Gradual regime shifts in spatially extended ecosystems

Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative-state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. Using a minimal model, we study gradual state transitions and show that they precede abrupt transitions. We propose indicators to probe gradual regime shifts, and suggest that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local vegetation mortality caused by repeated droughts.     

Bistability and regular spatial patterns in arid ecosystems

A variety of patterns observed in ecosystems can be explained by resource–concentration mechanisms. A resource–concentration mechanism occurs when organisms increase the lateral flow of a resource toward them, leading to a local concentration of this resource and to its depletion from areas farther away. In resource–concentration systems, it has been proposed that certain spatial patterns could indicate proximity to discontinuous transitions where an ecosystem abruptly shifts from one stable state to another. Here, we test this hypothesis using a model of vegetation dynamics in arid ecosystems. In this model, a resource–concentration mechanism drives a positive feedback between vegetation and soil water availability. We derived the conditions leading to bistability and pattern formation. Our analysis revealed that bistability and regular pattern formation are linked in our model. This means that, when regular vegetation patterns occur, they indicate that the system is along a discontinuous transition to desertification. Yet, in real systems, only observing regular vegetation patterns without identifying the pattern-driving mechanism might not be enough to conclude that an ecosystem is along a discontinuous transition because similar patterns can emerge from different ecological mechanisms.     

Beyond Turing: The response of patterned ecosystems to environmental change

Spatially periodic patterns can be observed in a variety of ecosystems. Model studies revealed that patterned ecosystems may respond in a nonlinear way to environmental change, meaning that gradual changes result in rapid degradation. We analyze this response through stability analysis of patterned states of an arid ecosystem model. This analysis goes one step further than the frequently applied Turing analysis, which only considers stability of uniform states. We found that patterned arid ecosystems systematically respond in two ways to changes in rainfall: (1) by changing vegetation patch biomass or (2) by adapting pattern wavelength. Minor adaptations of pattern wavelength are constrained to conditions of slow change within a high rainfall regime, and high levels of stochastic variation in biomass (noise). Major changes in pattern wavelength occur under conditions of either low rainfall, rapid change or low levels of noise. Such conditions facilitate strong interactions between vegetation patches, which can trigger a sudden loss of half the patches or a transition to a degraded bare state. These results highlight that ecosystem responses may critically depend on rates, rather than magnitudes, of environmental change. Our study shows how models can increase our understanding of these dynamics, provided that analyses go beyond the conventional Turing analysis.     

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 effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation

In a rapidly changing world, quantifying ecosystem resilience is an important challenge. Historically, resilience has been defined via models that do not take spatial effects into account. These systems can only adapt via uniform adjustments. In reality, however, the response is not necessarily uniform, and can lead to the formation of (self‐organised) spatial patterns – typically localised vegetation patches. Classical measures of resilience cannot capture the emerging dynamics in spatially self‐organised systems, including transitions between patterned states that have limited impact on ecosystem structure and productivity. We present a framework of interlinked phase portraits that appropriately quantifies the resilience of patterned states, which depends on the number of patches, the distances between them and environmental conditions. We show how classical resilience concepts fail to distinguish between small and large pattern transitions, and find that the variance in interpatch distances provides a suitable indicator for the type of imminent transition. Subsequently, we describe the dependency of ecosystem degradation based on the rate of climatic change: slow change leads to sporadic, large transitions, whereas fast change causes a rapid sequence of smaller transitions. Finally, we discuss how pre‐emptive removal of patches can minimise productivity losses during pattern transitions, constituting a viable conservation strategy.     

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.     

Do herbivores cause habitat degradation or vegetation state transition? Evidence from the tundra

Range expansion and increasing densities of large herbivores are held responsible for large-scale habitat degradation in a wide range of natural and semi-natural ecosystems. Herbivore-driven ecosystem changes frequently represent predictable transitions from one vegetation state to another. Whether such predictable changes justify the value judgement 'habitat degradation' may be debatable as this strongly depends on individual perspective.
To further the debate on herbivore-driven habitat degradation, I apply the concept of alternative stable states to arctic tundra as a framework to capture predictable stepwise vegetation transitions in which the productivity and hence herbivore-carrying capacity increases with grazing pressure. Specifically, evidence is provided that large parts of the tundra biome can be in either of three relatively discrete vegetation states and that changes in reindeer/caribou density are responsible for sudden, predictable but often reversible state transitions. From this, it appears that the relatively rapidly emerging vegetation changes do not necessarily equate to habitat degradation, but in many cases reflect predictable vegetation change. Acknowledgement of the existence of predictable state transitions in tundra ecosystems may help to evaluate the observed radical vegetation changes occurring throughout the reindeer/caribou range.     

Linking habitat modification to catastrophic shifts and vegetation patterns in bogs

Paleoecological studies indicate that peatland ecosystems may exhibit bistability. This would mean that these systems are resilient to gradual changes in climate, until environmental thresholds are passed. Then, ecosystem stability is lost and rapid shifts in surface and vegetation structure at landscape scale occur. Another remarkable feature is the commonly observed self-organized spatial vegetation patterning, such as string-flark and maze patterns. Bistability and spatial self-organization may be mechanistically linked, the crucial mechanism being scale-dependent (locally positive and longer-range negative) feedback between vegetation and the peatland environment. Focusing on bogs, a previous model study shows that nutrient accumulation by vascular plants can induce such scale-dependent feedback driving pattern formation. However, stability of bog microforms such as hummocks and hollows has been attributed to different local interactions between Sphagnum, vascular plants, and the bog environment. Here we analyze both local and longer-range interactions in bogs to investigate the possible contribution of these different interactions to vegetation patterning and stability. This is done by a literature review, and subsequently these findings are incorporated in the original model. When Sphagnum and encompassing local interactions are included in this model, the boundaries between vegetation types become sharper and also the parameter region of bistability drastically increases. These results imply that vegetation patterning and stability of bogs could be synergistically governed by local and longer-range interactions. Studying the relative effect of these interactions is therefore suggested to be an important component of future predictions on the response of peatland ecosystems to climatic changes.     

Precursors of state transitions in stochastic systems with delay

Ecosystem dynamics may exhibit alternative stable states induced by positive feedbacks between the state of the system and environmental drivers. Bistable systems are prone to abrupt shifts from one state to another in response to even small and gradual changes in external drivers. These transitions are often catastrophic and difficult to predict by analyzing the mean state of the system. Indicators of the imminent occurrence of phase transitions can serve as important tools to warn ecosystem managers about an imminent transition before the bifurcation point is actually reached. Thus, leading indicators of phase transitions can be used either to prepare for or to prevent the occurrence of a shift to the other state. In recent years, theories of leading indicators of ecosystem shift have been developed and applied to a variety of ecological models and geophysical time series. It is unclear, however, how some of these indicators would perform in the case of systems with a delay. Here, we develop a theoretical framework for the investigation of precursors of state shift in the presence of drivers acting with a delay. We discuss how the effectiveness of leading indicators of state shift based on rising variance may be affected by the presence of delays. We apply this framework to an ecological model of desertification in arid grasslands.     

Persistent versus transient tree encroachment of temperate peat bogs: effects of climate warming and drought events

Peatlands store approximately 30% of global soil carbon, most in moss‐dominated bogs. Future climatic changes, such as changes in precipitation patterns and warming, are expected to affect peat bog vegetation composition and thereby its long‐term carbon sequestration capacity. Theoretical work suggests that an episode of rapid environmental change is more likely to trigger transitions to alternative ecosystem states than a gradual, but equally large, change in conditions. We used a dynamic vegetation model to explore the impacts of drought events and increased temperature on vegetation composition of temperate peat bogs. We analyzed the consequences of six patterns of summer drought events combined with five temperature scenarios to test whether an open peat bog dominated by moss (Sphagnum) could shift to a tree‐dominated state. Unexpectedly, neither a gradual decrease in the amount of summer precipitation nor the occurrence of a number of extremely dry summers in a row could shift the moss‐dominated peat bog permanently into a tree‐dominated peat bog. The increase in tree biomass during drought events was unable to trigger positive feedbacks that keep the ecosystem in a tree‐dominated state after a return to previous ‘normal’ rainfall conditions. In contrast, temperature increases from 1 °C onward already shifted peat bogs into tree‐dominated ecosystems. In our simulations, drought events facilitated tree establishment, but temperature determined how much tree biomass could develop. Our results suggest that under current climatic conditions, peat bog vegetation is rather resilient to drought events, but very sensitive to temperature increases, indicating that future warming is likely to trigger persistent vegetation shifts.     

A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system

A quantitative, physiology-based model of the ascending arousal system is developed, using continuum neuronal population modeling, which involves averaging properties such as firing rates across neurons in each population. The model includes the ventrolateral preoptic area (VLPO), where circadian and homeostatic drives enter the system, the monoaminergic and cholinergic nuclei of the ascending arousal system, and their interconnections. The human sleep-wake cycle is governed by the activities of these nuclei, which modulate the behavioral state of the brain via diffuse neuromodulatory projections. The model parameters are not free since they correspond to physiological observables. Approximate parameter bounds are obtained by requiring consistency with physiological and behavioral measures, and the model replicates the human sleep-wake cycle, with physiologically reasonable voltages and firing rates. Mutual inhibition between the wake-promoting monoaminergic group and sleep-promoting VLPO causes ;;flip-flop' behavior, with most time spent in 2 stable steady states corresponding to wake and sleep, with transitions between them on a timescale of a few minutes. The model predicts hysteresis in the sleep-wake cycle, with a region of bistability of the wake and sleep states. Reducing the monoaminergic-VLPO mutual inhibition results in a smaller hysteresis loop. This makes the model more prone to wake-sleep transitions in both directions and makes the states less distinguishable, as in narcolepsy. The model behavior is robust across the constrained parameter ranges, but with sufficient flexibility to describe a wide range of observed phenomena.     

Metastable states, phase transitions, and persistent neural activity

A new mathematical model to describe the spiking rate of a neural population is derived, which considers both the mean and the variance of the activity. Bifurcation analysis identifies a critical interval of parameter values in which the standard bistability regime coexists with an additional third attractor corresponding to the metastable state of bounded mean activity and high variance. To understand the structure of spatio-temporal activity in the metastable state, we study a simple discrete-time model of binary elements with random noise locally coupled on the grid, which produces rich dynamics including metastability. A critical value of the noise amplitude is identified; in the vicinity of this value the system is flexible and can easily generate transitions between UP and DOWN metastable states, either autonomously or in response to a control process. These metastable states and phase transitions provide a proper basis for the modelling of persistent neural activity reported in many experimental studies.     

Dynamic bistable switches enhance robustness and accuracy of cell cycle transitions

Bistability is a common mechanism to ensure robust and irreversible cell cycle transitions. Whenever biological parameters or external conditions change such that a threshold is crossed, the system abruptly switches between different cell cycle states. Experimental studies have uncovered mechanisms that can make the shape of the bistable response curve change dynamically in time. Here, we show how such a dynamically changing bistable switch can provide a cell with better control over the timing of cell cycle transitions. Moreover, cell cycle oscillations built on bistable switches are more robust when the bistability is modulated in time. Our results are not specific to cell cycle models and may apply to other bistable systems in which the bistable response curve is time-dependent.     

Bistability in fluctuating environments. Implications in tumor immunology

We analyze under different environmental conditions the occurrence of bistability, i.e. of two simultaneously stables steady states, in a biological model system which describes the immunological interactions of neoplastic-target cells and cytotoxic-effector cells. As a result of environmental fluctuations such complex biological systems may undergo drastic modifications of their steady state properties. In particular, when the variance of fluctuations increases around a well defined mean value, transition phenomena appear which are absent in the usual bifurcation diagrams. The properties of the non-fluctuating systems can no longer be considered as a first approximation of the properties of the real system. Interestingly, in the case of the model, these transitions correspond to a rejection of tumor cells.     

Outbreaks by canopy-feeding geometrid moth cause state-dependent shifts in understorey plant communities

The increased spread of insect outbreaks is among the most severe impacts of climate warming predicted for northern boreal forest ecosystems. Compound disturbances by insect herbivores can cause sharp transitions between vegetation states with implications for ecosystem productivity and climate feedbacks. By analysing vegetation plots prior to and immediately after a severe and widespread outbreak by geometrid moths in the birch forest-tundra ecotone, we document a shift in forest understorey community composition in response to the moth outbreak. Prior to the moth outbreak, the plots divided into two oligotrophic and one eutrophic plant community. The moth outbreak caused a vegetation state shift in the two oligotrophic communities, but only minor changes in the eutrophic community. In the spatially most widespread communities, oligotrophic dwarf shrub birch forest, dominance by the allelopathic dwarf shrub Empetrum nigrum ssp. hermaphroditum, was effectively broken and replaced by a community dominated by the graminoid Avenella flexuosa, in a manner qualitatively similar to the effect of wild fires in E. nigrum communities in coniferous boreal forest further south. As dominance by E. nigrum is associated with retrogressive succession the observed vegetation state shift has widespread implications for ecosystem productivity on a regional scale. Our findings reveal that the impact of moth outbreaks on the northern boreal birch forest system is highly initial-state dependent, and that the widespread oligotrophic communities have a low resistance to such disturbances. This provides a case for the notion that climate impacts on arctic and northern boreal vegetation may take place most abruptly when conveyed by changed dynamics of irruptive herbivores.     

Climate change causes critical transitions and irreversible alterations of mountain forests

Mountain forests are at particular risk of climate change impacts due to their temperature limitation and high exposure to warming. At the same time, their complex topography may help to buffer the effects of climate change and create climate refugia. Whether climate change can lead to critical transitions of mountain forest ecosystems and whether such transitions are reversible remain incompletely understood. We investigated the resilience of forest composition and size structure to climate change, focusing on a mountain forest landscape in the Eastern Alps. Using the individual‐based forest landscape model iLand, we simulated ecosystem responses to a wide range of climatic changes (up to a 6°C increase in mean annual temperature and a 30% reduction in mean annual precipitation), testing for tipping points in vegetation size structure and composition under different topography scenarios. We found that at warming levels above +2°C a threshold was crossed, with the system tipping into an alternative state. The system shifted from a conifer‐dominated landscape characterized by large trees to a landscape dominated by smaller, predominantly broadleaved trees. Topographic complexity moderated climate change impacts, smoothing and delaying the transitions between alternative vegetation states. We subsequently reversed the simulated climate forcing to assess the ability of the landscape to recover from climate change impacts. The forest landscape showed hysteresis, particularly in scenarios with lower precipitation. At the same mean annual temperature, equilibrium vegetation size structure and species composition differed between warming and cooling trajectories. Here we show that even moderate warming corresponding to current policy targets could result in critical transitions of forest ecosystems and highlight the importance of topographic complexity as a buffering agent. Furthermore, our results show that overshooting ambitious climate mitigation targets could be dangerous, as ecological impacts can be irreversible at millennial time scales once a tipping point has been crossed.     

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.     

Abrupt shifts in African savanna tree cover along a climatic gradient

Aim To describe patterns of tree cover in savannas over a climatic gradient and a range of spatial scales and test if there are identifiable climate‐related mean structures, if tree cover always increases with water availability and if there is a continuous trend or a stepwise trend in tree cover. Location Central Tropical Africa. Methods We compared a new analysis of satellite tree cover data with botanical, phytogeographical and environmental data. Results Along the climatic transect, six vegetation structures were distinguished according to their average tree cover, which can co‐occur as mosaics. The resulting abrupt shifts in tree cover were not correlated to any shifts in either environmental variables or in tree species distributions. Main conclusions A strong contrast appears between fine‐scale variability in tree cover and coarse‐scale structural states that are stable over several degrees of latitude. While climate parameters and species pools display a continuous evolution along the climatic gradient, these stable structural states have discontinuous transitions, resulting in regions containing mosaics of alternative stable states. Soils appear to have little effect inside the climatic stable state domains but a strong action on the location of the transitions. This indicates that savannas are patch dynamics systems, prone to feedbacks stabilizing their coarse‐scale structure over wide ranges of environmental conditions.     

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.     

Bistability in cerebellar Purkinje cell dendrites modelled with high-threshold calcium and delayed-rectifier potassium channels

 Phase-plane analysis of the ionic currents underlying dendritic plateau potentials was carried out to study the nonlinear dynamics and steady-state transfer properties of the dendritic tree in cerebellar Purkinje cells. The results of an analysis of the P-type calcium and delayed rectifier potassium channel system are presented in this study. These channels constitute a simple system that can support bistability and plateau potentials. By requiring both the steady-state current-voltage curve and nullclines to mimic basic plateau potential properties, we obtained well-defined ranges of specific conductance that can support bistability. Hysteresis was found to be surprisingly prevalent in this simple ion-channel system. Using the steady-state current voltage relationship, we derive concise, algebraic expressions for the voltage and current thresholds of state transitions as functions of specific conductance. The significance of bistability in this ion-channel system is discussed with respect to the generation of plateau potentials in Purkinje cells dendrites and the role of the cerebellum in motor control. Received: 13 October 1993/Accepted in revised form: 21 March 1995