the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics
Ulrike Feudel
Abstract. The climate system as well as ecosystems might undergo relatively sudden qualitative changes in the dynamics when environmental parameters or external forcings vary due to anthropogenic influences. The study of these qualitative changes, called tipping phenomena, requires the development of new methodological approaches that allow modeling, analyzing, and predicting observed phenomena in nature, especially concerning the climate crisis and its consequences. Here we briefly review the mechanisms of classical tipping phenomena and investigate in more detail rate-dependent tipping phenomena which occur in non-autonomous systems characterized by multiple timescales. We focus on the mechanism of rate-induced tipping caused by basin boundary crossings. We unravel the mechanism of this transition and analyze, in particular, the role of such basin boundary crossings in non-autonomous systems when a parameter drift induces a saddle-node bifurcation in which new attractors and saddle points emerge, including their basins of attraction. Furthermore, we study the detectability of those bifurcations by monitoring single trajectories in state space and find that depending on the rate of environmental parameter drift, such saddle-node bifurcations might be masked or hidden and they can be detected only if a critical rate of environmental drift is crossed. This analysis reveals that quasi-stationary saddle points in non-autonomous multistable systems are the organizing centers of the global dynamics and need much more attention in future studies.
Ulrike Feudel
Status: final response (author comments only)
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RC1: 'Comment on npg-2023-7', Anonymous Referee #1, 11 Apr 2023
The manuscript "Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics" by Ulrike Feudel is an outstanding contribution to the field of tipping phenomena and their characterization in complex systems as the climate and ecosystems. It presents a concise but complete review on the mechanisms of classical tipping phenomena, especially focusing on rate-dependent tippings in non-autonomous multi-scale systems. The manuscript is clear and well written, the length and the references are appropriate, and the topic is timely and of wide interest. I have some minor recommendations to improve some missing concepts (for a wider audience) as outlined below.
General comment
As a general comment I would suggest to stress a bit more the question of "predictability" of tippings, especially in relation with the nature of tipping phenomena, since it is a central issue in many natural system and in the description with deterministic-stochastic approaches of natural phenomena.
Furthermore, another important issue to be highlighted could be the role of processes operating at different scales in changing the topology and the geometry of fixed point and attractors, as well as, the role of symmetries and scale-invariance (turbulence is one of the possible examples). Some possible suggested (optional) references are highlighted below.Suggested (optional) references
Alberti T. et al. Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems. Chaos. 2023 Feb;33(2):023144. doi: 10.1063/5.0106053.
Bastiaansen R. et al. Climate response and sensitivity: time scales and late tipping points. Proc. R. Soc. A. 479:20220483 (2023) http://doi.org/10.1098/rspa.2022.0483.
Charó G. et al. Noise-driven topological changes in chaotic dynamics. Chaos 31, 103115 (2021) https://doi.org/10.1063/5.0059461.
Dubrulle B. Multi-Fractality, Universality and Singularity in Turbulence. Fractal Fract. 2022, 6(10), 613; https://doi.org/10.3390/fractalfract6100613.
Pierini, S., Ghil, M. Tipping points induced by parameter drift in an excitable ocean model. Sci Rep 11, 11126 (2021). https://doi.org/10.1038/s41598-021-90138-1.
Line-by-line comments
- I would suggest to use a uniform nomenclature for "timescale", i.e., "timescale" or "time scale".
- Lines 37-49: I would suggest to add some general references to the different outlined points.
- Line 48: to avoid repetition I would suggest to change "than the timescale" with "than that".
- Line 81 and Line 84: "i.e." → "i.e.,".
- Line 98: after "vertical path" I would add the notation dsi for clarity.
- Line 99: the same after "horizontal path" please add dpi.
- Lines 145-146: I would suggest to list some other possible types of bifurcations.
- Line 157: "systems state" → "system's state".
- Line 254: please add parenthesis around X(2) or delete around X(3) for consistency.
- Eq. (4): the role of a rate-dependent b is like the albedo feedback in energy-balance models. It would be interested to mention, if this is the case, as a possible example in the context of the present paper.
- Figure 4: in the caption should be "Eqs. 4" and a description of the cyan line (fixed point X(3)?) is missing.
- Lines 273-275: this corresponds to a quadric map in Eqs. 4. Does this mean that an attractor and/or a closed basin is missing/forbidden?
- Line 322: "ε1" should be "εi".
- Figs. 5, 6, 8, 9, 10: please add for clarity what are the filled colored areas in yellow, green, blue, and violet.
- Figure 6: the white line is not clearly visibile, I would suggest to change its color.
- Lines 389-390: is this an effect of the variations of the nature of the basin correspondi to X1~3, X2~3?
- Line 402: how tippint probabilities are evaluated?
- Figure 7: I would suggest to also label the different basins in previous figures for consistency and clarity.
- Section 3.3: it is clear that "drastic" effects are more evident for changing rate of environmental than timescale of the process, this is crucial for predictability and particularly true for climate change. A few explanatory/additional lines on this would be desirable.
- Line 551: Code availability → the link seems not to work.
- Line 570: reference "Bastiaansen et al., 2020" is duplicated.
Citation: https://doi.org/10.5194/npg-2023-7-RC1 -
RC2: 'Comment on npg-2023-7', Anonymous Referee #2, 29 Apr 2023
The article reviews tipping phenomena in deterministic non-autonomous dynamical systems. It illustrates the behaviour of trajectories when crossing basin boundaries, and the non-autonomous basins of attraction, and the phenomenon of “hidden” stable states. Although these phenomena are known in the dynamical systems community, I believe that the study works well in reviewing them, drawing attention to non-autonomous behaviour that is relevant in the context of environmental change, but that has been much less discussed compared to “static” / bifurcation-related tipping points. In my opinion, the writing is quite clear and informative, and the content and links to previous literature are thorough. I also believe that the article will be useful for the readership of the journal.
I only have a number of minor comments and suggestions.
The paper explains tipping phenomena so clearly that it could be a chance to also explain a few necessary basics of dynamical systems theory, in particular, what a “manifold” and “saddle points” are (terms are used a lot but not explained). This could make the paper more accessible for a broader readership.The manuscript cites the relevant literature as far as I can judge. There is a new study by Ritchie et al. about a very similar topic which could be cited: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.052209,
Title: Why “in ecosystems and climate”? The systems presented are so general that they are not restricted to these fields. Environmental change is of course a good context and illustrates the relevance of non-autonomous tipping, so the examples in the text are useful, but I am sceptical if the title should make the scope more narrow than it actually is.
Comments about all figures showing the state space (X1 versus X2), i.e., Fig. 5, 6, 8, 9, 10:
• Overall, in all the phase space figures, less intense / candy type colouring of the basins might look more appealing and also improve the visibility of the trajectories.
• The green nullcline is often hard to see, especially when combined with a green basin.
• The “*” (fixed points) are visible but too small to see that they are actually a “*”.
• Figures might be more understandable if one could see the whole vector field, not a number of trajectories.
• Arrows indicating the direction of the trajectories (or vectors) would help a lot! Otherwise, it takes a moment to even understand that they are trajectories, and even longer to infer the direction.
• One could remove the distinction of white and dashed / continuous lines. I did not find them so important, but they add complexity that could distract from the more essential things.
• “trajectories are plotted in black during the parameter drift and in white after the drift when c is constant” (Fig. 6, 10) – the white lines are very hard to see and quite short anyway, they could just be black as well, as mentioned above
• The captions are difficult to digest, I suggest to reorder the information, starting with general features displayed (colour= non-autonomous basins, lines=trajectories and nullclines, left vs right figure), and only then provide details (dashed / dotted / white), possibly outsourcing some numbers to the text; the captions could also mention which system the figue displays (which Equations / system no.)
• Sub-figures could be named a, b, …, not left / right.
• Fig. 5: It could make sense if nullclines for X1 and X2 have different colours in each panel, instead of magenta left and green right (these could not be confused anyway); Fig. 5 could also include 1-2 examples of trajectories from the initial condition to the equilibrium, with arrows
• Fig. 8: what is magenta here was white in the previous figures. If you keep what is now displayed in white (should change the colour, see above), I suggest to use the same colour – maybe red to make it visible.Fig. 2,3: tick labels are too large
Fig. 4-11: tick labels are a bit small
Fig. 5-12: a, blines 97-99:
“These disturbances correspond to the displacement of the state from the valley of the fixed stability landscape, depicted as a vertical path of perturbations in Fig. 2. On the other hand, disturbances in the system parameters or external forcings change the stability landscape. Both types of disturbances are possible and have very different effects (Schoenmakers and Feudel, 2021).” Is this always clearly separable? For example, perturbing water density (state variable) and applying freshwater flux (forcing) in an ocean model could be argued to be essentially the same thing.line 125: “flow patterns in the ocean” => ocean circulation would probably be the more typical expression.
line 127-128: “In those systems, bi- or even multistability has been discovered”. Could be misleading. There are some (often simple) models that show multistability, but complex ESMs usually do not, and there are large uncertainties.
line 127-128, line 133-134: “even multistable” Strictly speaking, “multistability” includes bistability (so “or even” is not really adequate).
line 130-131: “The other stable state would be a reverse circulation pattern.” This may be the case in very simple models like Stommel’s model, but as far as I know not in higher-complexity models. There, the alternative state is an “off” state, often with still some (weak) overturning, but no flow reversal. I suggest the author checks and adjusts this.
140-141: “As a result, the system tips or collapses from a coral-dominated into an algae-dominated
reef (Holbrook et al., 2016).” Again, my impression is that there is much more uncertainty than this sentence suggests. I’m not so familiar with this research field, but I believe that the question whether coral reefs display several stable states is inconclusive.line 158-159 (+line 171, ...): “There is a vast literature on noise-induced transitions in many different science disciplines since they have been studied since the 80’s (Horsthemke and Lefever, 1984;...)”
There is a confusion of terms here that unfortunately is very common in the “tipping points” / dynamical systems literature. A noise-induced transition refers to a (possibly radical) change in the pdf of the state when the noise intensity (a parameter of the stochastic system) is changed. This is also the definition used in Horsthemke and Lefever 1984, as well as Kuehn 2011, which the author cites here. An example of this phenomenon would be stochastic resonance (in case an oscillating forcing is also applied in addition). A noise-induced transition is therefore not the same as N-tipping (Ashwin et al., 2012), or “noise-induced tipping” as the author calls it here, which just describes a single event in a single realisation of the system. It could be a good opportunity to clarify this common misunderstanding in the paper, or at least the paper should avoid the misleading term of “noise-induced transitions”line 161: “...living in the sediment of the North Sea at the end of the 90s” is a bit confusing (order of words). The 90ies refers to the regime shift mentioned earlier I suppose.
Fig. 3: The cyan colour is hard to see; it is unclear how the potential is accelerated (to what side) – to the left according to the trajectory of the ball, but the arrows point to the right; it is not visible where the ball would cross an unstable equilibrium in case e (see point about absence of basin-crossing above).
Fig. 3 and line 204-214: I am a bit confused why a large excursion in phase space without crossing any equilibrium point counts (no basin crossing) as a tipping here. Isn’t this definition a bit arbitrary and too qualitative? If the ball never leaves its basin and does not even cross an unstable equilibrium (or basin boundary), what excursion is large enough to constitute a tipping?
line 224-232: I find the case of spatially extended systems particularly interesting. These systems are often neglected in other “tipping point” related literature, so I here see the opportunity that this paper could add something. Some more text about this case with an example might thus be a good idea. This could be merged with Sect. 3.5 which treats diffusive systems (but also too briefly).
Sect 3.1: A figure with the equilibria of the population model would be a nice addition to the equations, maybe as part of Fig. 4.
Caption of Fig. 4: why does it refer to “Eqs. 5”? Isn’t this system described by Eq. 4?
lines 314-333: Fig. 5 is mentioned only very late, could refer to it as soon as the concept of nullclines is introduced; I think this would help to get the idea.
The step from Fig. 5 to 6 could be illustrated a bit more, to support intuitive understanding. For example, one could show a few intermediate steps here, with snapshots of the system’s state in between (one trajectory in each X2-hemisphere might be enough), and the (“frozen”) basins of attraction. This would help readers to understand why some fixed points are not reached by the trajectory even though they exist in the final “frozen” stability landscape.
line 366: “However, comparing the non-autonomous basin of attraction B̃ at the final value c = 0.1 after the parameter drift and the frozen-in basin B reveals that the new boundaries of the basin of attraction after the drift are different from the stable manifolds of the corresponding saddle point as the saddle point does not even lie on the boundary of the non-autonomous basin of attraction B̃“ - good statement, I suggest to point out where we see this in the figures (Fig. 6 right versus Fig. 5 right I guess; by the way, naming the Figs with a and b might be good).
lines 387-403: The hidden basins are an interesting phenomenon. Though I like the state-space figures, they may not be ideal to illustrate this phenomenon. What about a figure displaying the parameter c on the horizontal versus some state variable on the vertical? Then I guess that it could be seen why some stable stable states are hidden when c is varied slowly, and why they could be realised under certain faster shifts in c. These two cases could be displayed as trajectories in the parameter-state space.
line 402 + Fig. 7b: It may be a bit misleading to refer to “tipping probabilities”. This term makes me think of a stochastic system with always the same parameters and initial conditions but different noise realisations. e.g., see https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.052209
But here, the outcome depends only on the distribution of the initial conditions. I would rather call it “frequency” or “fraction” of tipping trajectories.line 423-426: A bit short. Some more description and explanation in words about what differences we see between the plots, and why, would be great. Also, it’s not clear to me why we need to epsilons. Could we not just remove epsilon 1 in this system?
Eqs. 11-13. Again, why two epsilons? Would one not be enough to capture a time scale separation? As you write yourself in lines 450-454, only the ratio matters.
Fig. 10: It could help to have one figure showing the static system similar to Fig. 5 for the coupled system.
You could call Eq. 8-10 system 1, and Eq. 11-13 system 2; it would help when refering to these.
Equations 6-7 and 8-9 are identical; could remove one pair.
Fig. 12: Due to the overlap, it is hard to see which two basins change together. One might use a combination of colour and line type to solve this.
Sect 3.5 appears only as an afterthought, though diffusive coupling plays an important role in many models of many different systems. I suggest to expand this section, adding a similar analysis as for models 1 and 2 above, or at least explain in some more detail how and why the results differ.
line 497: focused (one s)
line 519: “The slower…” (capital)
line 535-537: “either the trajectory “meets” the saddle point directly or it approaches first the neighborhood of its moving stable manifold and travels along it until the saddle point is reached for the crossing.” - This is a very qualitative distinction, isn’t it?
I am not sure if “frozen-in case” is the best wording for the case of a static stability landscape. It should be checked what the most common name is in previous literature and stick to it, to avoid confusion. If there is no common name, it might be worth to coin a good term here, and put some thought into it. “frozen” does convey the meaning but also seems a bit colloquial to me.
Citation: https://doi.org/10.5194/npg-2023-7-RC2
Ulrike Feudel
Ulrike Feudel
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