the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Rateinduced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics
Ulrike Feudel
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 Final revised paper (published on 03 Nov 2023)
 Preprint (discussion started on 28 Mar 2023)
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RC1: 'Comment on npg20237', Anonymous Referee #1, 11 Apr 2023
The manuscript "Rateinduced 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 ratedependent tippings in nonautonomous multiscale 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 deterministicstochastic 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 scaleinvariance (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 Lorenz63 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. Noisedriven topological changes in chaotic dynamics. Chaos 31, 103115 (2021) https://doi.org/10.1063/5.0059461.
Dubrulle B. MultiFractality, 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/s41598021901381.
Linebyline comments
 I would suggest to use a uniform nomenclature for "timescale", i.e., "timescale" or "time scale".
 Lines 3749: 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 ds_{i} for clarity.
 Line 99: the same after "horizontal path" please add dp_{i}.
 Lines 145146: 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 ratedependent b is like the albedo feedback in energybalance 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 273275: 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: "ε_{1}" should be "ε_{i}".
 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 389390: is this an effect of the variations of the nature of the basin correspondi to X_{1}~3, X_{2}~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/npg20237RC1 
AC1: 'Reply on RC1 Response to Reviewer 1', Ulrike Feudel, 05 Jul 2023
Response to Reviewer 1:
I would like to thank the reviewer for his/her careful reading of the manuscript and pointing out some interesting references to be added. I have taken almost all suggestions of the reviewer into account and hope that the manuscript is now acceptable.
Here is a description to all the changes to improve the manuscript according to the suggestions of the reviewer.
“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 deterministicstochastic approaches of natural phenomena.“
I would like to thank the reviewer for the suggestion to additionally address the question of predictability in the review part of the manuscript. I have included a whole paragraph devoted to this question at the end of Sec. . This paragraph includes a short discussion of developments of methodology to identify early warning signals when approaching critical transitions. In addition, I shortly recall some approaches of how to predict critical transition including ideas of how to prevent them. This paragraph reads:
{\it {\bf Prediction of tipping points and early warning signals}}
In the course of climate change it becomes more and more important to find appropriate methods to predict tipping points and to identify early warning signals. One method that has been developed in the physics and chemistry literature is critical slowing down (CSD) of the restoring forces when a transition is approached \citep{Heinrichsetal1981,Ganapathrisubramanianetal1983,Tredicceetal2004,Schefferetal2009}. Resulting from these smaller restoring forces that bring the system back to its stable state after a perturbation, the response to inevitable noise is amplified, leading to a rising standard deviation \citep{Suroviatkina2005,Carpenter2006} and a lag1 autocorrelation when approaching the bifurcation \citep{Heldetal2004,Dakosetal2008}. These methods—critical slowing down and noise amplification—have become extremely popular over the last decade as possible earlywarning signals. Besides those methods other statistical approaches have been developed to estimate how close we are to tipping points in the climate system and in ecology \citep{Lenton2011,Lentonetal2012,Fanetal2020} or how probable noise and rateinduced transitions are \citep{RitchieSieber2017}. They have been used to estimate the proximity of several tipping points in climate such as e.g. the melting of the Greenland ice sheet \citep{BoersRypdal2021}, the collapse of the Atlantic Meridional Overturning \citep{Boers2021} or the loss of the Amazon rainforest \citep{Boultonetal2022}. Despite these various applications of early warning signals they have also been critically discussed from various perspectives \citep{Ditlevsenetal2010,Boettigeretal2012,Wagneretal2015}.
“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 scaleinvariance (turbulence is one of the possible examples). Some possible suggested (optional) references are highlighted below.”
I have added some suitable references suggested by the reviewer to emphasize more the impact of different scales on the topological structure in state space. This is now included in the introduction as follows:
These timescale separations lead to partly unexpected behaviors. Such scale dependence has been studied in the literature in different contexts such as e.g. climate sensitivity \citep{Bastiaansenetal2022}, tipping in excitable systems \citep{PieriniGhil2021,Vanselowetal2022}, overshooting and reversing tippings \citep{Ritchieetal2021,Ritchieetal2023} and the topological structure of invariant sets in complex systems including their characteristics like fractal dimensions \citep{Albertietal2023,Charoetal2021}.
I would like to thank the reviewer for the numerous corrections of English in the manuscript All the suggestions concerning notation and language have been taken into account. But there have been specific questions, which I answer again onebyone:
“Lines 3749: I would suggest to add some general references to the different outlined points.”
Since these are very general statements, I find it strange to assign them to some authors.
“Eq. (4): the role of a ratedependent b is like the albedo feedback in energybalance models. It would be interesting to mention, if this is the case, as a possible example in the context of the present paper.”
I would like to thank the reviewer for this interesting thought. The energybalance model looks indeed quite similar. However, to see that we would have the same behavior here, would need a thorough investigation and this cannot be explained in a halfsentence. Therefore, I omitted it here, but will undertake this investigation later.
“Lines 273275: this corresponds to a quadratic map in Eqs. 4. Does this mean that an attractor and/or a closed basin is missing/forbidden?”
After the saddlenode bifurcation only one attractor is indeed left and this is the attractor corresponding to extinction. To avoid here a bifurcationinduced tipping, the ramping interval has been restricted to a value of b below the saddlenode bifurcation.
“Lines 389390: is this an effect of the variations of the nature of the basins corresponding to X_1=3, X_2=3?”
I thank the reviewer for noticing that the explanation is not clear here. I have added it to the text now as follows:
The reason for this behavior is twofold. One reason is the separation between the intrinsic dissipative timescale and the timescale of environmental change. The nonautonomous basins of attraction are computed starting from a grid of initial conditions, but all of them have already converged to the neighborhood of the attractor at $X_1 \simeq 3$, $X_2 = 3$, when the attractor $(X_1 \simeq 1, X_2 = 3)$ appears in the saddlenode bifurcation due to a faster intrinsic timescale. On the other hand, the example studied describes either chemical concentrations or abundances of species, which have to be positive or equal to zero. This restriction of possible valid initial conditions are another reason, why none of the used initial conditions can reach the new attractor. They can only do so in the frozenin case, but not beyond a critical rate of environmental change.
“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.”
This explanation has been added as follows:
“Line 551: Code availability → _the link seems not to work.”
Now the link should work, I corrected the mistake.

RC2: 'Comment on npg20237', Anonymous Referee #2, 29 Apr 2023
The article reviews tipping phenomena in deterministic nonautonomous dynamical systems. It illustrates the behaviour of trajectories when crossing basin boundaries, and the nonautonomous 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 nonautonomous behaviour that is relevant in the context of environmental change, but that has been much less discussed compared to “static” / bifurcationrelated 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 nonautonomous 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= nonautonomous 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.)
• Subfigures 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 12 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. 411: tick labels are a bit small
Fig. 512: a, blines 9799:
“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 127128: “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 127128, line 133134: “even multistable” Strictly speaking, “multistability” includes bistability (so “or even” is not really adequate).
line 130131: “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 highercomplexity 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.
140141: “As a result, the system tips or collapses from a coraldominated into an algaedominated
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 158159 (+line 171, ...): “There is a vast literature on noiseinduced 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 noiseinduced 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 noiseinduced transition is therefore not the same as Ntipping (Ashwin et al., 2012), or “noiseinduced 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 “noiseinduced 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 basincrossing above).
Fig. 3 and line 204214: 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 224232: 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 314333: 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 X2hemisphere 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 nonautonomous basin of attraction B̃ at the final value c = 0.1 after the parameter drift and the frozenin 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 nonautonomous 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 387403: The hidden basins are an interesting phenomenon. Though I like the statespace 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 parameterstate 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 423426: 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. 1113. Again, why two epsilons? Would one not be enough to capture a time scale separation? As you write yourself in lines 450454, 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. 810 system 1, and Eq. 1113 system 2; it would help when refering to these.
Equations 67 and 89 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 535537: “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 “frozenin 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/npg20237RC2 
AC2: 'Reply on RC2 Response to Reviewer 2', Ulrike Feudel, 05 Jul 2023
Response to reviewer 2
I would like to thank the reviewer for the careful reading and the numerous suggestions to improve the manuscript. He/she has also made many suggestions to improve the figures which I have also taken into account. However new figures I could not supply since my request for a prolongation of the deadline was not answered. Instead I extended many of the explanations and formulated everything in words.
I hope that the manuscript is now much better readable. I will now answer all the concerns of the reviewer pointbypoint except for the corrections in English language, which I have all taken into account.
“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 terms are now explained as follows:
The unstable state (red ball) located on the hill of the stability landscape marks the basin boundary. This boundary separates the two basins of attraction, i.e., the two set of initial conditions which all converge to one of the respective attractors. In higher dimensional systems these unstable states on the boundary are of saddle type, possessing stable and unstable manifolds. The stable manifolds are hypersurfaces in state space whose dimension is equal to the number of stable directions or stable eigenvalues of the corresponding Jacobian matrix of the saddle, while the unstable manifolds correspond to hypersurfaces determined by the number of unstable directions or eigenvalues. In the special case of a twodimensional system the saddle steady state has two eigenvalues one stable and one unstable and the corresponding stable and unstable manifolds are onedimensional. The stable manifolds along which trajectories move towards the saddle make up the basin boundaries.
I would like to thank the reviewer for this suggestion, now the main basics in nonlinear dynamics are explained in a separate paragraph as follows:
“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,
I have included now this interesting reference.
“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 nonautonomous 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.“
The reviewer is correct in saying that the material presented is much more general than the title says. The title is chosen according to the readership of this journal, which are in general from geosciences. We are working already on a more general review for the broad readership of physicists, which will be submitted elsewhere.
“Comments about all figures showing the state space (X1 versus X2), i.e., Fig. 5, 6, 8, 9, 10”
I would like to thank the reviewer for the suggestions made to improve the figures. All figures are completely new and took almost all the suggestions into account, in particular the size of the labels, the number of trajectories shown, the color coding, the size of the symbols of steady states, using a) and b) instead of left right. I also changed completely the captions to explain all things shown in the figure. However, I could not realize some of the suggestions. These are:
It is impossible to include the whole vector field since it would only represent the vector field at one particular time instant, but I wanted to show the entire time evolution. This works best with showing the whole time evolution of trajectories including the parameter drift. To avoid too much confusion, I reduced the number of trajectories to 4 `and omitted the indication of the different time intervals which the trajectories go through (to the first and the second saddlenode bifurcation and the stop of the drift, which was indicated by dashed, dotted and white lines). It was meant to illustrate how far the trajectory got in state space, when the bifurcations occur. But I agree with the reviewer, that the most important parts, the dotted and white lines were barely visible so that the original purpose was not well illustrated. Now all trajectories are shown as full black lines. Since arrows were difficult to include and did not look well (since I was not plotting a vector field) I have now indicated the starting point of the trajectories. I changed the colors of the nullclines from candycolors to darker ones, but I did not change the colors in different panels. The colors represent at which time instant the nullclines are plotted, since they are all timedependent, one color corresponds to the starting point $t=0$, the other to the end point $t=T{end}$. This was only explained in the text, but I included it now also in the caption to make that very clear. The color coding of the nonautonomous basins of attraction is now also mentioned in the caption.
“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.
Of course not and I mentioned that already in the text in the next sentence line 104106. But I cannot follow fully the argumentation of the reviewer, because in his example one can separate the two, since changing the freshwater flux is the cause, but the change of the density is the response. Following the reviewer, one can also say that a perturbation of density (e.g. by noise) could be interpreted as fluctuations of the freshwater flux, which would also be a possible interpretation. However, I think that this discussion could also be very misleading since it is closely related to the example and could lead to confusion of the reader. Therefore, I omitted it here.
“line 125: “flow patterns in the ocean” => ocean circulation would probably be the more typical expression.”
I changed it in the text, since the cited papers indeed only include ocean circulation.
“line 127128: “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 127128, line 133134: “even multistable” Strictly speaking, “multistability” includes bistability (so “or even” is not really adequate).”
“line 130131: “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 highercomplexity 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.”
I agree with the reviewer, that in this part the formulations were a bit sloppy. All facts the reviewer mentions are of course clear to me. Indeed, bistability is part of multistability. I wanted to make a distinction between bistability and “true” multistability with more than 2 attractors coexisting on purpose, since the latter is related to much more complex behavior and is much less understood. I have changed the text to highlight this difference in another way and added a very recent paper on that by Lohmann et al. 2023.
I have rewritten this part completely and it reads now:
In those systems, bi or even multistability, i.e. the coexistence of more than 2 stable states for the same environmental conditions, has been discovered. For the AMOC, mostly two different stable flow patterns exist: one of them can be considered as a conveyor belt transporting heat to the Northern latitudes, releasing this heat to the atmosphere, forming North Atlantic Deep Water (NADW) which is transported back to the Southern latitudes at considerable depth. This would be the state, where the heat transfer to the north in ``on''. The other stable state is related to an ``off'' state. This bistability can give rise to a possible breakdown of the AMOC, which has been discussed employing several conceptual models \citep{Stommel1961,Rahmstorf1996,Rooth1982,Woodetal2019}. In those conceptual models often the second state is related to a reverse circulation. In large ocean circulation models this bistability has also been confirmed \citep{Weijeretal2012}, with an ``off'' state which does not relate to a reverse circulation but to a very weak circulation northwards. In large ocean circulation models it has been shown, that the system can even exhibit the coexistence of several different flow patterns related to different spatial patterns of heat transfer to the atmosphere \citep{Rahmstorf1995}. This occurrence of multistability has been confirmed recently with other highresolution models \citep{Mehlingetal2022,Lohmannetal2023}.
140141: “As a result, the system tips or collapses from a coraldominated into an algaedominated 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.
I agree with the reviewer that one should formulate more carefully, though Holbrook et al. use the terminology of tipping points in their study. That the collapse occurs and what are the possible reasons for it, is widely accepted in the ecological literature based on observations. However, that this can be interpreted as a tipping in the dynamical systems sense, is questionable as the existence of thresholds in ecology in general is controversially debated in ecology. I have added a corresponding citation Hillebrand et al. 2021. Therefore, I cut the word “tips” in the sentence. This part reads now:
Examples of alternative states in ecosystems have been discussed in the literature (cf. \citep{Folkeetal2004} and references therein), though the existence of thresholds in ecology is controversely debated \citep{Hillebrandetal2021}. A prominent example in which such transitions from one stable state to another are nowadays already observed are tropical coral reefs, which are found to be overgrown with green algae due to climate change and other anthropogenic and nonanthropogenic influences. As a result, the system collapses and exhibits a shift from a coraldominated into an algaedominated reef \citep{Holbrooketal2016}.
“There is a confusion of terms here that unfortunately is very common in the “tipping points” / dynamical systems literature. A noiseinduced 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 noiseinduced transition is therefore not the same as Ntipping (Ashwin et al., 2012), or “noiseinduced 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 “noiseinduced transitions”.”
I would like to thank the reviewer for pointing out this confusion in the two notations, what a noiseinduced transition is. In the literature indeed different descriptions are used for the phenomenon called “noiseinduced transition” To outline this a bit further I have changed the text accordingly and also cite again Ashwin whose definition of Ntipping often used is again different. The text reads now:
Though this description sounds like a result of a single trajectory, it needs a stochastic description, since one has to study ensembles of trajectories with different realizations of the noise and probability distributions (pdfs) over the state space. There is a vast literature on noiseinduced transitions in many different science disciplines and the notion of noiseinduced transitions differs across the literature. While many studies classify noiseinduced transitions as qualitative changes in the aforementioned pdf with the noise strength as the bifurcation parameter \citep{HorsthemkeLefever1984,Kuehn2011}, other works focus directly on the transition from one stable state to another mentioned under the influence of noise as outlined above. Ashwin et al. \citet{Ashwinetal2012} define Ntipping as a system which leaves the neighborhood of a quasistationary state due to the influence of fluctuations.
“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.”
This is a mistake in English, which I have corrected now. Indeed, the regime shift has been observed in the 90s.
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 basincrossing above).
Fig. 3 and line 204214: 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?
On purpose I have not provided an illustration for the basin crossing, since this is discussed later in the manuscript in more detail. Fig.3 is meant to illustrate the other case of rateinduced tipping which is not related to a basin boundary crossing. It is related to the mechanism discussed in Wieczorek et al. 2011 and Vanselow et al. 2019, where the rateinduced tipping occurs without the existence of an alternative state. It occurs in slowfast systems and requires a bended critical manifold. The exact mathematical conditions are outlined in Wieczorek et al. 2011. I have extended the explanation of this part of Fig. 3 to make this more clear. It reads now:
Since all aforementioned tipping mechanisms are related to the coexistence of alternative states, rateinduced tipping can also occur when there is only one stable state present and the system is characterized by different timescales (slowfast system). The dynamics of such systems can be described by so called critical manifolds in case of a perfect timescale separation or slow manifolds, when the timescale separation is finite. In case of a complex structure of the critical manifolds, for instance when these manifolds have stable and unstable parts which meet in a fold, then a rateinduced crossing of this fold can make the trajectory visit very different parts of the state space far away from the original stable state and perhaps even dangerous for the system. This mechanism of rateinduced critical transitions is illustrated in Fig.~\ref{tippingmech}d,e where the whole stability landscape is moved at a certain rate. Suppose that the stability landscape in Fig.~\ref{tippingmech}d is pulled with a certain rate towards the observer. Consequently, the ball will no longer be located in the minimum of the valley but will be displaced to the left. The restoring forces will start acting, and the ball begins to "roll” to catch the moving minimum. If the pulling rate is slow, then the stable state (ball) follows, or we say it {\it tracks} the minimum of the stability landscape. By contrast, in Fig. \ref{tippingmech}e, the rate of "pulling away" the stability landscape is much faster or comparable with the timescale of the restoring forces. In this case, the ball lands in a completely different region in state space, leaving the minimum's proximity and leading to qualitatively different dynamics. This large excursion in state space corresponds to rateinduced tipping since the system visits very different parts of the state space with qualitatively different behavior. If the change in the environmental conditions stops, this visit to a different state will be transient and, finally, the system returns to the stable quasistationary state, which has moved. This transient dynamics could lead to qualitatively different states, like population collapse in predatorprey systems \cite{Vanselowetal2019} or population outbreaks \cite{Vanselowetal2022b}.
“line 224232: 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).”
I have extended this paragraph with an example according to the suggestion of the reviewer. A merging with Sect. 3.5 does not fit, since the description of tipping in spatially extended systems deals with tipping in partial differential equations and in particular with pattern formation, while Sect. 3.5 deals with a simplified version of a kind of network approach with just two nodes. The extended paragraph reads now:
Several examples have been studied by Meron and coworkers in dryland vegetation models \cite{Zelniketal2013,Beletal2012,Zelniketal2018}. These models of different complexity study the interplay between vegetation and soil water. Besides the homogeneous states ``bare soil'' and ``full vegetation cover'' there exist depending on the environmental conditions, in general the precipitation level, different patterns like holes in the vegetation, stripes and spots of vegetation. These patterns can coexist and fronts separating the different pattern can occur. The speed of the fronts determines the speed with which one patterns is exchanged by the other leading to a gradual tipping between different patterns in the whole area.
“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.”
Fig. 4 could have been extended by a second panel showing the potential landscape at the beginning and the end of the parameter drift. However, the time for preparing the figure was not sufficient before the deadline.
“Caption of Fig. 4: why does it refer to “Eqs. 5”? Isn’t this system described by Eq. 4?”
I would like to thank the reviewer for pointing out this mistake. It has been corrected.
“lines 314333: 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 whole paragraph has been rearranged and reads now:
Rescaling the time in terms of $\tau=\varepsilon_1 t$, it turns out that only the ratios between the intrinsic timescales of the different subsystems $\varepsilon_2/\varepsilon_1$ and the ratios between the timescale of transport or coupling and the intrinsic timescale (like, e.g., $c/\varepsilon_1$) are important. Therefore, we will continue the analysis with the rescaled equations:
\begin{eqnarray}
{\dot X}_1 &=& (X_1s_1)(X_1s_2)(X_1s_3) + cX_2,\\\
{\dot X}_2 &=& \varepsilon(X_2s_1)(X_2s_2)(X_2s_3),
\label{masterslave}
\end{eqnarray}
where $\varepsilon$ and $c$ are the corresponding ratios.
System 2 appears as a driver or master for system 1. We analyze the dynamics in the most intuitive way, we use the concept of nullclines, which are given by the algebraic equations ${\dot X}_1=f_1(X_1, X_2)=0$ and ${\dot X}_2=f_2(X_2)=0$. While for the driver system 2, the nullclines are given by straight lines at the values of the three steady states of system 2, the nullcline of system 1 is represented by the cubic function $X_2 =(X_1s_1)(X_1s_2)(X_1s_3)/c$. The intersection points of $f_1=0$ and $f_2=0$ are the steady states of the masterslave system. Their stability can be computed from the eigenvalues of the corresponding Jacobian. An illustration of two possible situation is given in Fig.~\ref{BasinsMasterSlave}a,b. Depending on the internal parameters $\varepsilon$ and the coupling strength $c$, the system possesses two, three, or four stable, steady states in the considered parameter range of coupling strength $c$. The shown two cases serve as the beginning [Fig.~\ref{BasinsMasterSlave}a] and the end [Fig.~\ref{BasinsMasterSlave}b] point of the parameter drift along a linear ramp. We are fixing all parameters (frozenin case) and compute the attractors and their corresponding basins of attraction by choosing a grid of initial conditions in a specified region of state space and integrating them all in parallel until they reach the attractor. This allows us to compute also the relative size of the basins of attraction ${\mathcal B}_{\mathcal A}$ as the quotient of the number of initial conditions converging to attractor ${\mathcal A}$ divided by the total number of initial conditions taken into account \citep{Feudeletal1996}. Fig.~\ref{BasinsMasterSlave} shows that the state space is ``partitioned'' into different basins of attraction indicated by different colors with basin boundaries separating them. In the frozenin case, the basin boundaries are invariant sets that cannot be crossed by trajectories and, hence, represent rigid boundaries in state space for the trajectories.
“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.”
Since I am using the same measure as introduced in Kaszas et al. 2019 I have kept the name and inserted also the formula to avoid confusion, though I agree with the reviewer that “fraction” would be more appropriate. I have inserted this now in the text:
We follow here the approach introdcued in \citep{Kaszasetal2019} and compute the tipping probabilities $\mathcal P{\mathcal A1A2}$. We calculate first the relative size of the nonautonomous basin of attraction ${\mathcal {\tilde B}({\tilde A}2)}$ for the quasisteady state ${\tilde{A}2}$ at the end of the parameter drift simulation $T{end}$ and take its intersection with the frozenin basin of attraction $\mathcal B$ of the frozenin attractor ${\mathcal A}1$ at the beginning of the simulation. To compute a proper part of the initial conditions we have to normalize this quantity by the frozenin basin of attraction ${\mathcal B}({\mathcal A}1)$ at the beginning of the simulation. Loosely speaking, we calculate that fraction of initial conditions which would have converged to the attractor ${\mathcal A}1$ in the frozenin case, but which during the parameter drift tip to the basin of attraction of the moving quasisteady state ${\tilde A}2$. In mathematical terms this can be expressed as follows:
\begin{equation}
{\mathcal P}{\mathcal {\tilde A}1{\tilde A}2} = \frac{\mathcal {\tilde B}({\tilde A}2) \bigcap {\mathcal B}({\mathcal A}1)}{{\mathcal B}({\mathcal A})1)}
\end{equation}
The continuous change of the tipping probabilities shown in Fig.~\ref{TipProbUni} indicates that we observe partial tipping.
“line 423426: 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. 1113. Again, why two epsilons? Would one not be enough to capture a time scale separation? As you write yourself in lines 450454, only the ratio matters.”
I have changed the whole notation in the new version of the manuscript and rescaled time by $\epsilon1$ as discussed earlier in Sect. 3.4. This explanation in Sect. 4 is now removed, since the whole manuscript deals now only with the ratios of the parameters.
“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. 810 system 1, and Eq. 1113 system 2; it would help when refering to these.
Equations 67 and 89 are identical; could remove one pair.”
“Fig. 12: Due to the overlap, it is hard to see which two basins change change together. One might use a combination of colour and line type to solve this.”
I tried it out, but it did not look any better, so I left the figures as they were. The line types are a combination of the two when overlapping, but I agree, it is hard to see.
“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.”
Since there is no qualitative difference between the results in Sec. 3.3 and 3.4 compared to Sec. 3.5. I expanded the section by an explanation why there are quantitative differences. Instead of presenting only the example with a timescale separation I have now added also the case of no timescale separation to compare with. Here is the explanation added:
For ecological systems, this would be the appropriate coupling when considering two populations in two different habitats coupled through species migration. The same coupling would be used for coupled chemical systems, where diffusion is assumed to be the most important spatial transport process. Following the same protocol of numerical simulations with the same parameter values, we observe that qualitatively we observe the same behavior as for the other coupling schemes with one important difference. The masking effect occurs for much larger $T_r$, i.e., a much slower rate of environmental change, if the two systems are identical and no timescale separation occurs. This is due to the fact, that the effective coupling is much smaller than in cases of a masterslave coupling or a mutual coupling. The coupling strength is multiplied by the difference of the two variables in system 1 and 2 instead of the variable itself and this diminishes the effect of the coupling. Therefore, the overall effect for identical systems is much smaller than in the other cases (Fig.~\ref{TipProbDiff}a,b. By contrast, if we introduce the timescale separation shown in Fig.~\ref{TipProbDiff}c,d, we note, that the masking effect is again stronger now taken over by the timescale separation. This emphasizes again the role of the timescale separation between coupled systems.
“line 535537: “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 agree with the reviewer that this is just a qualitative description. To make this more quantitative and precise, other methods are needed than simulations. This research is already under way, but will be published elsewhere.

AC2: 'Reply on RC2 Response to Reviewer 2', Ulrike Feudel, 05 Jul 2023