The notion that small changes can have large consequences in the climate or ecosystems has become popular as the concept of tipping points. Typically, tipping points are thought to arise from a loss of stability of an equilibrium when external conditions are slowly varied. However, this appealingly simple view puts us on the wrong foot for understanding a range of abrupt transitions in the climate or ecosystems because complex environmental systems are never in equilibrium. In particular, they are forced by diurnal variations, the seasons, Milankovitch cycles and internal climate oscillations. Here we show how abrupt and sometimes even irreversible change may be evoked by even small shifts in the amplitude or time scale of such environmental oscillations. By using model simulations and reconciling evidence from previous studies we illustrate how these phenomena can be relevant for ecosystems and elements of the climate system including terrestrial ecosystems, Arctic sea ice and monsoons. Although the systems we address are very different and span a broad range of time scales, the phenomena can be understood in a common framework that can help clarify and unify the interpretation of abrupt shifts in the Earth system.
The climate and ecosystems usually respond rather linearly to changes in external conditions. However, there are also occasions when at some point a sudden and large shift to a very different state occurs. An important cause of such behavior can be a positive feedback which leads to a self-amplifying change, making the system very sensitive to external conditions. Such feedback-induced threshold behavior is often referred to as a tipping point1. If the feedback is very strong, alternative stable states can exist under the same external conditions (Fig. 1a). In models, the parameter values at which the system’s dynamics suddenly changes are called bifurcation points. For example, when the system is driven over a bifurcation point where a current equilibrium ceases to exist, an abrupt and irreversible shift toward a different equilibrium can occur. Tipping points and alternative states exist in many simple models, for example representing ocean circulation2, global ice coverage3,4, Arctic sea ice5, vegetation-atmosphere interaction6, terrestrial ecosystems7, and the East Asian and Indian monsoons8,9. Moreover, there are sometimes also observational indications that multiple states exist in reality10, and there is evidence that abrupt shifts have occurred in the past11,12,13.
In the conventional illustration of a tipping point it is assumed that the driver (representing external conditions), varies monotonically and gradually, so that the system stays close to a stable equilibrium (black curve in Fig. 1a). However, many environmental conditions change so rapidly that the system’s state cannot catch up with the changing equilibrium. For example, global warming can drive the Greenland ice sheet into a regime where it is not sustainable anymore14. However, the ice sheet can take millennia to melt completely, which gives us a window of “borrowed time”15 in which we could save the ice-sheet should the temperature decrease again in time14. Figure 1 illustrates the two possibilities of a saved (blue) and a collapsing system (red). As the excursion beyond the threshold takes more time for the red trajectory, the system shifts to the realm of the lower stable branch, while the blue trajectory returns toward its old state.
As natural systems are permanently exposed to environmental change, they are usually not in equilibrium. The external fluctuations that drive them are usually simplified as stochastic, but in reality there often is a dominant timescale as well. This is particularly obvious if the change in external conditions consists in a periodic oscillation, for example the very regular annual and diurnal cycles of solar insolation, and the quasi-periodic changes in the Earth’s orbital parameters. Also, the climate system’s internal variability shows modes with specific frequency ranges, most importantly the El Niño Southern Oscillation (ENSO)16.
Such oscillations will affect parts of the climate system or ecosystems that have intrinsic thresholds or bifurcations: a gradual change in the mean, amplitude or frequency of periodic drivers can trigger abrupt change in these systems. In the following, we first illustrate these phenomena with a simple mathematical system. Subsequently, we point out how models and observations indicate the existence of these phenomena in many elements of the Earth system, spanning a vast range of time scales. We illustrate some consequences of this view for monsoon systems, sea ice and terrestrial ecosystems, and argue how a general view on tipping points can contribute to a conceptual understanding of different tipping elements in the Earth system.
Tipping points in periodically forced systems
To understand the occurrence of tipping points in periodically forced systems, we take a very simple dynamical system as a useful conceptual model example, where the evolution of state x over time t in this system is determined by
The black lines in Fig. 1a represent this system’s equilibria (dx/dt = 0) for constant D, which we refer to as the equilibrium system. D is an external (prescribed) driver representing environmental conditions, which makes the system non-autonomous (meaning that it explicitly depends on time). In case the driver D changes slowly enough, the system follows a stable equilibrium branch which ends at a bifurcation point. After the driver passes this point the system switches to the remaining stable equilibrium – the case of a classical tipping point.
We now investigate the behavior of the system in response to a driver that oscillates around a mean Dm with amplitude Da and period T.
Our conceptual model is hence determined by Eqs 1 and 2, and is well-known as the overdamped limit of the Duffing oscillator. We integrate the system for different sets of the three parameters, Dm, Da and T, until it reaches a stable attractor in form of a periodic oscillation. Like the driver, the system’s solution is periodic in time with period T (also involving harmonics that can arise close to a bifurcation point)17,18. Figure 2 illustrates how the periodic solutions of the system can change when one of the three parameters is varied. Varying Dm (Fig. 2a) is similar to the classical scenario of a tipping point, only that the stable solutions oscillate in time: if Dm is increased across the bifurcation, the upper branch of stable solutions disappears and the system has to shift to the lower branch. Interestingly, such tipping-point behavior can also be obtained by increasing the driver’s period T or its amplitude Da, without any change in Dm (Fig. 2b–d). For example, when the driver is fast (small T) the system regularly passes the bifurcation point of the equilibrium system but returns in time to keep the system close to the upper branch. Therefore, there can be two alternative periodic solutions. However, when the driver becomes slower, the system destabilizes and propels itself toward the only remaining solution around the lower branch. Figure 2e illustrates this forced transition as a blue trajectory in the phase space spanned by driver D and state x.
A similar result is obtained when Da is increased (Fig. 2c,f) because the relaxation toward the lower branch becomes too strong. A systematic sampling of the parameter space spanned by Dm, Da and T shows that Da and T essentially have a very similar effect in the model (Figs S1,S2) as they both reduce the Dm-range in which multiple solutions occur. When both become large, the dependence on initial conditions is destroyed (Fig. S3). This is due to another type of abrupt transition, which occurs at Da = 0.8 in Fig. 2d. At this point, the system reaches a regime where it makes a full cycle involving both branches of the equilibrium system, flipping up and down between the branches (Fig. 2g). As the system’s response becomes very slow at the bifurcation point of the equilibrium system, the transition to the alternative state occurs after the bifurcation point is reached19. Nonetheless, if the driver’s amplitude is large enough, two abrupt shifts occur within each full cycle of the system under constant parameter conditions. When this regime becomes active, the amplitude of the solution increases rapidly with Da. In contrast to the bifurcations mentioned previously, such sudden amplitude change is reversible: for intermediate Da in Fig. 2d, there are no multiple solutions because all movement occurs along one continuous stable branch far from the second one above. When Da increases, the driver suddenly encloses both bifurcation points. During a cycle, the system now switches to the upper branch, but falls down again later on the other side. It should be noted that the amplitude change is only a sharp transition if T is sufficiently large (compare Fig. 2c with 2d). Otherwise, the time is too short for the system to get close to the alternative branch to which it is suddenly attracted in a short period during each cycle.
This reminds us that Fig. 2a–d displays only asymptotic solutions where the system ends up after an infinite time. In similarity to tipping points in an equilibrium system, the time it takes do make a transition from one to the other solution (blue trajectories in Fig. 2e–g), depends on the response time of the system. This response time can be idealized as a single relaxation time τ. For example, in the equilibrium system given by Eq. 1 and constant D, the relaxation time can be shown to be
where xeq is a stable equilibrium and lies on either of the continuous black lines in Fig. 1a.
The ratio between τ and the driver’s period T determines the abruptness of every one of the transitions discussed above. Three regimes can be distinguished: when T ≫ τ, the system closely follows the stable state of the equilibrium system. This limit case is usually addressed with the classical tipping point concept. When T ≪ τ, the system cannot follow the rapid forcing at all and remains static. It can then be described with an equivalent time-independent system. In the intermediate regime, where the time scales of the system and driver are similar, irreversible amplitude or timescale-induced shifts can occur.
Consequences for tipping elements in the Earth system
Mathematically, bifurcations in time-independent and periodically forced systems are equivalent20. However, transitions that are triggered by changes in periodic forcing have not been explicitly included in the concept of climate tipping points which usually build on the simple paradigm of stable equilibria (and their loss of stability at a tipping point)1,21. This stands in contrast to the fact that the climate system is interspersed with a number of internal and external oscillations that may cause such transitions on a vast range of time scales. This seems plausible because hardly any oscillation remains the same over time. For example, the seasonal cycle of solar insolation varies due to the Earth’s precession, which in turn is modulated by eccentricity. Similarly, the diurnal insolation cycle depends on the seasons, the Earth’s orbital parameters and its rotation rate. Moreover, modes of internal climate variability often have a distinct range of frequencies, and affect ecosystems and other components of the climate system. Besides these naturally occurring phenomena, the human interference with the climate system is another reason why such oscillations might change. For example, a human impact on the amplitude and frequency of ENSO can be expected22. Even the amplitude and timing of the annual cycle can change in many variables when the balance of climate feedbacks shifts under increased greenhouse gas concentrations, for example associated with changes in Arctic cloud cover, or the loss of sea ice23. Figure 3 gives a non-exhaustive overview of (quasi-) periodic oscillations and the nonlinear systems they affect. The possibility of tipping points induced by quasi-periodic drivers emerges from a large body of literature on these systems, but is hardly discussed in such process-based studies or in review articles on tipping points.
Starting our overview with small scales, tipping point phenomena have a long tradition in the nonlinear world of fluid dynamics24,25. For example, it has lately been found that the structure of the nocturnal atmospheric boundary layer over land can show two alternative regimes26,27. The question which regime will be established can crucially depend on the conditions during the evening28. On a much larger scale, marine boundary layer clouds show distinct regimes in observations and multiple equilibria in specific models29,30. It therefore seems rewarding to study how the diurnal cycle of solar insolation31,32 or the annual cycle of sea surface temperatures can affect the long-term structure of the lower atmosphere.
Particularly important for the lives of many people are the abrupt shifts in monsoon systems occurring within the seasonal and orbital cycles. Similarly, the current loss of summer sea ice reminds us that sea ice is nonlinear system that is subject to a marked seasonal cycle. Moreover, multiple equilibria are supposed to exist in several marine and terrestrial ecosystems including lakes33, marine sea-grass ecosystems34, tropical forests, and drylands35. In terrestrial ecosystems, rainfall is often the most important environmental driver, and the longevity and severance of droughts has long-term consequences. We will discuss the cases of monsoons, sea ice and terrestrial ecosystems in more detail below.
Finally, an iconic example of a tipping point is the Atlantic Meridional Overturning Circulation (AMOC) which can show multiple states in models of different complexity2,36. Probably related to the AMOC are the quasi-regular Dansgaard-Oeschger events which have been argued to be caused by a regular driver37. A subject where the role of periodic drivers has been discussed explicitly are glacial-interglacial cycles. In particular, it has been shown that they can conceptually be understood as forced shifts in a multi-stable landscape38. Due to the inertia of the cryosphere and lithosphere, the climate is not in equilibrium with the orbital forcing. Small insolation differences from one orbital cycle to the next can thus cause very different climates39,40. Such a conceptual view can also be beneficial for understanding other components of the Earth system and can inspire research questions. We will illustrate this by focusing on three example cases.
Monsoon systems cover a large part of the subtropical land areas and supply many people with water. The most important monsoons occur in India, East Asia, West Africa, Australia and North and South America. The classical view of monsoon systems is that subtropical heat lows in summer induce a low-level circulation from the tropical ocean towards higher latitudes that supplies the subtropical land with moisture41. Due to geographical differences between the monsoon regions, each monsoon system is usually studied on its own. However, observations from several monsoon systems show an intriguing asymmetry in the annual cycle of precipitation: while the onset of the monsoon rainfall in spring tends to occur rather abruptly, its retreat in autumn occurs much more gradually. Abrupt monsoon onsets have been observed in West Africa42, the southwest Indian coast43, and in East Asia, where several abrupt stages of monsoon onset have been identified44. This behavior stands in contrast to the gradual and symmetric annual cycle of solar isolation that drives the monsoon evolution.
There are many potential reasons for abrupt monsoon onsets. First, meteorological theory predicts that switches from a zonal to a meridional flow can occur due to hydrodynamic instabilities when the atmosphere is sufficiently heated away from the equator45,46. In particular, the mechanism has been found to operate in a regional model of West African monsoon climate47 and is supported by observations48. Second, air-sea interactions involving wind, evaporation, clouds and moisture advection can provide positive feedbacks which can rapidly enhance the monsoon after its onset49,50. One important positive feedback results from the advection of moisture by the low-level monsoon flow. When this moisture condenses over land, the latent heat release further enhances the circulation. This feedback is at the heart of the monsoon model by51 that produces abrupt monsoon shifts, and whose simplified version8 we will apply below. Third, the distribution of land surface properties such as albedo and orography modulate the monsoon onset52,53,54. Fourth, the seasonal cycle is superimposed by faster oscillations, causing periods of rapid change42,55. Such oscillations can make the monsoon onset more abrupt because they tend to be phase-locked to the annual cycle56,57.
Despite the complexity and geographical differences between the monsoon systems, the widespread existence of sudden monsoon onsets invites describing them within the common concept of tipping points. In addition, the annual cycle of solar insolation is a cyclic driver of these systems. One could therefore speculate that monsoons are periodically driven bistable systems. As the monsoon circulation adjusts very quickly to its boundary conditions it would be best described as a system with a very small ratio τ/T, similar to the example system in Fig. 2d and g. One could therefore think of a typical annual cycle as a complete loop over both branches of the bistable system (the large amplitude solutions in Fig. 2d). It then follows that a small shift in external conditions could bring the system into a regime where the bifurcation point of the monsoon onset is not reached anymore in a particular year. Indeed, past abrupt transitions between episodes of strong and weakened (“failed”) East Asian monsoon have been identified in Chinese cave records13. These switches occur due to the oscillations of the Earth’s orbit that modulate summer insolation on the northern hemisphere. In this regard, the behavior of monsoons within a year and on timescales of hundred thousands of years is remarkably similar.
Model example: sudden onset and failure of monsoons
To conceptualise the link between abrupt monsoon transitions on different time scales we use the simple model of Levermann, et al.8 which describes an idealized summer monsoon that can switch between an “on” and “off” state (see Supplementary Information). The mechanism responsible for the abrupt behavior in the model is the moisture-advection feedback. It has recently been challenged that this feedback would be strong enough to support such nonlinearity58,59,60. While the underlying mechanism is hence uncertain, it is clear that abrupt monsoon switches both in the seasonal cycle and in response to orbital forcing do occur. By applying the tipping point framework we speculate that a common mechanism is behind the jumps within the annual cycle and the abrupt transitions on millennial time scales. While the Levermann model suggests the moisture-advection feedback as the cause, any other positive feedback would essentially yield the same qualitative behaviour.
The most important parameters that determine the model’s state are the atmospheric humidity over the ocean qo and the radiative balance of the atmospheric column over land R. Without any alterations to the model we generalize R here to also include the sensible heat flux at the Earth’s surface and call the resulting quantity R*. Figure 4 shows the precipitation P associated with the summer monsoon as a result of qo and R*. The threshold that separates the “on” and “off” states can be seen as a sharp boundary between the blue area (no monsoon rainfall) and other colors (monsoon rainfall of different magnitude).
We now impose an annual cycle on the parameters qo and R*. The model is only valid for summer conditions and cannot describe any precipitation not associated with the summer monsoon. We therefore set P to zero in other seasons, knowing that no monsoon rainfall occurs during that time. In order to obtain an asymmetric annual cycle with an abrupt onset like in observations, we apply two strategies:
We follow the approach by9 by introducing a small bistable regime at the critical threshold (the shaded region with a width of 0.5 g/kg at the boundary).
We introduce a phase lag of one month between the two cyclic drivers, making oceanic moisture qo lag the seasonal cycle in R*. This approach may be justified by the different memories of ocean and atmosphere: while the atmosphere can adjust to the solar insolation within days, the oceanic temperature which controls specific humidity lags the insolation cycle by many weeks. The resulting annual cycle of qo and R* is shown as a red orbit in the parameter plane of Fig. 4, rotating counter-clockwise.
The annual cycle of rainfall that results from the trajectory of qo and R* is shown in the inset of Fig. 4. As soon as the “off” state disappears in early summer at the right border of the shaded bistable region, an abrupt monsoon onset occurs. Later in the summer, R* and qo decline and rainfall decreases with them, until it drops to zero when the “on” state disappears at the left border of the shaded bistable region. An alternative situation with a different mean of qo and R* is shown by the dashed orbit and dashed annual precipitation cycle, where precipitation is zero during the whole year. The transition from the regime where a monsoon circulation establishes each year to the regime where it does not establish constitutes a generalized tipping point. The monsoon collapse occurs when the threshold value of qo and R* is not reached anymore in summer.
Arctic sea ice
It has been hypothesized whether global warming can cause abrupt and irreversible sea-ice loss in the Arctic61. The most important argument for this idea is the positive feedback between surface albedo and sea-ice loss (ice-albedo feedback). It has been found that the existence of bifurcations in sea ice models depends on the description of spatial differences and the seasonal cycle62,63. While future ice loss is reversible in comprehensive models, these models also indicate that Arctic winter sea ice can be very sensitive to warming64. Moreover, a comprehensive model shows three alternative stable sea-ice states on a completely land-free Earth65,66. One of these states is a globally ice-covered planet, resembling the “snowball” conditions that occurred during the Neoproterozoic. This state also occurs in at least one other complex model under present-day boundary conditions67.
Hence, it is plausible that tipping points in sea ice have occurred in the Earth’s past, triggered by changes in the continental configuration, Milankovitch forcing, or the seasonal insolation cycle. In this regard, Eisenman68 provides an insightful parameter analysis of a simple Arctic sea ice model that is driven by the annual cycle of solar and terrestrial radiation. Small variations in these fluxes can lead to abrupt changes in the annual cycle of sea ice. Compared to the previous systems, Arctic sea ice and the surface ocean constitute a slow system that is not in equilibrium with the annual cycle of insolation. This can be seen from the fact that the annual cycle of Arctic sea ice volume lags the insolation cycle by approx. three months. The analysis by Eisenman68 was performed with a model that already included the annual cycle of short- and long-wave insolation. The concept of tipping points in periodically forced systems suggests that the bifurcations in the model originate from multiple states under certain fixed values of insolation, as the following model results show.
Model example: abrupt Arctic winter sea ice loss
The model by Eisenman68 calculates an energy balance for a well-mixed box of ocean water, covered with ice of a single thickness (see Supplementary Information). When ice is present, the enthalpy E is negative and proportional to the ice thickness. In the absence of ice, E is positive and proportional to the ocean temperature. The bifurcation parameter Lm represents the annual mean outgoing long-wave radiation budget at the surface (not including temperature feedbacks) and becomes smaller when the climate warms. For values of Lm between approx. 65 W/m2 (which roughly represents the present-day climate) and 50 W/m2, two stable solutions can be found: a seasonally ice-covered ocean and an ice-free ocean (Fig. 5a).
The solutions in Fig. 5 are periodic in time because of the prescribed annual cycle in the long-wave budget L, and downwelling short-wave radiation, S. Figure 5b–d again displays the annual cycle for three distinct values of Lm (red lines). In addition, we show to what state the sea-ice system would evolve if there was no annual cycle (green and blue lines). To this end, we fix S and L to the values corresponding to a particular day in a year and integrate the model forward in time, starting from large positive and negative E. For most days in the year, only one asymptotic solution exists. For example, under permanent summer conditions, all ice melts away and the ocean becomes so warm that the outgoing long-wave radiation balances the absorbed radiation (Fig. 5, green line). Under permanent winter conditions, the ice grows to an infinite thickness, because there is no negative feedback in the model to limit the ice growth. In such cases of failed convergence we artificially set the enthalpy to −7 GJ/m2 (corresponding to a thickness of approx. 23 m). As the ice-ocean system cannot fully adjust to the fast insolation changes in the course of a year, the amplitude of its annual cycle is very small compared to the difference of its steady state solutions. The solutions overlap only during a short period in spring and an even shorter period in autumn, in the rest of the year only one solution exists. Nonetheless, the existence of multiple solutions still carries over to the periodically forced system for a certain parameter regime.
The bifurcations in this default model (Fig. 5a) can thus be understood from the change of the asymptotic solutions with Lm. In a cold climate, the summer is too short and insolation is too low to melt enough ice within one summer; a perennial ice cover is therefore the only solution of the periodically forced system. Under global warming (decreasing Lm), the asymptotic solution moves to larger E and the melting season becomes longer (Fig. 5c); this is where the periodically forced system has multiple solutions, balancing between the cold and warm attractors. Finally, the winter becomes too short for the formation of sea ice thick enough to survive the following summer, and the ice-free solution is the only one that remains (Fig. 5d).
In conclusion, the fact that a certain comprehensive model does not show multiple solutions under the same external conditions does not automatically suggest that tipping points are not possible. This has two reasons: First, tipping points can remain hidden in a parameter range related to periodic forces that is not explored when varying one parameter (such as CO2) alone. As Eisenman’s single column model above shows, the existence of multiple equilibria under fixed seasonal conditions in a model is a ‘smoking gun’ pointing to a strong positive feedback. Such a feedback indicates that the system is prone to tipping points even if the multistability is not apparent in the seasonally forced (non-autonomous) system. Changes in the insolation forcing (for instance due to changes in the annual cycle of cloud cover) can hence bring the system into a regime where abrupt changes are possible. The same can occur due to changes in the intrinsic relaxation time of the system, for instance related to the fact that the growth rate of sea ice depends on its thickness and hence the background climate69,70. Second, abrupt shifts can also occur in systems without multiple states. Specifically, the freezing point of water acts as a natural threshold for the formation of sea ice, making ice cover very sensitive on temperature in a certain regime and prone to reversible abrupt shifts64. The quasi-periodic changes in insolation related to Milankovitch forcing might therefore switch the Arctic back and forth between an ice-free and a seasonal ice regime (in analogy to the monsoon switches outlined above). A similar switch-mechanism determining the growth and melt of ice sheets has also been discussed as an explanation of ice ages during the Pleistocene71.
Multiple equilibria are thought to exist in several marine and terrestrial ecosystems including lakes33, marine sea grass ecosystems34, tropical forests, and drylands35. As an example we consider the bistability of tropical forest versus savannah. For instance, satellite observations across continents show a multimodality in the probability distribution of tree cover under the same precipitation regime10,72. The likely cause is the competition between tropical grasses and trees and their different vulnerability to fire73,74: in a savannah ecosystem, fire occurs frequently and as grasses can quickly re-establish burned areas they outcompete the slower-growing tree saplings. In contrast, where a forest has established, fire is less likely to occur and the high tree cover can persist at the expense of grasses. Dynamic vegetation models indeed indicate that large subtropical land areas fall into a precipitation regime where several stable states are possible75,76.
However, precipitation is not constant but varies on multi-annual time scales. In the tropics, rainfall over land is often controlled by the magnitude and distribution of sea surface temperature (SST). Tropical SSTs show characteristic modes of variability like the El Nino Southern Oscillation (ENSO), Pacific decadal oscillation (PDO) and Atlantic multidecadal oscillation (AMO) which affect tropical rainfall. For example, high Atlantic SSTs can lead to severe droughts in the Amazon like those observed in recent years77, and changes in ENSO activity strongly affect Amazon rainfall and vegetation78. Proxy-based reconstructions suggest that ENSO amplitude was low during the mid-Holocene, and relatively high in recent decades79. Moreover, tree rings indicate that ENSO amplitude and frequency have changed substantially on decadal to centennial time scales during the last 1,000 years16,80. The time scales of such changes are on the same order of magnitude than the time scale of tree growth which suggests the possibility that terrestrial tipping points can be triggered by changes in ENSO variability81. It is important in this context that a dieback caused by a large-scale drought can occur within a much shorter time than regrowth – in contrast to our example model above, the system is not symmetric. This time scale asymmetry suggests that forests are not safe from a drought-induced collapse even if the long-term mean rainfall is very large.
Moreover, precipitation is not strictly external, but interacts with vegetation. Such atmosphere-vegetation interaction is believed to be particularly large in Northern Africa due to positive surface-albedo and water cycle feedbacks82,83. It has therefore been investigated if these feedbacks can give rise to multiple equilibria and whether the end of the Green Sahara some 5,500 years ago can be interpreted as a tipping point6,12,84. Moreover, rainfall in Northern Africa shows large decadal variations. Climate models show that these variations are caused by SST changes in several ocean basins and that the Sahel drought of the 1970s and early 1980s was caused by such modes of natural climate variability85,86. The time scale of these inter-annual fluctuations is of similar magnitude than the time scale of responses in grass coverage in such drylands. It is therefore plausible that rainfall fluctuations can lead to locally amplified shifts between a desert and a grass-covered state (representing a case like Fig. 2c and f).
Building on a conceptual model by Brovkin, et al.6, Zeng, et al.87 showed such abrupt shifts in a vegetation model with oscillating precipitation forcing. We reproduce their result in our Fig. S4 as another explicit model example (see Supplementary Information for model description). Despite the model differences, the results are directly comparable to our conceptual model above (Fig. 2c,d). At small amplitude, two alternative vegetation states exist. As soon as the amplitude becomes large enough, a bifurcation occurs, leading to an irreversible vegetation increase when starting from a desert state as an initial condition. At even larger amplitude, the rainfall variations now cover both equilibria and the mean vegetation cover drops to an intermediate average state while having much larger amplitude. Hence, a small change in the amplitude of rainfall fluctuations can result in a vegetation collapse or increase. As in the generic conceptual model above, the abruptness of the second (reversible) type of the transition depends on the timescale of the forcing relative to the relaxation time of the system (top versus bottom panel).
Toward a more general view on tipping points
For the sake of simplicity, our example models are a very idealized representation of reality. For example, alternative equilibria are arguably less likely or at least harder to find in more complex models88. Also, low-order models can often be misleading when spatial dimensions come into play. Taking space into account can often suppress multiple equilibria, but can also lead to new critical phenomena like pattern formation35. Despite the relevance of abrupt shifts in the Earth system, the connection between regime transitions in spatially extended systems (like the onset of turbulence in fluids)24 and bifurcations in ordinary differential equations is not straightforward25, and neither is such theoretical knowledge well applied to the specific physical systems mentioned above. Nonetheless, nonlinearities and critical thresholds are arguably present in all Earth system components, providing a basis for abrupt shifts. At the same time, other yet unrecognized tipping points could exist in the Earth system.
In more realistic systems the drivers also have a more continuous spectrum. In models, the fast time scales in such spectra are often described by stochastic terms. The amplitude and period of the periodic oscillations we assumed above can be seen as an analogy for the typical magnitude and duration of random external fluctuations. Small changes in these statistical properties can lead to large and sometimes counter-intuitive changes in the distribution of a system’s state89,90. In this regard, the concept of tipping points triggered by changes in periodic forcings is one representation of the fundamental question how vulnerable components of the Earth system are to environmental change. Theoretical progress has been made in recent decades to generalise concepts from bifurcation theory, for example by defining pullback attractors for deterministic non-autonomous systems91, and random attractors for stochastic systems92,93. Unfortunately, such theoretical insights do not always feed back to our knowledge on tipping points in the Earth system.
One particular challenge in this regard is to investigate the applicability of statistical stability indicators that could provide an early warning of abrupt change. In climate research, this concept has mostly been explored in the static context of classical tipping points94,95,96. In the case of the systems above, classical indicators like rising variance and autocorrelation are not necessarily the most useful approach18,97. It is therefore important to explore alternative approaches to early warnings of abrupt change. Although conceptual models cannot replace studies based on observations and process-based modelling, they can help raise questions for future research. For example, it may be beneficial to compare abrupt monsoon changes on different time scales and in different regions: Are there common mechanisms behind the abrupt onsets in different monsoon systems on the planet? Are these mechanisms also responsible for abrupt monsoon shifts in palaeorecords? Is the expectation to see early warning signals before the occurrence of a certain shift supported by current process understanding? Will stronger El Niño events have irreversible consequences for tropical forest and savannah ecosystems? Was sea ice involved in a tipping point in the Earth’s history, and what was the role of periodic forcings? To explore such questions, it would be beneficial to further develop models of low to intermediate complexity in addition to existing Earth system models. Such a model hierarchy is most established in the case of sea ice models, but patchy in the case of ecosystems, and lacking in the case of monsoons. A suite of models that are easy to use would also better connect research attempts to understand different monsoon systems.
On top of that, studies on isolated tipping elements may even not be sufficient, as different Earth system components are linked in their stability behaviour and thus do not act in isolation. For example, sea-ice loss and the meridional ocean circulation affect each other65,66,98, and transitions between vegetation states depend on the global background climate involving the ice sheets84 and the Atlantic overturning circulation84,99. In light of the large uncertainties, the candidates and suspected mechanisms of tipping points will certainly evolve with new knowledge. As the Earth system is interspersed with feedbacks and natural thresholds, and features oscillations and environmental fluctuations on a vast range of time scales, it seems plausible that non-equilibrium tipping points have already occurred in the past and could occur again in the future.
All analysed data can be produced with the simple models as explained in the Supplementary Information, and can also be obtained from the lead author (S.B.).
Lenton, T. M. et al. Tipping elements in the Earth’s climate system. Proceedings of the National Academy of Sciences of the United States of America 105, 1786–1793, https://doi.org/10.1073/pnas.0705414105 (2008).
Stommel, H. Thermohaline convection with two stable regimes of flow. Tellus 13, 224–230 (1961).
Budyko, M. I. The effect of solar radiation variations on the climate of the Earth. Tellus 21, 611–619 (1969).
Sellers, W. D. A global climatic model based on the energy balance of the earth-atmosphere system. Journal of Applied Meteorology 8, 392–400 (1969).
Eisenman, I. & Wettlaufer, J. S. Nonlinear threshold behavior during the loss of Arctic sea ice. Proceedings of the National Academy of Sciences of the United States of America 106, 28–32, https://doi.org/10.1073/pnas.0806887106 (2009).
Brovkin, V., Claussen, M., Petoukhov, V. & Ganopolski, A. On the stability of the atmosphere-vegetation system in the Sahara/Sahel region. J. Geophys. Res. 103, 31613–31624 (1998).
van Nes, E. H., Hirota, M., Holmgren, M. & Scheffer, M. Tipping points in tropical tree cover: linking theory to data. Global change biology 20, 1016–1021, https://doi.org/10.1111/gcb.12398 (2014).
Levermann, A., Schewe, J., Petoukhov, V. & Held, H. Basic mechanism for abrupt monsoon transitions. Proceedings of the National Academy of Sciences of the United States of America 106, 20572–20577 (2009).
Schewe, J., Levermann, A. & Cheng, H. A critical humidity threshold for monsoon transitions. Climate of the Past 8, 535–544, https://doi.org/10.5194/cp-8-535-2012 (2012).
Hirota, M., Holmgren, M., Van Nes, E. H. & Scheffer, M. Global resilience of tropical forest and savanna to critical transitions. Science 334, 232–235, https://doi.org/10.1126/science.1210657 (2011).
North Greenland Ice Core Project members. High-resolution record of Northern Hemisphere climate extending into the last interglacial period. Nature 431, 147–151, https://doi.org/10.1038/nature02805 (2004).
deMenocal, P. et al. Abrupt onset and termination of the African Humid Period: rapid climate responses to gradual insolation forcing. Quaternary Science Reviews 19, 347–361 (2000).
Wang, Y. et al. Millennial- and orbital-scale changes in the East Asian monsoon over the past 224,000 years. Nature 451, 1090–1093, https://doi.org/10.1038/nature06692 (2008).
Ridley, J., Gregory, J. M., Huybrechts, P. & Lowe, J. Thresholds for irreversible decline of the Greenland ice sheet. Climate Dynamics 35, 1049–1057, https://doi.org/10.1007/s00382-009-0646-0 (2010).
Hughes, T. P., Linares, C., Dakos, V., van de Leemput, I. A. & van Nes, E. H. Living dangerously on borrowed time during slow, unrecognized regime shifts. Trends in ecology & evolution 28, 149–155, https://doi.org/10.1016/j.tree.2012.08.022 (2013).
Li, J. et al. Interdecadal modulation of El Niño amplitude during the past millennium. Nature Climate Change 1, 114–118, https://doi.org/10.1038/nclimate1086 (2011).
Wiesenfeld, K. Noisy precursors of nonlinear instabilities. J. Stat. Phys. 38, 1071–1097 (1985).
Williamson, M. S., Bathiany, S. & Lenton, T. M. Early warning signals of tipping points in periodically forced systems. Earth System Dynamics 7, 313–326 (2016).
Tredicce, J. R. et al. Critical slowing down at a bifurcation. American Journal of Physics 72, 799, https://doi.org/10.1119/1.1688783 (2004).
Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. Vol. 112 (Springer, 2004).
Alley, R. B. et al. Abruptclimate change. Science 299, 2005–2010, https://doi.org/10.1126/science.1081056 (2003).
Cai, W. et al. ENSO and greenhouse warming. Nature Climate Change 5, 849–859, https://doi.org/10.1038/nclimate2743 (2015).
Bintanja, R. & van der Linden, E. C. The changing seasonal climate in theArctic. Scientific reports 3, 1556, https://doi.org/10.1038/srep01556 (2013).
Barkley, D. et al. The rise of fully turbulent flow. Nature 526, 550–553, https://doi.org/10.1038/nature15701 (2015).
Fowler, A. C. Homoclinic bifurcations for partial differential equations in unbounded domains. Studies in Applied Mathematics 83, 329–353 (1990).
van de Wiel, B. J. H., Moene, A. F., Steeneveld, G. J., Hartogensis, O. K. & Holtslag, A. A. M. Predicting the Collapse of Turbulence in Stably Stratified Boundary Layers. Flow, Turbulence and Combustion 79, 251–274, https://doi.org/10.1007/s10494-007-9094-2 (2007).
Monahan, A. H., Rees, T., He, Y. & McFarlane, N. Multiple Regimes of Wind, Stratification, and Turbulence in the Stable Boundary Layer. Journal of the Atmospheric Sciences 72, 3178–3198, https://doi.org/10.1175/jas-d-14-0311.1 (2015).
van Hooijdonk, I. G. S., Donda, J. M. M., Clercx, H. J. H., Bosveld, F. C. & van de Wiel, B. J. H. Shear Capacity as Prognostic for Nocturnal Boundary Layer Regimes. Journal of the Atmospheric Sciences 72, 1518–1532, https://doi.org/10.1175/jas-d-14-0140.1 (2015).
Randall, D. A. & Suarez, M. J. On the dynamics of stratocumulus formation and dissipation. Journal of the Atmospheric Sciences 41, 3052–3057 (1984).
Bretherton, C. S., Uchida, J. & Blossey, P. N. Slow manifolds and multiple equilibria in stratocumulus-capped boundary layers. Journal of Advances in Modeling Earth Systems 2, Art #14, https://doi.org/10.3894/james.2010.2.14 (2010).
Turton, J. D. & Nicholls, S. A study of the diurnal variation of stratocumulus using a multiple mixed layer model. Quarterly Journal of the Royal Meteorological Society 113, 969–1009 (1987).
Caldwell, P., Bretherton, C. S. & Wood, R. Mixed-Layer budget analysis of the diurnal cycle of entrainment in Southeast Pacific stratocumulus. Journal of the Atmospheric Sciences 62, 3775–3791 (2005).
Scheffer, M., Hosper, S. H., Leijer, M.-L., Moss, B. & Jeppesen, E. Alternative equilibria in shallow lakes. Trends in Ecology and Evolution 8, 275–279 (1993).
van der Heide, T. et al. Positive Feedbacks in Seagrass Ecosystems: Implications for Success in Conservation and Restoration. Ecosystems 10, 1311–1322, https://doi.org/10.1007/s10021-007-9099-7 (2007).
Rietkerk, M., Dekker, S. C., De Ruiter, P. C. & Van de Koppel, J. Self-organized patchiness and catastrophic shifts in ecosystems. Science 305, 1926–1929 (2004).
Hawkins, E. et al. Bistability of the Atlantic overturning circulation in a global climate model and links to ocean freshwater transport. Geophysical Research Letters 38, L10605, https://doi.org/10.1029/2011gl047208 (2011).
Rahmstorf, S. Timing of abrupt climate change: A precise clock. Geophysical Research Letters 30, 1510, https://doi.org/10.1029/2003gl017115 (2003).
Paillard, D. The timing of Pleistocene glaciations from a simple multiple-state climate model. Nature 391, 378–381 (1998).
Calov, R. & Ganopolski, A. Multistability and hysteresis in the climate-cryosphere system under orbital forcing. Geophysical Research Letters 32, https://doi.org/10.1029/2005gl024518 (2005).
Abe-Ouchi, A. et al. Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume. Nature 500, 190–193, https://doi.org/10.1038/nature12374 (2013).
Webster, P. J. et al. Monsoons: Processes, predictability, and the prospects for prediction. Journal of Geophysical Research: Oceans 103, 14451–14510, https://doi.org/10.1029/97jc02719 (1998).
Sultan, B. & Janicot, S. Abrupt shift of the ITCZ over West Africa and intra-seasonal variability. Geophysical Research Letters 27, 3353–3356, https://doi.org/10.1029/1999gl011285 (2000).
Ananthakrishnan, R. & Soman, M. K. The onset of the southwest monsoon over Kerala: 1901–1980. Journal of Climatology 8, 283–296 (1988).
Ueda, H., Ohba, M. & Xie, S.-P. Important Factors for the Development of the Asian–Northwest Pacific Summer Monsoon*. Journal of Climate 22, 649–669, https://doi.org/10.1175/2008jcli2341.1 (2009).
Plumb, R. A. & Hou, A. Y. The response of a zonally symmetric atmosphere to subtropical forcing: threshold behavior. Journal of the Atmospheric Sciences 49, 1790–1799 (1992).
Eltahir, E. A. B. & Gong, C. Dynamics of wet and dry years in West Africa. Journal of Climate 9, 1030–1042 (1996).
Hagos, S. M. & Cook, K. H. Dynamics of the West African Monsoon Jump. Journal of Climate 20, 5264–5284, https://doi.org/10.1175/2007jcli1533.1 (2007).
Cook, K. H. Role of inertial instability in the West African monsoon jump. Journal of Geophysical Research: Atmospheres 120, 3085–3102, https://doi.org/10.1002/2014jd022579 (2015).
Minoura, D., Kawamura, R. & Matsuura, R. A Mechanism of the onset of the South Asian Summer Monsoon. Journal of Meteorological Society of Japan 81, 563–580 (2003).
Ueda, H. Air-sea coupled process involved in stepwise seasonal evolution of the Asian Summer Monsoon. Geographical Review of Japan 78, 825–841 (2005).
Zickfeld, K., Knopf, B., Pethoukov, V. & Schellnhuber, H. J. Is the Indian summer monsoon stable against global change? Geophysical Research Letters 32, https://doi.org/10.1029/2005gl022771 (2005).
Sultan, B. & Janicot, S. The West African monsoon dynamics. Part II: the “preonset” and “onset” of the summer monsoon. Journal of Climate 16, 3407–3427 (2003).
Ramel, R., Gallée, H. & Messager, C. On the northward shift of the West African monsoon. Climate Dynamics 26, 429–440, https://doi.org/10.1007/s00382-005-0093-5 (2006).
Abe, M., Hori, M., Yasunari, T. & Kitoh, A. Effects of the Tibetan Plateau on the onsetof the summer monsoon in South Asia: The role of the air-sea interaction. Journal of Geophysical Research: Atmospheres 118, 1760–1776, https://doi.org/10.1002/jgrd.50210 (2013).
Kawamura, R. A mechanism of the onset of the Australian summer monsoon. Journal of Geophysical Research 107, https://doi.org/10.1029/2001jd001070 (2002).
Wang, B. & Xu, X. Northern Hemisphere summer monsoon singularities and climatological intraseasonal oscillation. Journal of Climate 10, 1071–1085 (1997).
Lin, H. & Wang, B. The time–space structure of the Asian–Pacific Summer Monsoon: a fast annual cycle view. Journal of Climate 15, 2001–2019 (2002).
Boos, W. R. & Storelvmo, T. Near-linear response of mean monsoon strength to a broad range of radiative forcings. Proceedings of the National Academy of Sciences of the United States of America 113, 1510–1515, https://doi.org/10.1073/pnas.1517143113 (2016).
Boos, W. R. & Storelvmo, T. Reply to Levermann et al.: Linear scaling for monsoons based on well-verified balance between adiabatic cooling and latent heat release. Proceedings of the National Academy of Sciences of the United States of America 113, E2350–2351, https://doi.org/10.1073/pnas.1603626113 (2016).
Levermann, A., Petoukhov, V., Schewe, J. & Schellnhuber, H. J. Abrupt monsoon transitions as seen in paleorecords can be explained by moisture-advection feedback. Proceedings of the National Academy of Sciences of the United States of America 113, E2348–2349, https://doi.org/10.1073/pnas.1603130113 (2016).
Lindsay, R. W. & Zhang, J. The thinning of Arctic sea ice, 1988–2003: Have we passed a tipping point? Journal of Climate 18, 4879–4894 (2005).
Moon, W. & Wettlaufer, J. S. On the existence of stable seasonally varying Arctic sea ice in simple models. Journal of Geophysical Research: Oceans 117, C07007, https://doi.org/10.1029/2012jc008006 (2012).
Wagner, T. J. W. & Eisenman, I. How climate model complexity influences sea ice stability. Journal of Climate 28, 3998–4014, https://doi.org/10.1175/jcli-d-14-00654.1 (2015).
Bathiany, S., Notz, D., Mauritsen, T., Brovkin, V. & Raedel, G. On the potential for abrupt Arctic winter sea-ice loss. Journal of Climate 29, 2703–2719 (2016).
Ferreira, D., Marshall, J. & Rose, B. Climate determinism revisited: multiple equilibria in a complex climate model. Journal of Climate 24, 992–1012, https://doi.org/10.1175/2010jcli3580.1 (2011).
Rose, B. E. J., Ferreira, D. & Marshall, J. The role of oceans and sea ice in abrupt transitions between multiple climate states. Journal of Climate 26, 2862–2879, https://doi.org/10.1175/jcli-d-12-00175.1 (2013).
Marotzke, J. & Botzet, M. Present-day and ice-covered equilibrium states in a comprehensive climate model. Geophysical Research Letters 34, L16704, https://doi.org/10.1029/2006gl028880 (2007).
Eisenman, I. Factors controlling the bifurcation structure of sea ice retreat. Journal of Geophysical Research: Atmospheres 117, D01111, https://doi.org/10.1029/2011jd016164 (2012).
Thorndike, A. S., Rothrock, D. A. & Maykut, G. A. & Colony, R. The thickness distribution of sea ice. Journal of Geophysical Research 80, 4501–4513, https://doi.org/10.1029/JC080i033p04501 (1975).
Bathiany, S. et al. Statistical indicators of Arctic sea-ice stability – prospects and limitations. The Cryosphere 10, 1631–1645 (2016).
Gildor, H. & Tziperman, E. Sea ice as the glacial cycles’ climate switch: role of seasonal and orbital forcing. Palaeogeography 15, 605–615 (2000).
Staver, A. C., Archibald, S. & Levin, S. A. The global extent and determinants of savanna and forest as alternative biome states. Science 334, 230–232 (2011).
Higgins, S. I., Scheiter, S. & Sankaran, M. The stability of African savannas: insights from the indirect estimation of the parameters of a dynamic model. Ecology 91, 1682–1692 (2010).
Staver, A. C. & Levin, S. A. Integrating theoretical climate and fire effects on savanna and forest systems. The American naturalist 180, 211–224, https://doi.org/10.1086/666648 (2012).
Higgins, S. I. & Scheiter, S. Atmospheric CO2 forces abrupt vegetation shifts locally, but not globally. Nature 488, 209–212, https://doi.org/10.1038/nature11238 (2012).
Lasslop, G., Brovkin, V., Reick, C. H., Bathiany, S. & Kloster, S. Multiple stable states of tree cover in a global land surface model due to a fire-vegetation feedback. Geophys. Res. Lett. 43, 6324–6331, https://doi.org/10.1002/2016gl069365 (2016).
Lewis, S. L. The 2010 Amazon drought. Science 331, 554 (2011).
Hilker, T. et al. Vegetation dynamics and rainfall sensitivity of the Amazon. Proceedings of the National Academy of Sciences of the United States of America 111, 16041–16046, https://doi.org/10.1073/pnas.1404870111 (2014).
Carré, M. et al. Holocene history of ENSO variance and asymmetry in the eastern tropical. Pacific. Science 345, 1045–1048 (2014).
Li, J. et al. El Niño modulations over the past seven centuries. Nature Climate Change 3, 822–826, https://doi.org/10.1038/nclimate1936 (2013).
Holmgren, M. & Scheffer, M. El Niño as a window of opportunity for the restoration of degraded arid ecosystems. Ecosystems 4, 151–159, https://doi.org/10.1007/s100210000065 (2001).
Charney, J., Stone, P. H. & Quirk, W. J. Drought in the Sahara: a biogeophysical feedback mechanism. Science 187, 434–435 (1975).
Rachmayani, R., Prange, M. & Schulz, M. North African vegetation–precipitation feedback in early and mid-Holocene climate simulations with CCSM3-DGVM. Climate of the Past 11, 175–185, https://doi.org/10.5194/cp-11-175-2015 (2015).
Claussen, M., Kubatzki, C., Brovkin, V. & Ganopolski, A. Simulation of an abrupt change in Saharan vegetation in the mid-Holocene. Geophysical Research Letters 26, 2037–2040 (1999).
Biasutti, M., Held, I. M., Sobel, A. H. & Giannini, A. SST forcings and Sahel rainfall variability in simulations of the twentieth and twenty-first centuries. Journal of Climate 21, 3471–3486, https://doi.org/10.1175/2007jcli1896.1 (2008).
Rodríguez-Fonseca, B. et al. Interannual and decadal SST-forced responses of the West African monsoon. Atmospheric Science Letters 12, 67–74 (2011).
Zeng, N., Hales, K. & Neelin, J. D. Nonlinear dynamics in a coupled vegetation–atmosphere system and implications for desert–forest gradient. Journal of Climate 15, 3474–3487 (2002).
Bathiany, S. et al. Beyond bifurcation – using complex models to understand and predict abrupt climate change. Dynamics and Statistics of the Climate System 1, https://doi.org/10.1093/climsys/dzw004 (2016).
Timmermann, A. & Lohmann, G. Noise-induced transitions in a simplified model of the thermohaline circulation. Journal of Physical Oceanography 30, 1891–1900 (2000).
Bathiany, S., Claussen, M. & Fraedrich, K. Implications of climate variability for the detection of multiple equilibria and for rapid transitions in the atmosphere-vegetation system. Climate Dynamics 38, 1775–1790, https://doi.org/10.1007/s00382-011-1037-x (2012).
Kloeden, P. E. & Schmallfuss, B. Asymptotic behaviour of nonautonomous difference inclusions. Systems and Control Letters 33, 275–280 (1998).
Ghil, M., Chekroun, M. D. & Simonnet, E. Climate dynamics and fluid mechanics: Natural variability and related uncertainties. Physica D: Nonlinear Phenomena 237, 2111–2126, https://doi.org/10.1016/j.physd.2008.03.036 (2008).
Arnold, L. Random Dynamical Systems. (Springer, 1998).
Kleinen, T., Held, H. & Petschel-Held, G. The potential role of spectral properties in detecting thresholds in the Earth system: application to the thermohaline circulation. Ocean Dynamics 53, 53–63, https://doi.org/10.1007/s10236-002-0023-6 (2003).
Held, H. & Kleinen, T. Detection of climate system bifurcations by degenerate fingerprinting. Geophysical Research Letters 31, L23207, https://doi.org/10.1029/2004gl020972 (2004).
Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461, 53–59, https://doi.org/10.1038/nature08227 (2009).
Medeiros, E. S., Caldas, I. L., Baptista, M. S. & Feudel, U. Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles. Scientific reports 7, 42351, https://doi.org/10.1038/srep42351 (2017).
Sévellec, F., Fedorov, A. V. & Liu, W. Arctic sea-ice decline weakens the Atlantic Meridional Overturning Circulation. Nature Climate Change 7, 604–610, https://doi.org/10.1038/nclimate3353 (2017).
Tjallingii, R. et al. Coherent high- and low-latitude control of the northwest African hydrological balance. Nature Geoscience 1, 670–675, https://doi.org/10.1038/ngeo289 (2008).
This work was carried out under the program of the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW). We are grateful to Chris Huntingford for his constructive comments that helped us to improve the manuscript. We would also like to acknowledge Michel Crucifix, Henk Dijkstra, and Peter Cox for their helpful comments. S.B. is eternally grateful to Nina Engelhardt and the University of Edinburgh for the inspiring working conditions.
The authors declare no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
About this article
Cite this article
Bathiany, S., Scheffer, M., van Nes, E.H. et al. Abrupt Climate Change in an Oscillating World. Sci Rep 8, 5040 (2018). https://doi.org/10.1038/s41598-018-23377-4
Physical Review E (2020)
Physics Letters A (2020)
Current Environmental Health Reports (2019)
Journal of Theoretical Biology (2019)
The Scandinavian Journal of Economics (2019)