Population dynamics of Wolbachia symbionts playing Dr. Jekyll and Mr. Hyde

Dear authors,

The three exports who reviewed your paper during the first round have looked into your revised version of the manuscript and have provided me with comments. This time, I also read and reviewed your paper in detail (see my comments, written in Markdown, below), and found many opportunities for improvement. I therefore ask that you revise the manuscript to take into account my own and the other reviewer's comments. Please also provide a detailed point-by-point response to all comments. After this, I will be able to further consider the preprint for recommendation.

I look forward to receiving your revised manuscript.

Best regards,

Jorge Peña

# Review of "Positive fitness effects help explain
the broad range of Wolbachia prevalences in natural populations"

## Major comments

1. One of the main merits of this paper is that it revisits previous models of cytoplasmatic incompatibility in the literature, and fills the gaps in some theoretical understanding of this phenomenon. In this respect, it is unfortunate that you are not able to prove analytically the stability of the equilibria of the haplodiploid models, and instead are forced to run a numerical analysis. I encourage you to devote some time to see if you can make some analytical progress. Perhaps necessary and sufficient conditions are difficult to obtain, but either necessary or sufficient conditions can be obtained more easily? Or perhaps a different strategy might prove useful? For instance, you seem to rely quite a bit on the manipulation of the explicit solutions of the equilibria. Perhaps working directly with implicit expressions leads to some progress (see, e.g., the next comment)?

2. In line 314, you say: "In all cases that we explored in the haplodiploid models, the infection frequency was higher in females than in males". I think that this statement is actually easy to prove for the female-killing effect. To see this, note that, at equilibrium, $\Delta p_M = 0$ and hence, by setting the second line of Eq. 4 to zero and solving for $\hat{p}_M$, $\hat{p}_M = \frac{\hat{p}_F ft}{\hat{p}_F f + 1-\hat{p}_F}$. It follows from this equation that the ratio $\hat{p}_F/\hat{p}_M$ is equal to $\frac{\hat{p}_F f + 1-\hat{p}_F}{\hat{p}_F f t} = \frac{1}{t}+\frac{1}{\hat{p}_F ft} - \frac{1}{ft}$, which can be rewritten as $\frac{1}{t} \left[1+\frac{1}{f} \left(\frac{1}{\hat{p}_F}-1\right)\right]$. From this, it is then easy to see that, for any interior equilibrium $\hat{p}_F \in (0,1)$, the term inside round brackets is positive, and hence that the term within square brackets is greater than one. As $t \in (0,1]$, it follows that the whole ratio $\hat{p}_F/\hat{p}_M$ must be greater than one and hence that $\hat{p}_F > \hat{p}_M$ holds, finishing the proof. Could you double check that this is a valid proof and, if so, include it in the paper? I wonder also if a similar reasoning allows for a similar proof for the case of the masculinization effect (I did not check).

3. Figures. The authors should pay more attention to the figures and their captions, as I think that they can still be much improved.

    3.1. In particular, for Fig. 1 I suggest (i) that the x range is always [0,1] (currently panels b, d, and f have smaller ranges), and (ii) that the label of the y axis in panels a and b do not have the additional $\times 10^2$ or $\times 10^3$ (instead, show a rescaled y axis). Also in this Fig., a better visualization strategy seems to be not to impose limits on the x and y ranges, but rather to let matplotlib decide on this.

    3.2. Figures 4 and 6 should have a legend describing what the different shades of the areas refer to (this is currently done in the caption).

    3.3. The image quality of Figure 7 should be improved (it currently looks pixelated).

    3.4. All mathematical symbols in the figures should be rendered with LaTeX, e.g., the f=0.8 on top of Fig. 7A should be rendered as $f=0.8$, the t on the y axis should be rendered as $t$, etc.

    3.5. I urge you to write the names of the variables in the labels of the axis of all figures. For example, the label of the x axes of the panels of Fig. 2 should read "infected proportion, $p$" (or similar), the y axis labels should read "change in infected proportion, $\Delta p$" (or similar), etc.

    1.6. The y label of panel a of Fig. 5 should be $p_M$, not $p_F$. Here, as per the comment above, these labels should be enhanced by providing the name of the variable the symbols refer to.

4. As per the last comment in the first page of the review of reviewer 1, mathematical formulas are parts of a sentence, and punctuation and grammar rules apply. Please revise carefully your manuscript in light of this comment.

5. In your response to reviews, you reply to all (but for the first, about a figure that you introduced) the general comments by reviewer 2 by saying that they are interesting questions but that they are beyond the scope of the present paper. At the very least (if you indeed found the comments interesting) you should devote some lines of the Discussion to these possible extensions and potential future work.

6. Appendices. I think that it would be worthwhile reorganizing the appendices in the following way: Appendix A: Diplodiploid model (both equilibria and linear stability analysis). Appendix B: Haplodiploid models. B.1 Female-killing, B.2 Masculinization.

## Minor comments

1. l. 31: "without" -> "without a"
2. l. 49: "frequency-dependence" -> "frequency dependence"
3. l. 65: "on-going" -> ongoing
4. l. 74: "CI inducing symbiont" -> "CI-inducing symbiont"
5. l. 79: "lower" -> "a lower"
6. l. 86: "(0.5-1)" -> "between one half and one"
7. l. 88: "be close" -> "be too close"
8. l. 90: "in wide" -> "in a wide"
9. l. 91: "without the invasion barrier" -> "without an invasion barrier"
10. l. 92: "to different" -> "to the results of different"
11. l. 95: "in diplodiploid" -> "in a diplodiploid"
12. l. 118: denominator of the fraction appearing in the last line of eq. 1: This can be also written as the expression appearing in the denominator of Eq. 1 of Engelstadter 2009. Unless you have a good reason to prefer writing the expression as you do (if so, explain) I suggest you stick to the expression reported in Engelstadter 2009, so that it is more clear that the model is the same.
13. l. 123: $B=-1+f-L$ -> $B=f-1-L$
14. l. 132: $r=-B/2A$ -> $r=-B/(2A)$
15. l. 134: "with $0 < t, L \leq 1$": From this, it seems you are assuming $t \in (0,1]$, $L \in (0,1]$. But then, p=1 can be an equilibrium for $t=1$, and this is not reported below. Also, it would be clearer if you write $0 < t \leq 1$, $0 < L \leq 1$.
16. l. 138: "shows this" -> "shows that this"
17. l. 144: "range (0.5-1)" "interval $(1/2,1)$"
18. l. 145: "based on" -> "given"
19. l. 146: $r>0.5$ -> "$r>1/2$"
20. l. 148: "The conclusion arises that" -> "Hence,"
21. l. 162: "poor spread" -> "selection against"
22. l. 162: "better success" -> "selection for"
23. l. 166: "we get $\lim_{p \to 0} \Delta p = pf t − p = (f t − 1)p$": $\lim_{p \to 0}$ is missing at the beginning of the final expression. The limit $\lim_{p \to 0} \Delta p$ is zero, not $(f t − 1)p$.
24. l. 175: "$f t$ is clearly below 1" -> "$f t$ is below 1"
25. l. 189: "$t \in]0,1]$": Rather use $($ for open interval, as it is more standard (see one of the comments of reviewer 3).
26. l. 205: "compare Fig. 2c,e" -> "compare Fig. 2c and/to Fig. 2e"
27. l. 215: "(; also shown in Fig. S1": remove ;
28. l. 217: "Figure 2b shows examples" -> "Figure 2c shows examples"
29. l. 221: "Higher values of $L$ lead to equilibria with a higher prevalence of Wolbachia" (and the rest of the paragraph): Is there a proof of this statement? If yes, refer to it. If not, prove and show the proof in the appendix.
30. l. 230: "instead, higher $f$ increases the frequency". Prove.
31. l. 234: "curves in 2e" -> "curves in Fig. 2e"
32. l. 248: "Effect [...] are shown" -> "The effect [...] is shown"
33. caption of Fig. 4: "non-trivial stable equilibrium $\hat{p}_2 > 1/2$" -> "non-trivial stable equilibrium with invasion threshold $\hat{p}_2 > 1/2$"; "and stable equilibrium $\hat{p}_2 < 1/2$" ->  "and non-trivial stable equilibrium $\hat{p}_2 < 1/2$"
34. l. 269: in "Leptopilina type", write the quotation marks as ``'' instead of "" so that they are nicely rendered in LaTeX. Apply these changes also below and in other parts of the manuscript.
35. l. 297: "that this applies for" -> "that this applies to"
36. l. 298: "but only for" -> "but only to"
37. l. 301: "(examples of equilibria: Fig. 5)" -> "(for examples of equilibria, see Fig. 5)"
38. l. 329: "they can occur only if f > 1." Try to avoid as much as possible writing formulas in the Discussion. In this case, you can simply state the inequality in words.
39. l. 341: "is unstable. See also" -> "is unstable; see also"
40. l. 343: "eciency of transmission and relative fecundity" -> "eciency of transmission, and relative fecundity"
41. l. 356: "when studying" -> "when studying the"
42. l. 358: ""if the proportion infected daughters" -> ""if the proportion [of] infected daughters"
43. l. 359: "than the proportion daughters" -> "than the proportion [of] daughters"
44. p. 25: "it is simple to see that" -> "it is easy to see that"
45. p. 26: "(i.e. the square brackets)" -> "(i.e. the expression within square brackets)"
46. p. 26 after eq. A2: "$f_{lim}$" -> "$f_{\text{lim}}$"
47. l. 454: "egg into haploid that" -> "egg into a haploid that"
48. p. 27: "i.e. we show with contradiction" -> "i.e. we show
by contradiction"
49. p. 27: "Thus, $r_F >= 1/2$" -> $r_F \geq 1/2$"
50. p. 27: You need to write something to introduce the last equation.
51. p. 28: "the slope of that function $−2fLt(1 + f (t − 1)) < 0$ and the root" -> "the slope of that function is negative, and the root satisfies"
52. l. 461: "to equal condition" -> "to the equivalent condition"
53. l. 466: "and the root $\hat{k} \geq 1$" -> "and $k \geq 1$ holds"
54. l. 468: "As A4 is equivalent with" -> "As (A4) is equivalent to the condition"
55. p. 29. There's a missing period at the end.
56. l. 517: "Since $ft > 0$ always" -> "Since $ft > 0$ always holds"
57. right after eq. B6: "multiply with" -> "multiply both sides of the inequality by"
58. l. 540: "The left-hand side of this equation is called the Jacobian matrix." There are several problems with this statement. First, this is not an equation, this is an inequality. Second, the left-hand side is not a Jacobian matrix, but (as I understand from the context) the absolute value of the (real) dominant eigenvalue of the Jacobian matrix. Here, the notation you use for this is more distracting than helpful. I'd go for something like $|\lambda| < 1$ (or $|\zeta| < 1$, since you are already using this notation) and then would explain in words what this $\lambda$ (or $\zeta$) is, having described the Jacobian matrix $M$ as you do below.
59. l. 542: "We will denote the dominant eigenvalues by $\zeta$." You denoted it below instead by $\lambda_1$.
60. p. 34, last line: "have the following general shape": Write down the $g$ and $h$ functions.
61. p. 36 "We can rewrite the term under the square roots in
equilibria above as follows": Couldn't you already write the expressions for the equilibrium in this form from the beginning? It would save some space.
62. l. 556: "$h(\hat{p}_F) = \hat{p}_M/\hat{p}_F$". You defined $h$ in p. 34 as a function of two variables, but only one variable is given here.
63. Equations between l. 556 and l. 558: The left hand side is not evaluated at equilibrium, but the right hand side is. Correct.
64. l. 560: Multiply by minus one and rearrange the term in square brackets on the right hand so that the expression looks similar to eq. B9 already at this point.
65. l. 572: "I" -> "we"
66. p. 38: "the term before square root": It is unclear which term this refers to. Rewrite.
67. l. 604: "whenever they occurred in" -> "whenever they occurred within a"

Dear authors,

I have now received three expert reviews on your preprint. As you will see below, all reviewers agree that your manuscript addresses an interesting question, that the paper is well written, and that the analysis seems to be correct. However, they also raise several questions and comments that suggest different ways of improving the analysis and presentation of your results. Having read your paper, I agree with the reviewers' assessments, and see many opportunities for improvement.

In particular, and in line with some of the reviewers' comments, I would like you to pay particular attention to (i) the possibility of adding a figure illustrating cytoplasmic incompatibility and the workings of the model (as suggested by reviewers 2 and 3), (ii) being more concise in pages 7-10 (as suggested by reviewer 2), and (iii) the comment on the (unnecessarily complicated?) use of matrix calculus (suggested by reviewer 1).

I therefore ask that you revise the manuscript to take into account these and other reviewer's comments. Please also provide a detailed point-by-point response to all comments. After this, I will be able to further consider the preprint for recommendation.

In addition to the reviewers' specific comments, I have some of my own. First, I would like you to pay attention to a typographical detail. You often write 0.5 for either "1/2" or "one half". The form "1/2" looks nicer in math mode or in equations; the form "one half" looks nicer when writing sentences. For instance, in line 29 you could write "can produce low frequency ($<1/2$)" and in line 31 you could write "any stable equilibrium close to one half". Second, I found some of the choices for naming variables a bit strange. In the kind of population dynamics models that you study, time is usually denoted by (lower case) $t$, not (upper case) $T$, as you do in the manuscript. This choice might be due to the fact that you already use $t$ for one of the parameters of the model. My suggestion would be that you stick to the standard norm of using $t$ for time but use an alternative symbol for the transmission parameter (that you now represent by $t$). Third, you write down equations (1) and (2) in the form $p_{T+1}=g(p_T)$ for a given function $g$, but then in Fig. 1 you illustrate the dynamics by plotting $\Delta p_T$ as a function of $p_T$. I would stick to one or the other way of writing these recursions and would try to be consistent, i.e., I would either stick to the way equations (1) and (2) are written now, but then change Fig. 1 to show $p_{T+1}$ as a function of $p_T$, or I would keep Fig. 1 as it is but then rewrite equations (1) and (2) in difference form. When making one change or the other, please also try to be consistent with the way you write down the equations for the haplodiploid cases. Fourth, many equations in the Appendix seem to be copy-pasted directly from (I guess) Mathematica, leading to cumbersome expressions that are difficult to parse. As an example, the factor $f(-1+(-1+k)L(-1+t)t)$ appearing in p. 39 could be rewritten as $f[L(k-1)t(t-1)-1]$ (note also the use of square brackets, consider using e.g., $\left($ and $\right)$ instead of simply $($ and $)$), which is shorter and easier to parse. I encourage you to make these changes throughout the manuscript. Finally, although all reviewers agree that the manuscript is well written, I found some grammatical mistakes and typos---please revise the writing carefully and edit the manuscript accordingly.

I look forward to receiving your revised manuscript.

Best regards,

Jorge Peña