Review of MEE-24-03-215: “Spatial close-kin mark-recapture models applied to terrestrial species with continuous natal dispersal”

General comments
The submission addresses an important issue for CKMR studies: imperfect mixing in combination with non-random sampling. To tackle this issue, the authors developed a spatially-explicit lethal-sampling CKMR model that correct for imperfect mixing by incorporating natal dispersal distances and the spatial distribution of individuals. The performance of the four increasingly complex models was tested using agent-based simulation across six scenarios.
It is an amenable effort, especially as CKMR theory is complex and still relatively novel. We also appreciate the attention to the methodology, which was extensive and good to follow. It was an informative and enjoyable read, with an apparently sound methodology. Having said that, there are a several points that require attention, which I specify below:
1. The purpose of the Results and Discussion section. When I looked over the discussion for the first time, I was surprised by how short it was and how none of the results were discussed. Only after the first proper read-through, I realised that the discussion and interpretation of the results was included in the Results section. However, in general the Results section is supposed to only contain results, no interpretation, discussion, or contextualisation. This is also mentioned in the MEE authors guidelines (https://besjournals.onlinelibrary.wiley.com/hub/journal/2041210X/author-guidelines), which clearly state that the Result section should “state the results and draw attention in the text to important details shown in tables and figures” and the Discussion section should “[…] point out the significance of the results in relation to the reasons for doing the work, and place them in the context of other work.” This also comes back in the specific comments below.

1. I think the introduction could benefit from a bit more background. I think an overview of (some of) the CKMR applications, especially other CKMR simulation studies, is appropriate here. Also, I am missing a motivation for using agent-based simulation to test this statistical method. Again, see comments below.

2. Missing substantiation of simulation parameter values. It was unclear to me for most of the simulation parameters used in this study how they were set. Were they anchored on a case study, i.e., based on knowledge of a specific terrestrial species? Or was there some other motivation? Whatever way they were set, it should be clear to the reader. I also come back to this point a few times.

3. The function ‘dispersal.R’ was missing from the code repository. Therefore, I was unable to run the 3\_run\_CKMR.R script. This should be updated for the final release. Because of this, I was unable to run and check all the code. However, the code seemed clear and well documented.

Specific comments
1. Lines 34—36: “There is a … of harvest regulations.” – do you have a source for this statement?

1. Line 41: suggest adding at least one major reference for each of these methods.

2. Line 44: I am missing an overview or summary of CKMR applications to date and/or other CKMR studies that used agent-based stimulation. To my knowledge, an introduction is also supposed to summarise (the relevant part of) the current state of the field, a time-stamp to see the study in the context of its time. I am missing that in the current of the introduction. Perhaps an overview of pervious CKMR studies could fit nicely between the first and second paragraph of the Intro?

3. Line 62: You briefly mention that you are using simulation, but you don’t really mention why. I suggest adding some lines to address this and motivate the use of simulation to test (statistical) methods (e.g., https://doi.org/10.1002/sim.8086). Also, as mentioned before, I suggest providing a quick overview of previous simulation-based CKMR studies (there are only a few to date, e.g., Swenson et al 2024, Conn et al 2020, Petersma et al 2024 (DOI: 10.22541/au.171111996.68460083/v1; it has been accepted and should be published in E&E; soon with DOI: 10.1002/ece3.11352), and Sévêque et al 2024). This relates to the previous point.

4. Table 1: The variable RA is missing from this overview

5. Table 1: I would try to avoid multi-letter variable name sa_j, as it now looks like s multiplied by a_j. Why not just use a_j to denote sampled are around j, or use a Greek symbol like \alpha_j?

6. Table 1: In the definition of r you refer to a_j, but what is this? I don't think a_j is defined (should it be sa_j?)

7. Line 84: How is it possible to mate with two males? Does the female store the sperm, mix it, and then fertilise the eggs? I believe some animals do that, but I think some clarification would be good here.

8. Line 85: A normal distribution on reproduction means that animals could in theory have 10 offspring per litter, or even -1. I appreciate that it’s very unlikely, but did you truncate, for example, at 0 and 6, to make sure the numbers were always realistic?

9. Lines 88-89: Is this realistic dispersal behaviour realistic? It is missing substantiation.

10. Line 97: What are these numbers based on? It should be substantiated why certain parameters were chosen.

11. Line 101: \phi = 0.75. Again, why was this parameter set to this value? I stopped highlighting other parameters after this, but there are more parameter values that require some reasoning for how they were set.

12. Line 107: I appreciate that you need enough samples and thus sample 10% of the population each year for the last three years, but this likely means that sampling is no longer sparse. I'm not sure if this is a major issue in your study, but at least it could be mentioned somewhere as it means that the pseudo likelihood might not "behave" as a true likelihood any longer (See the last sentence of 2.2 in DOI: 10.1214/16-STS552)

13. Line 107: is the word “sampled” missing between “(ages 1+)” and “each”?

14. Line 108: does this mean that the sampled individuals are subtracted from the ones that die from natural causes, and thus total mortality does not increase in those last three years?

15. Line 138: I know you explain the variable K_ij in Table 1, but it should also be explained in text the first time it's introduced, which I believe is here. You do this for (almost) all other variables.

16. Line 148: maybe this a bit unfair since it is officially still in preprint, but I suggest also looking at DOI: 10.22541/au.171111996.68460083/v1. The revised version should be available in E&E; soon, and strongly highlights the sensitivity of CKMR to ageing error and thus seems relevant here.

17. Equation 4: I know you explain the variable N_\female,y_j in Table 1, but it should also be explained in text the first time it's introduced, which I believe is here. You do this for (almost) all other variables.

18. Line 203: Should this be p = 0.01? If not, why did you choose p=0.001?

19. Equation 7: I think this should be a sum rather than an integral. You are using discrete increments \Delta(d) right, not continuous? Also, in its current formulation you are summing the values 0 till d_max inclusive? Are you not overestimating the sample area then? Often when approximating an integral with a sum, you either sum the values at the lower end of an increment, the higher end, or in the centre. Whatever way you choose, I don’t think its correct to sum the point 0 AND the point d_max.

When integrating the increments are “infinitely” small and thus this does not matter, but when using discrete increments, you should be careful not summary too many points and thus overestimating sa_j. You probably did this correctly, but the current Equation needs some alteration.

1. Equation 10: ditto.

2. Line 241: “pseudo-likelihood detailed below”; I think you should reference the pseudolikelihood directly. This means that the equation in line 259 should not be presented in line and should get an equation number.

3. Line 258: I think you should explain what a pseudo-likelihood is, since it is not a common term, especially not for MEE readers. Maybe also a likelihood in general, and why we use a pseudo-likelihood here.

4. Line 259: I think this equation should not be presented inline and deserves its own equation number, like most other equations in the rest of the MS.

5. Lines 280—282: I think sentence belongs in the discussion.

6. Lines 296—312: I think most if not all of this section belongs in the discussion.

7. Line 315—328: Ditto.

8. Line 330: “abundance estimates” -- suggest adding that the abundance relates to year t = 28

9. Line 338: I think the Discussion could benefit from a summary of the study at the beginning of the first paragraph. See for an example the Discussion in Conn et al. 2020 (DOI: 10.1002/ece3.6296).

10. Line 339: “the most important assumption: -- Isn't this a bit strong? How about the assumptions by Bravington et al 2016, such as that an offspring can only have one father and one mother? I suggest changing to "one of the most important" or "one of the key assumptions" or something like that.

11. Line 247: “American black bears” and “black-tailed deer” – suggest adding the Latin names as well, similar to what you did for the caribou below.

12. Lines 361: “that reproductive output […] spatially and temporally homogeneous.” I think that you are assuming is that the EXPECTED reproductive output is homogeneous, not the actual reproductive output.