Reformulation of Trivers–Willard hypothesis for parental investment

The Trivers-Willard hypothesis (TWH) plays a central role in understanding the optimal investment strategies to male and female offspring. Empirical studies of TWH, however, yielded conflicting results. Here, we present models to predict optimal comprehensive multi-element parental strategies composed of primary sex ratio, brood size, resource allocation among offspring, and the resultant secondary sex ratio. Our results reveal that the optimal strategy depends on sex differences in the shape of offspring fitness function rather than in fitness variance. Also, the slope of the tangent line (through the origin) to the offspring fitness function can be used to predict the preferred offspring sex. We also briefly discuss links between the model and the empirical research. This comprehensive reformulation of TWH will offer a thorough understanding of multi-element parental investment strategies beyond the classical TWH.

I find this to be a really nice study that comprehensively demonstrates what our expectations should be for optimal sex allocation under various socio-ecological conditions. It represents a clear improvement on existing theory in this field, which has often only looked at a strongly limited subset of cases (e.g., the focus on males vs. females, where there are as many cases representing exceptions to the rule as adhering to the rule itself). Worse, it is a field that has long been muddled by vague verbal arguments (e.g., offspring with higher "variance in fitness" (variance over what?) should receive more investment "when parents are in high condition" (what is high?). I therefore applaud your efforts in building and analyzing this very general model, whose results aren't all that surprising once the model setup is properly understood -and this is a good thing. My main requests for changes are entirely presentational, and I do think this manuscript has a lot to gain on improving the clarity both of its structure, figures, and language.

Some general comments:
A) The numbering of references is not in the order in which they appear in the manuscript. (After reference 1 we are shown reference 5, then references 11 and 12…) Needs updating! B) Figure 1 and the Results section is very dense and difficult to navigate, although I appreciate that you've already done a massive job in narrowing all the results down to only the most important ones presented in the paper (c.f. the 50 pages of supplementary material…). I think a major improvement would be if just a bit more work was put in the presentation of the figures, and spending just slightly more time guiding the reader through the results. Specifically, splitting figure 1 with its 16 subpanels into several smaller figures, each with fewer subpanels, might help group certain results together. Many subpanels can be OK if they are all the same type of graph, or if there is a clear thread between them, but here there are very many different graph types, and finding the logical thread connecting them is very challenging. Most of the panels are useful on their own, but when the text introduces the subpanels in a non-chronological way (starting on line 92), and the reader needs to sift through all 16 panels across five rows of a figure to find the one you mention, it becomes unruly. Probably your best bet is to split the figure up into 1) and introductory one-panel figure of panel a; 2) a three-panel figure with panels b-d for model M1; 3) a six-panel figure with panels e-j for model M3; and 4) another six-panel figure with panels k-p for model M5. Another option is to group similar graph types together, so e.g. one figure with panels b, e & k; one with panels c, f & l, et cetera. I'm sure there are other logical ways of doing this as well.
Alternatively, a few more words on the figure in the form of "subheadings" over certain panels/rows might help. E.g., you state in the beginning of the figure legend that the figure shows "three examples chosen from the models presented in Supp Mat…", but it is not at all clear from the figure what panels belong to which examples, or what the examples even are. It's difficult even after digging through the lengthy figure legend, and the main text. Some simple visual formatting could help it be clear from first glance, for example the subheadings "Model M1: Fitness functions identical for both sexes" for panels b-d, and something like "Model M3: Low-variance sex requires little investment" and "Model M5: Low-variance sex requires more investment" (or something more accurate) respectively for panels e-j and k-p. C) I think just a little bit more discussion would strengthen the paper considerably. I understand that journal word limits might be at play, but the paper is currently quite short and I think it would really help empiricists interpret the importance of your work if you zoomed out a bit more at the end after finishing the technical bits. Currently the "Tangent line rule" is the last subheading in your "Results and discussion" section, but the final paragraph is clearly more general and could have a "Conclusions" subheading or something. It's good that you briefly address shortcomings/limitations of your model, but I also think you should mention some parts that are more relevant to empiricists rather than just modelers, as the journal is quite wide in scope. For example, what biological situations (ecological or social conditions) would perhaps create the different kinds of fitness functions you model (M3-M7)? When would the logistic curves be steeper or shallower, have earlier or later inflection points, have different upper asymptotes? When and why would sex differences in these arise? See an example of similar discussion in Haaland et al. Below is a rather long list of mostly nitpicky edits of style, grammar and wording. I hope you forgive me for this and realize that I did it because I really like the paper, consider it highly publishable, and therefore want it to be as clear and readable as possible in all aspects.
Specific comments: Line 28: Add "the": "The Trivers-Willard hypothesis (TWH)…" Line 31: Secondary sex ratios are an outcome of the parental strategy, not actually part of it, right? Perhaps add "resulting". Also, "The results" -> "Our results". "… and resulting secondary sex ratios. Our results reveal…". Line 35: Comms Biol guidelines specifically states to avoid words such as "novel". Perhaps reword slightly, e.g. "This comprehensive reformulation of TWH offers a thorough understanding…". Line 46: "(range of fitness function)": You haven't defined this function yet. Also, the historical use of "which offspring type provides the largest fitness variance" is quite vague, maybe you can highlight that here, since a strength of your model is that it offers a much more accurate criterion. So, maybe change this to something like "(typically defined as the range of offspring fitnesses as a function of parental investment)"? Line 51: Add "the": "the Trivers-Willard hypothesis (TWH)…". Line 53: Some missing plurals: "sex-specific costs of rearing, sex-specific growth curves,..". Line 55: Add "has": "While the theory has brought about…". Line 56: Change "or the" -> "and" to avoid that the two elements in the parenthesis sound like the same. "…primary sex ratio and sex-biased provisioning…". Also, "they" -> "these have", since you're referring to the empirical studies. "…these have produced…". Line 57: Contradicting -> contrasting. The empirically observed parental strategies differ, they don't contradict each other. (Models or predictions may contradict each other.) Also, missing "have". "…the contrasting parental strategies, we have built…" Line 60: Remove some superfluous words: "…for the understanding of the variation…" -> "…for understanding the variation…". Line 64: Slightly strange wording, small suggested removal and rewrite: "However, the typical models presented so far did not…" -> "However, existing models do not…" Line 67: Small grammar and wording changes. A missing s, an extra s, and specify that you're talking about offspring fitness functions: "…sex differences in the logistic (sigmoid) offspring fitness functions affect the…" Line 89: This subheading doesn't relate to the paragraph below it, which is a general introduction to the results section. Rather, move the subheading down one paragraph, to between line 94 and 95. Line 98: Remove a "the": "…the number of the offspring…" -> "…the number of offspring…". Line 100: Maybe change "are relevant to" to "comply with" or "adhere to"? Line 108: "…three types of the clutch…" -> "…three types of clutches…" Line 121: Slightly imprecise to say "maximum fitness is reached with relatively small investment…". I understand what you're trying to describe, but it's not quite correct because in M5, maximum fitness is reached later in Sex 1 than in Sex 2. What you write in the next sentence is more accurate (Sex 2 has a "slower initial increase"), so I suggest to change the wording to contrast more directly with that: "…assumes that fitness increases rapidly with relatively small investment…". Line 137: The middle part of this sentence, "the optimal primary SR was Sex 2-biased for the larger values of S (>89)", should also have a figure reference, since you're no longer talking about Fig. 1l. Which panel is this shown in -1m? Add reference, and also specify exactly what element of the panel you're talking about (as you nicely do for the previous result, "from 0 to 55; overlapping lines in Fig. 1l"). Line 155: Remove a "the", add an "s", change wording: "…indicating the interactive selection pressure…" -> "…indicating interacting selection pressures…". (Interactive means something one can interact with.) Line 163: Change "Sex 1 group over Sex 2 group" -> "Sex 1 offspring over Sex 2 offspring". You haven't really introduced calling Sex 1 and Sex 2 "groups" before… seems conceptually simpler to just stick to "offspring" here. Line 164: Maybe add specification of where we're looking: "orange-shaded plot in Fig. 1h". Also on line 165: "blue line in Fig. 1g". Line 175: Maybe change "the optimal parents" -> "optimally acting parents" Line 177 & 179: Maybe change "inversion" -> "switch" Line 178: "an" -> "the". "…investment towards the other sex…". Line 179 & 180: Missing article two places: "…e.g. the presence/absence of an intersection…" Line 182: Talking about both group and sex seems unnecessary? Removing "group" also avoids unfortunate double parenthesis… "(proportion of S toward each group (sex))" -> "(proportion of S toward each sex)" Line 186: Slightly imprecise wording: Change "…when the sex of the greater tangent line…" -> "…when the sex with the steeper tangent line…". Line 191: Slightly imprecise wording: Change "…maximum efficiency of fitness against a given investment…" -> "…maximum efficiency in terms of fitness gained per investment…". Line 198: "…similar to Fig. 1k" -> "…similar to those shown in Fig. 1k". Line 206: Remove s: "fitness function shape" Line 217: "The definitions" -> "Definitions". Line 224: "to specify the sex" -> "to specify the sexes" Line 247: Some grammar changes, remove several words: "The model simplifies the reality as it is assumed in the model that the mortality…" -> "The model simplifies reality as it assumes that mortality…" Line 257: I'm not sure what "differential values of fitness functions" means. Do you mean "different fitness function parameter values"? Some rewriting needed. Line 261: "…number of cared offspring…" -> "number of offspring to be cared for…". Line 264: "…some of the offspring within the type…" -> "…some of the offspring of a given type…" Line 265: By "optimal distributions" (plural s needed), do you mean "optimal allocation rules"? If so, change. Line 271: Perhaps specify a bit better what you're talking about here by adding model names and actually stating what the sex ratios are rather than just saying "the three" sex ratio. "We simulated broods with 14 different pairs of fitness functions (termed models M3 through M6 and M7-1 to M7-10), each pair tested in three different primary brood sex ratios: 0.3, 0.5 and 0.7.". Line 276: Remove "the" before "brood size reduction". Line 277: Remove starting parenthesis before "F(X)", as this one never gets a closing parenthesis, and it's not needed. Line 278: Misplaced comma. Remove comma before "F(X)", add comma after the summation term. "…defined as F(X)=sum f_b (x_i), where…". Line 285: Remove "a" before "decent condition". Also, is this an assumption you bake in to the calculation? Or just a statement justifying the lack of an offspring-quality effect at independence? In general it is unclear what point this paragraph is making. Line 286: Change "…is in proportion to…" -> "…is proportional to…". Line 291: Change "provision" -> "investment". You hardly talk about provisioning at all, investment is much more general. Line 297: Change "Let denote H…" -> "Let H denote…" Line 301-305: Formatting and punctuation makes it unclear what is happening here. Perhaps change the period on line 301 to a colon, and remove "(OP) To find". The abbreviation "OP" (for optimization problem?) is never returned to again and thus unnecessary. So you get the structure "The aim of the study is to solve the optimization problem: X* = (v_1, …, v_N) such that X* = argmax F(X) where F(X) = sum f_b_i (x_i) where b_i indicate¬s the type of fitness function." (Note an added "s" and a removed "the" on line 305. Line 308: Change "We used fmincon of MATLAB…" -> "We used the fmincon function in MATLAB..:". Line 309: Change "By comparing possible combinations x_i's" -> "By comparing different combinations of x_i's…". Line 310-312: Rewrite slightly for precision and readability. "…how many offspring should be cared for, and how investment should be allocated among Sex 1 and Sex 2 offspring in order to maximize total parental fitness.". Line 315-317: Rewrite slightly for precision, grammar and readability. "In the models where offspring of different sex have different fitness functions (M3-M7), we analyzed optimal strategies for Sex-1-biased, equal, and Sex-2-biased broods. In Sex-1-biased broods, …". Line: 318: Small suggested changes for grammar and precision: "In order for parents of Sex-1biased broods to optimize allocation of the total expendable care…". Line 322: Add missing comma and change "a" -> "the". "…sum(x_i) = S, where f_1 is the fitness function…". Line 323: Change "condition" -> "scenario". Line 326: Missing s. "Sex-2-biased broods"-Line 327: Remove "of a": "each pair of fitness functions". Change "constrained resources" -> "resource constraints". You can also spend a few more words describing where exactly these results are seen. For example something like: "E.g., Sex 1 offspring (red lines) receive most investment at high parental energetic states for equal or Sex-2-biased primary sex ratios, despite having a lower variance in fitness (n-p).".
Supplementary material: I haven't read the Supplementary Material in extensive detail, only used it for reference when needed while working through the main text, but in general it seems highly satisfactory. One suggestion for improvement may be to add a list of the numbered figures and tables (and possibly also theorems, lemmas and proofs?) with page numbers in or just after the table of contents? Such a list could also include a brief description ("header" for each element). This would make looking for a certain figure or table much easier.

Reviewer #2 (Remarks to the Author):
This study builds on the classic Trivers-Willard model of sex ratio and investment in animals by developing additional models to test this classic idea. The authors focus specifically on the role that different fitness functions (S) have on offspring investment and sex ratio. To me there is a bit of a disconnect between this approach and that of the TWH. The TWH argues that reproductive variance is key to parental sex investment. This study argues that the slope of the fitness function is key. They broach the idea of reproductive variance briefly, but I had trouble following how this is incorporated into the models. To basically argue that the fitness function of offspring is more important than the reproductive variance for each sex (as is done in the abstract, but nowhere else in the paper, the model has to explicitly address reproductive variance. More generally, it would help if the paper was written with a clearer set of predictions (and how those differ from the TWH). The Discussion sort of ends abruptly in the current version and could be expanded to more clearly compare and contrast the results of this study with TWH in more detail, as well other models on parental investment theory, clutch ratio evolution, etc. As it stands this manuscript covers a lot of ground but could do better integrating with the published literature and more clearly stating the important results in that context.

Specific comments
L33. Unclear what this means in a practical sense.

L73. What happens if you use a diminishing returns curve for S instead of sigmoidal function?
L89. It is unclear whether in this section, nests are composed of only one sex. L95. Do model results change if the clutch size changes (10 is quite high). How much of this result is simply due to producing so many offspring?
L100. This section has nothing to do with the TWH. It has to do with clutch size evolution, and seems out of place given the Introduction.
L128. Your abstract argues that these functions are more important that reproductive variance, but this section states otherwise.
L131. Isn't this argument just one of reproductive value? In other words, parents invest in chicks with higher reproductive value (S).

Response to Reviewers' Comments
We sincerely appreciate the endeavor and time of two anonymous reviewers that have been spent on comments and scrutiny of the manuscript. Due to considerate analysis and evaluation performed by the reviewers, we were able to find glitches or missing points, enabling us to augment the logical structure of the manuscript. The comments also improved the clarity of the paragraphs and visual data. We incorporated the comments from the reviewers into the main article and supplementary materials. As an important improvement, we added the possible empirical method that can corroborate the theoretical findings. The followings are the point-by-point response to the comments from the reviewers. We numbered the comments for smooth communication about our responses in the further process.

Comment: 1:
The numbering of references is not in the order in which they appear in the manuscript.
(After reference 1 we are shown reference 5, then references 11 and 12…) Needs updating! Response: We updated the reference as pointed out by the reviewer. something like "Model M3: Low-variance sex requires little investment" and "Model M5: Low-variance sex requires more investment" (or something more accurate) respectively for panels e-j and k-p.

Response:
In accordance with the suggestions by the reviewer, we divided the previous Figure 1 into four separate figures. We also added additional explanations to make them more clear with modified captions.
Comment 3: I think just a little bit more discussion would strengthen the paper considerably. I understand that journal word limits might be at play, but the paper is currently quite short and I think it would really help empiricists interpret the importance of your work if you zoomed out a bit more at the end after finishing the technical bits. Currently the "Tangent line rule" is the last subheading in your "Results and discussion" section, but the final paragraph is clearly more general and could have a "Conclusions" subheading or something. It's good that you briefly address shortcomings/limitations of your model, but I also think you should mention some parts that are more relevant to empiricists rather than just modelers, as the journal is quite wide in scope. Response: If the fitness is not monotonically increasing but decreases though S increases, then it is optimal not to spend all expendable care on the offspring. If the diminishing return is defined as decreasing but not the negative return, then the sigmoidal function satisfies this condition. This is because there is an upper bound of the function, and the slope becomes flat as the argument of the function increases.
Comment 70: L89. It is unclear whether in this section, nests are composed of only one sex.

Response:
We admit that there was inconsistency between the section name and its contents. We Therefore, the pattern shown in Fig. 1c will hold, and the change of the clutch size only will change the upper bound of the number of offspring receiving investment. We have mentioned this in the revised manuscript. Please have a look at pg. 3, lines 26-28.
Comment 72: L100. This section has nothing to do with the TWH. It has to do with clutch size evolution, and seems out of place given the Introduction.
Response: This section provides the basic structure of how the model works, i.e. the situation when all fitness functions are identical. The next step is the expansion of this model to two-sex broods. Also, this model can be viewed as a special situation when males and females have identical fitness functions. As mentioned by the reviewer, the discussion about the brood reduction hypothesis may at first appear additional and not in the same line with the Trivers-Willard hypothesis. However, we believe that this argument is in line with the general logic behind the Trivers-Willard hypothesis in that both theories of brood reduction and sex-biased allocation concern optimal parental allocation strategy into individual offspring.
Comment 73: L128. Your abstract argues that these functions are more important that reproductive variance, but this section states otherwise.

Response:
The sentence pointed out by the reviewer is about the model M3, in which the predictions follow the traditional TWH reasoning, also consistent with the "tangent-line rule of thumb." Additionally, we claim that the tangent line of the fitness function determines the optimal allocation rules. If the tangent of the Sex 1 is greater than that of Sex 2, then expectations from the classical Trivers-Willard hypothesis holds. If the tangent of the Sex 1 is smaller than Sex 2, then expectations from the classical Trivers-Willard hypothesis do not hold, and this is the reason we claim that the shape of the fitness function is the crucial factor. Table S4 of Supplementary Materials shows that it should be the tangent, rather than reproductive variance, that should be used to predict optimal parental strategy.
Comment 74: L131. Isn't this argument just one of reproductive value? In other words, parents invest in chicks with higher reproductive value (S).
Response: S, on the horizontal axis, is not reproductive value. S is the total investment that is being allocated among male and female offspring in a brood. The reproductive value is represented on the vertical axis in panels (a) in Fig. 3 and 4 presenting "fitness functions". To avoid this type of confusion we have rewritten this part to: "Sex-1-biased care for the parents with small S and Sex-2-biased care for those with large S as evidenced in Fig. 3d." Please see pg. 4, lines 22-24.
Comment 75: L162. But what does this mean in practical terms?
Response: Reviewer 1 also pointed out that more discussion of links between model and empirical studies is required. We provided a new part of the discussion entitled "Relevance of the model assumptions and predictions to empirical research" to accommodate the comments of the two Reviewers.
I want to thank the authors for the thorough responses to my comments from the previous round of review. I am now mostly happy with the paper, and have just a few more comments after rereading the revised manuscript. The new section on empirical predictions still needs some work, and one aspect of the figures and terminology still confuses me. Otherwise everything seems to be in order, nice work! About the figures (these are interrelated problems -make sure they're all fixed & consistent!): Fig. 3a and 4a: Why are the tangent lines labelled Tangent line 1 and 2 as they are? As far as I can tell, Tangent line 1 belongs to Sex 2 in both figures, so perhaps the labels 1 and 2 should be flipped around? If your intention was to have the steepest of the two tangents be tangent 1, this is the case in 3a but not in 4a. Figure legend, fig. 3a: What do you mean by saying "In this model, tangent line of Sex 1 is greater than that of Sex 2"? Do you mean steeper? If so, change to that, or "the slope of the tangent line of Sex 1 is greater than that of Sex 2." Or do you mean that the tangent line meets the fitness function at a greater investment x? Figure legend, fig. 4a: Here surely the tangent line of Sex 2 is steeper than that of Sex 1, although you write the opposite. However, the one labelled "tangent line 2" in the figure is steeper than the one labelled "tangent line 1". Please clarify. About the section "Relevance of model assumptions and predictions to empirical research", page 6: Line 30: Grammatical errors: "Similar issues can be rise to the measurement…"change to for example "Similar issues can arise regarding the measurement…" Line 33: "…or even the weather conditions (Öberg et al. 2015)": This seems out of place in what I gather has been a list of different ways parental investment has been measured. This last one is rather a factor that can affect the amount of parental investment given -not a proxy of parental investment itself, unless you are insinuating that the parents are controlling the weather conditions somehow. Rather, the proxy used in Öberg et al. is parental nest visitation rates, and it is these nest visitation rates that are affected by adverse weather. Line 36-39: Whether or not the tangent line of the male's fitness function is steeper than that of the female, surely depends on where (at what investment level x) the 'step' in the function occurs. If males only have a chance to get any fitness at very high parental investments, then the female's fitness function may have the steeper tangent line. But I note that in this paragraph (line 37, 39) you keep talking about which tangent line is 'greater'. Please note the wording here and throughout: the relevant comparison is which tangent line is steeper, or equivalently which tangent line has a greater slope. The phrase 'greater tangent line' is a bit meaningless. Line 36: Grammar error. Change "…then is be expected…" to "…then it is expected…" or "…then it would be expected".