence of fold bifurcations occurring in actual complex natural systems impacted by multiple stressors. Moreover, resilience of complex systems to change currently lacks clear operational measures with generic application. Here, we provide empirical evidence for the occurrence of a fold bifurcation in an exploited fish population and introduce a generic measure of ecological resilience based on the observed fold bifurcation attributes. We analyse the multivariate development of Barents Sea cod (Gadus morhua), which is currently the world's largest cod stock, over six decades (1949–2009), and identify a population state shift in 1981. By plotting a multivariate population index against a multivariate stressor index, the shift mechanism was revealed suggesting that the observed population shift was a nonlinear response to the combined effects of overfishing and climate change. Annual resilience values were estimated based on the position of each year in relation to the fitted attractors and assumed tipping points of the fold bifurcation. By interpolating the annual resilience values, a folded stability landscape was fit, which was shaped as predicted by theory. The resilience assessment suggested that the population may be close to another tipping point. This study illustrates how a multivariate analysis, supported by theory of critical transitions and accompanied by a quantitative resilience assessment, can clarify shift mechanisms in data‐rich complex natural systems.     

Rate of forcing and the forecastability of critical transitions

Critical transitions are qualitative changes of state that occur when a stochastic dynamical system is forced through a critical point. Many critical transitions are preceded by characteristic fluctuations that may serve as model‐independent early warning signals, implying that these events may be predictable in applications ranging from physics to biology. In nonbiological systems, the strength of such early warning signals has been shown partly to be determined by the speed at which the transition occurs. It is currently unknown whether biological systems, which are inherently high dimensional and typically display low signal‐to‐noise ratios, also exhibit this property, which would have important implications for how ecosystems are managed, particularly where the forces exerted on a system are anthropogenic. We examine whether the rate of forcing can alter the strength of early warning signals in (1) a model exhibiting a fold bifurcation where a state shift is driven by the harvesting of individuals, and (2) a model exhibiting a transcritical bifurcation where a state shift is driven by increased grazing pressure. These models predict that the rate of forcing can alter the detectability of early warning signals regardless of the underlying bifurcation the system exhibits, but that this result may be more pronounced in fold bifurcations. These findings have important implications for the management of biological populations, particularly harvested systems such as fisheries, and suggest that knowing the class of bifurcations a system will manifest may help discriminate between true‐positive and false‐positive signals.     

Moderate disturbances accelerate forest transition dynamics under climate change in the temperate–boreal ecotone of eastern North America

Several temperate tree species are expected to migrate northward and colonize boreal forests in response to climate change. Tree migrations could lead to transitions in forest types, but these could be influenced by several non‐climatic factors, such as disturbances and soil conditions. We analysed over 10,000 forest inventory plots, sampled from 1970 to 2018 in meridional Québec, Canada, to identify what environmental conditions promote or prevent regional‐scale forest transitions. We used a continuous‐time multi‐state Markov model to quantify the probabilities of transitions between forest states (temperate, boreal, mixed, pioneer) as a function of climate (mean temperature and climate moisture index during the growing season), soil conditions (pH and drainage) and disturbances (severity levels of natural disturbances and logging). We further investigate how different disturbance types and severities impact forests' short‐term transient dynamics and long‐term equilibrium using properties of Markov transition matrices. The most common transitions observed during the study period were from mixed to temperate states, as well as from pioneer to boreal forests. In our study, transitions were mainly driven by natural and anthropogenic disturbances and secondarily by climate, whereas soil characteristics exerted relatively minor constraints. While major disturbances only promoted transitions to the pioneer state, moderate disturbances increased the probability of transition from mixed to temperate states. Long‐term projections of our model under the current environmental conditions indicate that moderate disturbances would promote a northward shift of the temperate forest. Moreover, disturbances reduced turnover and convergence time for all transitions, thereby accelerating forest dynamics. Contrary to our expectation, mixed to temperate transitions were not driven by temperate tree recruitment but by mortality and growth. Overall, our results suggest that moderate disturbances could catalyse rapid forest transitions and accelerate broad‐scale biome shifts.     

Review on detection of critical transition in ecosystems

当一个存在多稳态的生态系统临近突变阈值点时, 外界条件即使发生一个微小变化, 也会引发生态系统的剧烈响应, 使之进入结构和功能截然不同的另一稳定状态, 这种现象称为重大突变(critical transition)。重大突变所导致的稳态转换总是伴随着生态系统服务的急剧变化, 可能对人类可持续发展产生重大影响。预测生态系统突变的发生非常困难, 但科学家在此领域的大量研究结果表明, 通过监测一些通用指标可以判断生态系统是否不断临近重大突变阈值点, 进而可以进行生态系统重大突变预警。该文对近年来生态系统重大突变检测领域所取得的成果进行总结与归纳, 论述了生态系统重大突变的产生机制及其后果, 介绍了生态系统突变预警信号提取的理论基础, 从时间和空间两个维度总结了近年来生态系统重大突变预警信号的提取方法, 概述了当前研究面临的挑战, 指出生态系统突变预警信号的检测应充分利用时空动态数据, 并且联合多个指标, 从多个角度进行综合预警, 此外, 还应重视生态系统结构与重大突变之间的关系, 增强生态系统突变预警能力。     

Ecosystems off track: rate‐induced critical transitions in ecological models

Theory suggests that gradual environmental change may erode the resilience of ecosystems and increase their susceptibility to critical transitions. This notion has received a lot of attention in ecology in recent decades. An important question receiving far less attention is whether ecosystems can cope with the rapid environmental changes currently imposed. The importance of this question was recently highlighted by model studies showing that elevated rates of change may trigger critical transitions, whereas slow environmental change would not. This paper aims to provide a mechanistic understanding of these rate‐induced critical transitions to facilitate identification of rate sensitive ecosystems. Analysis of rate sensitive ecological models is challenging, but we demonstrate how rate‐induced transitions in an elementary model can still be understood. Our analyses reveal that rate‐induced transitions 1) occur if the rate of environmental change is high compared to the response rate of ecosystems, 2) are driven by rates, rather than magnitudes, of change and 3) occur once a critical rate of change is exceeded. Disentangling rate‐induced transitions from classical transitions in observations would be challenging. However, common features of rate‐sensitive models suggest that ecosystems with coupled fast–slow dynamics, exhibiting repetitive catastrophic shifts or displaying periodic spatial patterns are more likely to be rate sensitive. Our findings are supported by experimental studies showing rate‐dependent outcomes. Rate sensitivity of models suggests that the common definition of ecological resilience is not suitable for a subset of real ecosystems and that formulating limits to magnitudes of change may not always safeguard against ecosystem degradation. Synthesis Understanding and predicting ecosystem response to environmental change is one of the key challenges in ecology. Model studies have suggested that slow, gradual environmental change beyond some critical threshold can trigger so‐called critical transitions and abrupt ecosystem degradation. An important question remains however whether ecosystems can cope with the ongoing rapid anthropogenic environmental changes to which they are currently imposed. In this study we demonstrate that in some ecological models elevated rates of change can trigger critical transitions even if slow environmental change of the same magnitude would not. Such rateinduced critical transitions in models suggest that concepts like resilience and planetary boundaries may not always be sufficient to explain and prevent ecosystem degradation.     

The sudden collapse of pollinator communities

Declines in pollinator populations may harm biodiversity and agricultural productivity. Little attention has, however, been paid to the systemic response of mutualistic communities to global environmental change. Using a modelling approach and merging network theory with theory on critical transitions, we show that the scale and nature of critical transitions is likely to be influenced by the architecture of mutualistic networks. Specifically, we show that pollinator populations may collapse suddenly once drivers of pollinator decline reach a critical point. A high connectance and/or nestedness of the mutualistic network increases the capacity of pollinator populations to persist under harsh conditions. However, once a tipping point is reached, pollinator populations collapse simultaneously. Recovering from this single community‐wide collapse requires a relatively large improvement of conditions. These findings may have large implications for our view on the sustainability of pollinator communities and the services they provide.     

Local facilitation, bistability and transitions in arid ecosystems

Arid ecosystems are liable to undergo sudden discontinuous transitions from a vegetated to a desert state as a result of human pressure and climate change. A predictive framework about the conditions under which such transitions occur is lacking. Here, we derive and analyze a general model describing the spatial dynamics of vegetation in arid ecosystems considering local facilitation as an essential process. We investigate the conditions under which continuous or discontinuous transitions from a vegetated to a desert state are likely to occur. We focus on arid ecosystems but our approach is sufficiently general to be applied to other ecosystems with severe environmental conditions. The model exhibits bistability and vegetation patchiness. High local facilitation decreases the risk of discontinuous transitions. Moreover, for arid ecosystems where local facilitation is a driving process, vegetation patchiness indicates proximity to a transition point, but does not allow distinguishing between continuous and discontinuous transitions.     

The impact of individual variation on abrupt collapses in mutualistic networks

Individual variation is central to species involved in complex interactions with others in an ecological system. Such ecological systems could exhibit tipping points in response to changes in the environment, consequently leading to abrupt transitions to alternative, often less desirable states. However, little is known about how individual trait variation could influence the timing and occurrence of abrupt transitions. Using 101 empirical mutualistic networks, I model the eco-evolutionary dynamics of such networks in response to gradual changes in strength of co-evolutionary interactions. Results indicated that individual variation facilitates the timing of transition in such networks, albeit slightly. In addition, individual variation significantly increases the occurrence of large abrupt transitions. Furthermore, topological network features also positively influence the occurrence of such abrupt transitions. These findings argue for understanding tipping points using an eco-evolutionary perspective to better forecast abrupt transitions in ecological systems.     

The dynamical implications of human behaviour on a social-ecological harvesting model

The dynamic aspects of human harvesting behaviour are often overlooked in resource management, such that models often neglect the complexities of dynamic human effort. Some researchers have recognized this, and a recent push has been made to understand how human behaviour and ecological systems interact through dynamic social-ecological systems. Here, we use a recent example of a social-ecological dynamical systems model to investigate the relationship between harvesting behaviour and the dynamics and stability of a harvested resource, and search for general rules in how relatively simple human behaviours can either stabilize or destabilize resource dynamics and yield. Our results suggest that weak to moderate behavioural and effort responses tend to stabilize dynamics by decreasing return times to equilibria or reducing the magnitude of cycles; however, relatively strong human impacts can readily lead to human-driven cycles, chaos, long transients and alternate states. Importantly, we further show that human-driven cycles are characteristically different from typical resource-driven cycles and, therefore, may be differentiated in real ecosystems. Given the potentially dramatic implications of harvesting on resource dynamics, it becomes critical to better understand how human behaviour determines harvesting effort through dynamic social-ecological systems.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

Interspecific facilitation and critical transitions in arid ecosystems

Climate change and intensified land‐use impose severe stress on arid ecosystems, resulting in relatively rapid degradation which is difficult to reverse. To prevent such critical transitions it is crucial to detect early warning signals. Increased ‘patchiness’– smaller and fewer vegetated patches – is thought to be such a signal, but the underlying mechanisms are still poorly understood. Facilitation between plants is known to be an important mechanism driving the patchiness of the vegetation, but we lack understanding of how interactions between plants change in response to combined effects of drought and consumer pressure – the main stressors in many arid ecosystems. Over the last decade numerous experimental studies have tested how intensity of facilitation between plants changes with increasing stress. The most recent synthesis predicts a decline in facilitation intensity at the severe end of a drought stress gradient. Adding consumer pressure may result in even earlier and faster declines in facilitation intensity. So far, studies on critical transitions and plant–plant interactions have developed separately. The relationship between stress and facilitation intensity has been overlooked in critical transition theory, while facilitation intensity may determine the position of a critical transition threshold. In this study, we incorporate experimental studies on the relation between stress and facilitation intensity into the critical transition framework, to improve our ability to predict critical transitions. Moreover, we propose that a decline in facilitation intensity at the severe end of a stress gradient may occur prior to a critical transition. Inclusion of consumer pressure will speed up this process, leading to earlier and faster degradation. In‐field monitoring of seedling–facilitator associations and declines in facilitator recruitment can indicate declines in facilitation intensity and may thus provide additional early warning signals for imminent critical transitions, besides increased patchiness.     

Invariant properties in coevolutionary networks of plant–animal interactions

Plant–animal mutualistic networks are interaction webs consisting of two sets of entities, plant and animal species, whose evolutionary dynamics are deeply influenced by the outcomes of the interactions, yielding a diverse array of coevolutionary processes. These networks are two‐mode networks sharing many common properties with others such as food webs, social, and abiotic networks. Here we describe generalized patterns in the topology of 29 plant–pollinator and 24 plant–frugivore networks in natural communities. Scale‐free properties have been described for a number of biological, social, and abiotic networks; in contrast, most of the plant–animal mutualistic networks (65.6%) show species connectivity distributions (number of links per species) with a power‐law regime but decaying as a marked cut‐off, i.e. truncated power‐law or broad‐scale networks and few (22.2%) show scale‐invariance. We hypothesize that plant–animal mutualistic networks follow a build‐up process similar to complex abiotic nets, based on the preferential attachment of species. However, constraints in the addition of links such as morphological mismatching or phenological uncoupling between mutualistic partners, restrict the number of interactions established, causing deviations from scale‐invariance. This reveals generalized topological patterns characteristic of self‐organized complex systems. Relative to scale‐invariant networks, such constraints may confer higher robustness to the loss of keystone species that are the backbone of these webs.     

Impacts of consumer–resource interaction transitions on persistence and long‐term interaction outcomes of random ecological networks

Despite the prevalence of context‐dependent interaction transitions in ecological systems, their impacts on persistence and interaction diversity have scarcely been explored in complex ecological networks. By using multispecies bi‐directional and unidirectional consumer–resource models, representing a continuum of interaction transitions (sign change of interaction outcomes), we investigated the effects of structural interaction transitions on persistence (the fraction of remaining species) and long‐term interaction outcomes in random ecological networks. We found that high interaction strength of exploiting resources generally decreased persistence, and high strength of providing resources increased persistence when the strength of exploiting resources was low in more complex networks; also, the networks with high persistence had a high proportion of mutualistic interactions relative to antagonistic interactions present initially and over the long term. The shifting of interaction strengths shaped the long‐term interaction compositions. Meanwhile, population dynamics, especially species extinction, affected the difference between initial and long‐term interactions. Based on classical consumer–resource theory, these results establish a transitional continuum of interaction outcomes in ecological networks and imply a theoretical association among interaction transition, community persistence and interaction diversity.     

Resilience indicators: prospects and limitations for early warnings of regime shifts

In the vicinity of tipping points—or more precisely bifurcation points—ecosystems recover slowly from small perturbations. Such slowness may be interpreted as a sign of low resilience in the sense that the ecosystem could easily be tipped through a critical transition into a contrasting state. Indicators of this phenomenon of ‘critical slowing down (CSD)’ include a rise in temporal correlation and variance. Such indicators of CSD can provide an early warning signal of a nearby tipping point. Or, they may offer a possibility to rank reefs, lakes or other ecosystems according to their resilience. The fact that CSD may happen across a wide range of complex ecosystems close to tipping points implies a powerful generality. However, indicators of CSD are not manifested in all cases where regime shifts occur. This is because not all regime shifts are associated with tipping points. Here, we review the exploding literature about this issue to provide guidance on what to expect and what not to expect when it comes to the CSD-based early warning signals for critical transitions.     

Financial Symmetry and Moods in the Market

This paper studies how certain speculative transitions in financial markets can be ascribed to a symmetry break that happens in the collective decision making. Investors are assumed to be bounded rational, using a limited set of information including past price history and expectation on future dividends. Investment strategies are dynamically changed based on realized returns within a game theoretical scheme with Nash equilibria. In such a setting, markets behave as complex systems whose payoff reflect an intrinsic financial symmetry that guarantees equilibrium in price dynamics (fundamentalist state) until the symmetry is broken leading to bubble or anti-bubble scenarios (speculative state). We model such two-phase transition in a micro-to-macro scheme through a Ginzburg-Landau-based power expansion leading to a market temperature parameter which modulates the state transitions in the market. Via simulations we prove that transitions in the market price dynamics can be phenomenologically explained by the number of traders, the number of strategies and amount of information used by agents, all included in our market temperature parameter.     

Slowing down in spatially patterned ecosystems at the brink of collapse

Predicting the risk of critical transitions, such as the collapse of a population, is important in order to direct management efforts. In any system that is close to a critical transition, recovery upon small perturbations becomes slow, a phenomenon known as critical slowing down. It has been suggested that such slowing down may be detected indirectly through an increase in spatial and temporal correlation and variance. Here, we tested this idea in arid ecosystems, where vegetation may collapse to desert as a result of increasing water limitation. We used three models that describe desertification but differ in the spatial vegetation patterns they produce. In all models, recovery rate upon perturbation decreased before vegetation collapsed. However, in one of the models, slowing down failed to translate into rising variance and correlation. This is caused by the regular self-organized vegetation patterns produced by this model. This finding implies an important limitation of variance and correlation as indicators of critical transitions. However, changes in such self-organized patterns themselves are a reliable indicator of an upcoming transition. Our results illustrate that while critical slowing down may be a universal phenomenon at critical transitions, its detection through indirect indicators may have limitations in particular systems.     

The nature of spatial transitions in the Arctic

Aim Describe the spatial and temporal properties of transitions in the Arctic and develop a conceptual understanding of the nature of these spatial transitions in the face of directional environmental change. Location Arctic tundra ecosystems of the North Slope of Alaska and the tundra‐forest region of the Seward Peninsula, Alaska Methods We synthesize information from numerous studies on tundra and treeline ecosystems in an effort to document the spatial changes that occur across four arctic transitions. These transitions are: (i) the transition between High‐Arctic and Low‐Arctic systems, (ii) the transition between moist non‐acidic tundra (MNT) and moist acidic tundra (MAT, also referred to as tussock tundra), (iii) the transition between tussock tundra and shrub tundra, (iv) the transition between tundra and forested systems. By documenting the nature of these spatial transitions, in terms of their environmental controls and vegetation patterns, we develop a conceptual model of temporal dynamics of arctic ecotones in response to environmental change. Results Our observations suggest that each transition is sensitive to a unique combination of controlling factors. The transition between High and Low Arctic is sensitive primarily to climate, whereas the MNT/MAT transition is also controlled by soil parent material, permafrost and hydrology. The tussock/shrub tundra transition appears to be responsive to several factors, including climate, topography and hydrology. Finally, the tundra/forest boundary responds primarily to climate and to climatically associated changes in permafrost. There were also important differences in the demography and distribution of the dominant plant species across the four vegetation transitions. The shrubs that characterize the tussock/shrub transition can achieve dominance potentially within a decade, whereas spruce trees often require several decades to centuries to achieve dominance within tundra, and Sphagnum moss colonization of non‐acidic sites at the MNT/MAT boundary may require centuries to millennia of soil development. Main conclusions We suggest that vegetation will respond most rapidly to climatic change when (i) the vegetation transition correlates more strongly with climate than with other environmental variables, (ii) dominant species exhibit gradual changes in abundance across spatial transitions, and/or (iii) the dominant species have demographic properties that allow rapid increases in abundance following climatic shifts. All three of these properties characterize the transition between tussock tundra and low shrub tundra. It is therefore not surprising that of the four transitions studied this is the one that appears to be responding most rapidly to climatic warming.