We thank the reviewer for the positive assessment of our work and the useful comments and suggestions. Below we provide a point by point response (in italics).

# Reviewer 1

This is an excellent review of empirical systems to which EWS methodologies have been applied, as well as a summary of the main methods.  I am not aware of anything in the existing literature that treats the subject as comprehensively as the authors have done here, and thus it adds significantly to existing knowledge.  The alluvial plots are very informative. 

I only have a few suggestions about the writing:

• Lines 95-102 and elsewhere: Although the focus of the paper is on bifurcations, the authors could mention that tipping points may also occur through a phase transition, and reference literature such as Hagstrom and Levin (https://arxiv.org/abs/2110.12287).

We mention this now in line 104 and add the reference.

• Table 2: Machine learning per secannot be classified as a non-CSD based approach, since ML can detect CSD (e.g. Bury et al 2021).  Machine learning is vast and putting it in one of these table cells would almost be like putting ‘mathematics’.  If the authors mean a particular approach based on machine learning, they could specify it. 

We corrected this in the table following the suggestion.

• The paper does not say much about structures underlying tipping points, aside from the potential well diagrams. I realize that the paper is not about dynamical systems, but it would be good to have a paragraph or two about types of bifurcations, and mention their implicit assumption that they are writing about fold bifurcations, since those lead to sudden shifts (although it’s also possible to get sudden shifts through other means, such as cobbling two transcritical bifurcations together).  

We have added a couple of sentences on this in lines116-124.

• Lines 105-110: this paragraph is confusing because noise-induced tipping is defined as occurring in the absence of environmental forcing (a gradual change in a system parameter value), but here the authors write that “However, as a system moves towards bifurcation, noise-induced tipping becomes more likely as it becomes easier for the system to leave its current basin of attraction when it is closer to the bifurcation”, which seems to conflate N-tipping and B-tipping. I think they are just talking about B-tipping here, since noise is also present in B-tipping (every system is subject to some level of noise).

We have rewritten this section.

• Alluvial plots: can you indicate the numbers associated with the width of the rivers? e.g. how many studies reported negative CSD results in ecology?  Or maybe mention in the caption how many studies the smallest and largest streams represent.  If the numbers are small, the readers should know this because it has bearing on the conclusions/speculations drawn in section 3.4.

This is a good suggestion. Unfortunately adding numbers to the flows makes the figures illegible. Instead we added numbers to the strata (column types) which we hope they can help to get an idea of the proportion the flows refer to.

• Line 645: The paragraph ends sudde 

Corrected.

We have a revised ms available for the reviewer.

We thank the reviewer for the positive assessment of our work and the useful comments and suggestions. Below we provide a point by point response (in italics).

# Reviewer 2

The paper sets up to offer a review of early warning signals of tipping points, and its development over the last two decades since introduced to ecology and climate sciences. The paper is well structured and a pleasurable read, I believe a timely contribution for scholars aiming to get into the science (and sometimes art) of detecting tipping points. That being said, I believe there is scope for improvements.

• First, the review misses important recent developments on early warnings. The first is critical speeding up (CSU), a phenomenon proposed as an alternative to detect stochastically driven transitions. The method has its history in physics applied to quantum problems, but at least a few studies have applied it ecological problems (e.g. Titus and Watson 2020 Theoretical Ecology). The second is average flux which is a non-equilibrium based early warnings that takes advantage of hysteresis, the different paths of recovery (see Xu et al PNAS 2022). Third, the fractal dimension (or Hurst exponent in the engineering literature); it is a common indicator as spatial early warning, but also used on time series. Geoffrey West work on scaling proposed as an explicit metric of resilience and it is related to percolation dynamics in networks, so I was surprised not seeing mentioned since you do have a section of resilience indicators in networks (West, 2017 Scale), and it has been applied to time series in ecology.

We totally agree that these are important contributions that we did not include. The reason is that we only focused on papers that analysed empirical data and were not theoretical papers or focused on simulated data. Titus and Watson 2020 is a purely theoretical paper and the same is true for Xu et al 2022. As far as the work on fractal dimension/ Hurst exponent as an early warning for tipping points, we would appreciate if the reviewer could point us to the specific papers.

The work by West 2017 is also interesting but we are not sure it is used an early warning. Similar work on scale has been proposed also by others (Peterson et al., 1998; Spanbauer et al., 2016). However referring to all this work goes beyond the scope of this paper. Yet, we now added average flux to Table 1 to the discussion the mentioned work.

• Second, the review misses some important modeling work on cascading effects and early warnings. Decker 2018 (Earth Syst Dinam) showed that when systems are connected there is no early warnings. They tested for coupled double-fold, fold-Hopf, Hopf-fold, and double Hopf bifurcations (in the context of climate tipping points). Thus figuring out EWS for connected systems is an open area of research. It links as well to the issue of sufficient observables in multidimensional systems, because if the system is connected and you can only observe some nodes or subsystems, how do you decide which ones to monitor? A paper by Carpenter’s group showed experimentally that tropics levels closer to the mechanisms responsible for the critical transition showed the early warning, while trophic levels distant from the mechanisms did not show a warning or not early enough (in Oikos 2014). When discussing cascading effects you are only focusing on the case of domino effects, where the occurrence of one critical transitions triggers a second. However, there are two other cases dealing with sharing drivers, and new feedbacks between two different types of transitions (see Rocha 2018 Science). In the case of multiple drivers, Dai et al (2015 PNAS) showed experimentally that multiple drivers acting at the same time can induce contradictory EWS e.g. increase in AR-1 and decrease in variance or viceversa. Some clarifications of under which conditions EWS fail or produce contradictory signals is missing, as well as strategies or cases that explain their emergence. In a bi-dimensional basin of attraction (x and y are dimensions of the basin, z is the potential), you can get a shallow basin on the x dimension (amenable for CSD) but narrowing on the y dimension (amenable for CSU), inducing contradictory signals.

These are excellent suggestions and points to be raised. Yet, as we only focus on the application of EWS in empirical data to get an overview of what has been done, where and what is found, it is difficult to cover the complete literature. We attempt in the discussion to touch upon the issue of multidimensionality and where to measure, the cascading effects and challenges to the performance of the EWS. However, a thorough treatment of such is beyond what we performed here. An in-depth understanding of the conditions that EWS fail or not and cases that explain their emergence needs to consider the peculiarities of the different data types as we looked across domains and the idiosyncracy of the case studies. Such analysis will require follow-up work on domain specific studies. We now add our reflection on these points in the discussion across the different subsections.

• On the light of the challenges and limitations of the previous paragraph, I believe the section of machine learning is a bit too optimistic. ML models are as good as their training set, if the training set is bias they will produce biased predictions. While you do call into the opportunities that new data and Earth observations are opening in that front, little is elaborated on current annotated datasets of true positives and true negatives necessary to train useful models. Thus far, most ML papers are trying to predict synthetic data (with some exceptions), which cast doubts into their usefulness in real world problems. CSD is an excellent technique for systems that one can reduce or treat as one-dimensional (e.g. ice melting) but it does not lend itself for high dimensional problems (e.g. ecological communities). Training ML models to optimize for CSD will by default miss transitions that obey other dynamics (n-tipping, r-tipping) or that occur in basins of attraction in higher dimensions. A word of caution is needed especially with black-box models. Intermediate complexity models such as random forests have been used in combination with EWS to explore the drivers or the type of tipping that trigger the EWS.

Thank you for the critical feedback. We have now modified this section stressing that one should be cautious of the potential shortcomings.

I hope the comments helps you improve the scope of the review. Below a couple of minor comments / typos.

• Specific commnents

Line



339. Yes please add examples

Done

1. Come instead of com.

Corrected

Peterson, G., Allen, C. R., and Holling, C. S.: Ecological resilience, biodiversity, and scale., Ecosystems, 1, 6–18, 1998.

Spanbauer, T. L., Allen, C. R., Angeler, D. G., Eason, T., Fritz, S. C., Garmestani, A. S., Nash, K. L., Stone, J. R., Stow, C. A., and Sundstrom, S. M.: Body size distributions signal a regime shift in a lake ecosystem, Proc. R. Soc. Lond. B Biol. Sci., 283, https://doi.org/10.1098/rspb.2016.0249, 2016.

We have a revised ms available for the reviewer.