the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Jan 2025
Global Stability and Tipping Point Prediction of the Coral Reef Ecosystem
Abstract. Coral reefs are highly diverse and important marine ecosystems, with the potential for multiple stable states and critical transitions between alternative states. Using landscape-flux theory, we explore the dynamics of these systems in the face of stochastic perturbations. We quantified the average flux as the non-equilibrium driving force and the entropy production rate as the nonequilibrium thermodynamic cost. We also compute the nonequilibrium free energy and time irreversibility from the cross-correlation functions as early warning signals for critical transitions. These early warning signals, characterized by turning points in the middle of two bifurcations, can provide predictions for both left and right bifurcations, much earlier than critical slowing down, a more traditional indicator that is evident only closer to the bifurcation point(s).
Status: final response (author comments only)
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RC1: 'Comment on egusphere-2025-57', Anonymous Referee #1, 25 Feb 2025
Dear authors,
Thank you for the interesting paper on early-warning signals in a coral-algae model. I read the manuscript with great care. As I am not a mathematician/physicist, the methods and results were tough for me to understand and more work is definitely needed on this manuscript to make it understandable for a broader (non-technical) audience. The mathematical concepts and their interpretation need to be explained in plain language. Below, I will first answer the ESD questions and then add some major comments. All my comments are in de supplemented pdf.
Answers to ESD journal-specific questions
In the full review and interactive discussion, the referees and other interested members of the scientific community are asked to take into account all of the following aspects:
- Does the paper address relevant scientific questions within the scope of ESD?
In theory it does. Early-warning signals are relevant, especially for corals which are thought to be sensitive to catastrophic shifts. However, the early-warning signals proposed in this paper seem to stem only from the model and it is unclear to me how these could be used in reality. The early-warning signals proposed in this paper are only compared to the notion of critical slowing down and the application of the proposed early-warning signals is not discussed in the manuscript. The concept itself (using the landscape flux) is not new. The lack of applicability of these concepts and the technical nature of the manuscript makes me question the relevance of this paper for the broader audience.
- Does the paper present novel concepts, ideas, tools, or data?
The concept is not new, but its application to a coral-algae system is, I believe.
- Are substantial conclusions reached?
The proposed early-warning signals are compared to one other early-warning signal (critical slowing down) and it is concluded that the newly proposed early-warning signals can detect a transition earlier than critical slowing down. I think there is more potential here, because the applicability of the proposed early-warning signals to real systems is not discussed, nor are they compared to other early-warning signals. It is also confusing to me whether the authors refer to noise-induced transitions or bifurcations (i.e., press disturbance/gradual parameter changes). Indeed, the landscape free flux and the other concepts seem to be able to pick up a bifurcation "earlier" than critical slowing down, but how is that related to noise-induced transitions? Moreover, there has been much critique on the use of critical slowing down as early-warning signal, so it is not a very new conclusion that there is an early-warning signal that is better than critical slowing down.
- Are the scientific methods and assumptions valid and clearly outlined?
This manuscript is very technical and really difficult to understand for someone like me who's well-informed about tipping and ecological models, but not about the mathematical analyses of nonlinear PDE systems. Work is thus needed to make the methods and assumptions clear for a non-technical audience. I believe the methods are valid. Again, the use of the landscape flux as early-warning signal is not new, but it is difficult for me to judge. One assumption that I have doubts about for this system, is the assumption that the noise is uncorrelated between the three components. In the coral-macroalgae system, I assume noise to be coming from external environmental forces which would impact the three components similarly. I also have doubts about the assumption of a spatially well-mixed system. For those two assumptions, I'd be interested to read more about the implications of these. I understand that the analyses become more complex if the noise is correlated, but how do you expect it to impact the results?
- Are the results sufficient to support the interpretations and conclusions?
Yes.
- Is the description of experiments and calculations sufficiently complete and precise to allow their reproduction by fellow scientists (traceability of results)?
I think that a mathematician or physicist that is very familiar with these kinds of analyses could replicate the results.
- Do the authors give proper credit to related work and clearly indicate their own new/original contribution?
No, I think there are very few papers cited. For example, you mention one critique of critical slowing down as early-warning signal, but there has been much more critique and I actually think that the only reason that researchers still refer to critical slowing down as a (useful) early-warning signal is its applicability in real systems.
- Does the title clearly reflect the contents of the paper?
I wouldn't say "the" coral reef system. And it should be clear in the title that this is about a model and only one type of model.
- Does the abstract provide a concise and complete summary?
I didn't understand most of the abstract because it is very technical.
- Is the overall presentation well structured and clear?
Yes. I think the manuscript is well-structured. The results as they are can be written more concisely and I didn’t understand the added value of some of the figures. There should be some explanations added for non-technical readers though. The properties/concepts that are derived from the model and used as early-warning signals are now mostly introduced in the results whereas they should be introduced in the methods. I think a table with an explanation of these properties would be help to keep an overview of everything that is calculated.
- Is the language fluent and precise?
There are some spelling/grammar mistakes, but not much. The language is precise enough, but too technical. Some things could be more precise such as whether it concerns noise-induced transitions or bifurcations and what does it mean that the proposed early-warning signals predict a transition 'earlier' than critical slowing down. What is earlier in this respect? How does it relate to real changes in a coral-algae system?
- Are mathematical formulae, symbols, abbreviations, and units correctly defined and used?
I did not find any errors. No units are given for the parameters.
- Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated?
I don’t think that all figures are necessary. There should be a clarification of the mathematical conceps used and also of the mathematical methods.
- Are the number and quality of references appropriate?
No, I think there are too few references. More background on early-warning signals in general would be nice, as well as on critical slowing down in particular and the criticism. Also, I'd like to read more background about the landscape flux as early-warning signal, its background and applications and possible criticism
- Is the amount and quality of supplementary material appropriate?
I could not find the supplementary figures and it was not clear to me from the main text what would be in those figures and what is their added value.
Major comments
- The paper is highly technical and therefore difficult to understand for a non-technical reader. To be appropriate for this journal non-technical explanations should be added to the paper. Otherwise, the paper might be more suitable for a technical journal.
- You compare the early-warning signals only to critical slowing down, whereas there are many more early-warning signals proposed in the literature. Also, critical slowing down has received much more criticism than listed in this paper.
- To me, the assumptions of a well-mixed space and non-correlated noise are not obviously appropriate for this system. Some discussion should be added on how these assumptions affect the results and in which domain/spatial extent they are appropriate.
- I was confused about the many different measures proposed, such as the free flux and the landscape flux and the intrinsic potential and the entropy production rate etc. These properties are calculated only in the results section without an explanation of what they are in normal words. Perhaps you can compile a table (in methods) with all these different concepts and add a non-technical explanation. I also don’t understand how these measures relate to real-world systems.
- In extension, I don’t understand how these early-warning signals are relevant for real coral systems. Since early-warning signals are only relevant in real systems, not in model systems, I think some explanation should be given with respect to the applicability of these early-warning signals.
- In general, the reference to real-world systems is lacking for me. How does a gradual shift in g relate to the real systems for example? And in h? The noise is assumed to be uncorrelated between the three variables, but what would the noise relate to with respect to real systems? What do the different early-warning signals relate to in real systems? Perhaps you can add the latter to the table.
- You sometimes refer to transitions in the sense of a bifurcation when continuing in a parameter (mostly g, and then also h at the end), but sometimes you refer to noise-induced transitions. I got confused as to whereas the early-warning signals relate to bifurcations or noise-induced transitions.
- Perhaps it would help a less technical reader if you’d present an equilibrium analysis of your essentially 2D system. The phase state could be in an appendix, and you could describe the equilibrium dynamics in words. Now, to my feeling, it jumps from the system to all these deduced properties such as the potential.
More detailed comments are in the supplemented PDF.
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RC2: 'Comment on egusphere-2025-57', Anonymous Referee #2, 24 Mar 2025
The paper introduces a method to approximate global stability in non-linear out-of-equilibrium systems, using as an example the case of coral reefs. The authors show that separating the gradient of the potential from the curl force serves as a proxy of changes in the basin of attraction, proposing another early warning for potential tipping points. Interestingly, their proposal seems to outperform other commonly used indicators such as the expected increase in autocorrelation and variance in critical slowing down.
One of the main contributions of the paper is the possibility of using the cross-correlation, entropy production rate, mean flow, or free energy as early warning given its performance with respect to critical slowing down. There are multiple disadvantages of critical slowing down, one of them is the fact that it only works with very low levels of noise [1]. Do you get equally good performances at higher levels of noise? It seems you’re using small fluctuations (D ~ 10^-5). If not, at what signal to noise ratio does the warning disappear? I believe these considerations are important to better understand the applicability of your proposal to real world data.
Other minor comments listed below.
## Specific comments
- Methods: I know the coral reef model used has only two stable states for simplicity. But I think is fair with the reader to point out some of the empirical work that has shown more attractions, up to 6 if I remember correctly: hard corals, turf algae, macro algae, soft corals, coralimorpharians, urchin barrens. See refs 2-3 for examples.
- Fig 5. Are the units of the colour bar always as x 10^-3? If so add the scale to all insets. In any case, why si the variation of the scale differs so widely across a, b and c?
## References
[1] www.doi.org/10.1111/j.1461-0248.2010.01439.x
[2] www.doi.org/10.1098/rstb.2013.0268
[3] www.doi.org/10.3354/meps07815
Citation: https://doi.org/10.5194/egusphere-2025-57-RC2
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