Dear Dr. Neumann,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

This is a clearly written and engaging study that combines careful field sampling with an intriguing conceptual discussion of stochastic variation and Fibonacci-related structure in floral traits. For PLOS ONE’s standards, I would require more rigorous specification and justification of the statistical modeling choices (including consideration of discrete alternatives and formal tests or simulations for the local peaks at 34 and 55) and clearer documentation of the sampling design and independence assumptions; I recommend, in addition, making the full ray flower count data easily accessible in machine-readable form and more explicitly distinguishing between what is demonstrated by the data and what is proposed as a broader conceptual hypothesis.

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Qinglin Meng, Ph.D.

Academic Editor

PLOS One

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Additional Editor Comments:

This is a clearly written and engaging study that combines careful field sampling with an intriguing conceptual discussion of stochastic variation and Fibonacci-related structure in floral traits. For PLOS ONE’s standards, I would require more rigorous specification and justification of the statistical modeling choices (including consideration of discrete alternatives and formal tests or simulations for the local peaks at 34 and 55) and clearer documentation of the sampling design and independence assumptions; I recommend, in addition, making the full ray flower count data easily accessible in machine-readable form and more explicitly distinguishing between what is demonstrated by the data and what is proposed as a broader conceptual hypothesis.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

Reviewer #1: Partly

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously? -->?>

Reviewer #1: No

Reviewer #2: No

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3. Have the authors made all data underlying the findings in their manuscript fully available??>

The PLOS Data policy

Reviewer #1: Yes

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English??>

Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1:

This paper combines empirical sampling of Bellis perennis ray flower numbers in Brandenburg with parametric modeling and a new conceptual term (“improbable recurrency”) to discuss how Fibonacci-like regularities may emerge from stochastic trait distributions. The figures nicely summarize spatial sampling, raw counts, within-population variation, and the fitted distribution, but some methodological and interpretive aspects need clarification for a PLOS ONE audience.

1. The counting protocol for ray flowers, defined as plucking and counting ligules, is described briefly but does not address potential observer error or within-head counting variability. Given that the argument hinges on single-unit differences in counts (e.g. 33 vs 34 or 54 vs 55), some indication of repeatability (e.g. re-counting a subset of capitula by the same or a second observer) would be useful to show that small differences are biological rather than measurement noise.

2. The analysis of spatial and temporal patterns in the “Results” section reports that there is “no spatial trend or autocorrelation” and that sampling date shows “no temporal trend (p = 0.36),” but the underlying statistical tests are not described. Clarifying whether these statements are based on regressions of mean ray flower number vs coordinates/date, variograms, Moran’s I, or other spatial statistics would make the claims about spatial and phenological independence more transparent.

3. Figures 2–4 are central to the argument but could be more analytically informative. For example, Figure 2 shows raw counts per integer value without confidence intervals, Figure 3 uses box-plots per population without explicitly marking sample sizes, and Figure 4 overlays the inverse gamma curve without any indication of uncertainty around the fit; adding sample sizes, error bars, or simulated envelopes would help readers visually gauge how unusual the local peaks at 34 and 55 actually are.

4. The discussion downplays the possibility that the observed local maxima are simply the result of finite sampling from a unimodal distribution, emphasizing instead the philosophical implications of “improbable recurrency.” To better align with PLOS ONE’s empirical focus, the authors should distinguish clearly between what the data and model demonstrably show (a good inverse gamma fit with small local deviations) and what is a broader interpretive or philosophical extrapolation, which might be better framed as a hypothesis rather than a conclusion.

5. The introduction and discussion draw on a wide and interesting set of references in evolutionary biology, phenotypic plasticity, and stochastic vs deterministic modeling, but they devote relatively little space to existing quantitative work specifically on floral trait distributions in Asteraceae and related phyllotactic systems. Strengthening the link to prior empirical studies on intra-specific variation in capitulum traits and phyllotactic disorder would better situate this dataset within the existing literature and clarify which aspects of the present study are confirmatory versus novel.

Reviewer #2:

The manuscript analyzes ray flower numbers in Bellis perennis across 34 populations and fits an inverse gamma probability density function to describe the distribution, with local deviations at Fibonacci numbers 34 and 55 interpreted as “improbable recurrency.” The study aims to link stochastic variation in floral traits with underlying deterministic numerical patterns such as the Fibonacci sequence.

1. The sampling design in “Material and methods” is not fully specified for statistical independence and representativeness: individuals were sampled within 50–1000 m² per population, but the within-plot selection procedure, avoidance of clonal individuals, and potential spatial clustering are not described. This is important, because treating 563 individuals as independent draws from a single distribution without accounting for population-level structure may underestimate uncertainty and obscure hierarchical variability between populations.

2. The choice of an inverse gamma distribution for discrete count data is only loosely justified by the GAMLSS AIC comparison, and no alternative discrete distributions (e.g. Poisson, negative binomial, Poisson–lognormal, zero-truncated models) are reported. Since the subsequent detection of local deviations at 34 and 55 depends critically on the assumed continuous reference model, the authors should clarify which families were compared, provide parameter estimates with uncertainty, and explain why a continuous inverse gamma is preferable to discrete count models for ray flower numbers.

3. The definition and use of the quantile residual measure RQn in equation [1] require more rigorous justification. Summing differences between sample and theoretical quantiles over all quantile sizes from 2–20 % is an ad hoc construction, and the interpretation of negative RQn as “positive deviations” from the inverse gamma is not standard; without a formal test (e.g. bootstrap envelopes or simulation under the fitted model), it is hard to distinguish genuinely unusual modes from sampling noise.

4. The interpretation of local peaks at 34 and 55 as evidence of “improbable recurrency” and deterministic Fibonacci-related structure is not backed by a quantitative assessment of how unlikely such peaks are under the fitted stochastic model. Given that counts at 34 and 55 are of similar magnitude to several neighboring values, a parametric bootstrap or simulation study under the inverse gamma model to estimate the distribution of local maxima would be needed before concluding that these peaks cannot arise by chance.

5. The conceptual claim that “improbable recurrency” represents deterministic components embedded in a fundamentally stochastic system currently rests on a single dataset from one species, one region, and one sampling season. Without replication across years, environments, or related Asteraceae species, it is difficult to know whether the observed pattern is robust, species specific, or even reproducible; this weakens the broader philosophical conclusions drawn about the co-constitutive nature of stochasticity and determinism in biological organization.

6. The statement that the distribution “is best described” by the inverse gamma pdf would benefit from more conventional model diagnostics, such as residual plots, goodness-of-fit tests, and information about overdispersion. Presently, the reader sees only a fitted curve overlaid on binned data and the qualitative comment that “slight deviations” exist, which is insufficient to judge model adequacy, especially when those deviations are central to the article’s main argument.

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what does this mean?). If published, this will include your full peer review and any attached files.

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Reviewer #1: No

Reviewer #2: No

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Dear Dr. Neumann,

Please submit your revised manuscript by Jun 15 2026 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

• A letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

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We look forward to receiving your revised manuscript.

Kind regards,

Sarah Jose, Ph.D.

Staff Editor

PLOS One

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If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

Reviewer #3: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions??>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: (No Response)

**********

3. Has the statistical analysis been performed appropriately and rigorously? -->?>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: (No Response)

**********

4. Have the authors made all data underlying the findings in their manuscript fully available??>

The PLOS Data policy

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: (No Response)

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English??>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: (No Response)

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Reviewer #1: All the comments have been addressed perfectly; no further comments. Strongly recommend acceptance for publication.

Reviewer #2: Thank you for addressing the reviewer concerns thoroughly and improving the manuscript; the revision is clear, technically consistent, and ready for publication.

Reviewer #3: This study PONE-D-25-51179 presents an interesting empirical investigation into the distribution of ray floret numbers in Bellis perennis, and I appreciate the attempt to introduce the concept of “improbable recurrency” to characterize the interplay between stochastic variation and deterministic patterns. The dataset of 563 individuals from 34 populations provides a valuable basis for examining intraspecific variability. I wish to disclose that my own expertise lies primarily in the developmental biology of Asteraceae, and I do not consider myself qualified to provide a definitive assessment of the mathematical and statistical modeling aspects of this manuscript. I therefore recommend that the editor seek an additional review from a mathematician or statistician with experience in distribution modeling to assess the appropriateness and correctness of the quantitative analyses presented in this paper.

The manuscript addresses a genuinely interesting question and contains a valuable dataset. However, the discussion must be placed in the context of known developmental mechanisms. Major revision is recommended.

Here, I draw the authors' attention to several points raised during the evaluation process that could significantly strengthen the manuscript if adequately addressed.

1. Absence of developmental context and recent literature

The manuscript treats Fibonacci numbers in Asteraceae flower heads primarily as a mathematical pattern to be verified or refuted statistically, but it does not engage with the substantial body of work on how such patterns arise developmentally. Particularly relevant is the study by Zhang et al. (2021, PNAS) on phyllotactic patterning in gerbera, which demonstrates that during early head development, incipient bract primordia emerge in Fibonacci numbers through a mechanism of intercalary insertion on an expanding rim, without requiring a fixed golden divergence angle. This process of “numerical canalization” explains why certain Fibonacci numbers (e.g., 13, 21, 34, 55) appear as robust attractors in Asteraceae inflorescences.

I strongly recommend the authors discuss their findings in light of this developmental framework. The local frequency peaks at 34 and 55 that the authors identify as “improbable recurrencies” are precisely the numbers that would be predicted to be the most strongly canalized developmental outcomes. This connection would transform the paper from a purely statistical observation into a biologically grounded explanation, and it would considerably strengthen the argument that these peaks represent genuine deterministic signals rather than random fluctuations.

2. The curious case of 42 and 47: additive combinations of Fibonacci numbers

During the evaluation of this manuscript, it was noted that the most frequent ray floret counts in the sample are 42 (mode) and 47 (second-highest peak), which can be expressed as 34 + 8 and 34 + 13 respectively — both sums of Fibonacci numbers (1,1,2,3,5,8,13,21,34,55,89). While the authors do not comment on this, the pattern is striking and deserves attention in the Discussion. If the developmental ground state tends toward 34 (a Fibonacci number), the frequent appearance of 42 and 47 could reflect an additive module that builds upon this foundation during later stages of floret initiation. Conversely, such post-hoc numerical patterns can easily arise by chance, so the authors should discuss this observation with appropriate caution. I do not insist that the authors endorse this interpretation, but a brief, balanced discussion would enrich the paper.

3. Clarification of plant material: single vs. double-flowered forms

double-flowered morphs can have dramatically altered ray floret numbers due to homeotic conversion of disc florets, which would introduce a severe structural bias into the distribution. Although double-flowered Bellis is not available in the wild, still I request that the authors add a clear statement in the Methods indicating that only wild-type, single capitula were sampled, and that any obviously aberrant garden escapes were excluded.

4. Terminology: The manuscript alternates between “ray flowers” and “ray florets.” Please use one term consistently.

5. Corrections: The abstract gives the mode as 42 and median as 46, but Figure 2 appears to show 42 and 47 as twin peaks. Confirm this data.

**********

what does this mean?). If published, this will include your full peer review and any attached files.

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Do you want your identity to be public for this peer review?  For information about this choice, including consent withdrawal, please see our Privacy Policy

Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

**********

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NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications.

Variations in ray flower numbers of Common Daisy (Bellis perennis L.) – the hidden cues of the Fibonacci order

PONE-D-25-51179R2

Dear Dr. Neumann,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Kind regards,

Jan Rychtář

Academic Editor

PLOS One

Additional Editor Comments (optional):

All reviewer's comments have been addressed

Reviewers' comments: