Abstract
The detection of male dimorphism has seen numerous statistical advances. Packard has recently criticized a widely used method, reanalyzing data from beetles and harvestmen using an alternative method. We disagree with Packard conclusions, probably due to different implicit definitions of male dimorphism. We consider that male dimorphism manifests in a distribution when it is significantly better described by a model with two values of central tendency (bimodality), rather than a model with only one (unimodality). Thus, while Packard suggests sigmoid allometries as alternatives to male dimorphism, we argue that such allometries are manifestations of mechanisms that generate bimodal distributions. Instead of focusing on this dichotomy, we propose an approach to test whether bimodality in a trait simply arises from its allometry by: (1) characterizing the trait static allometry, (2) simulating body size values based on original data parameters, and (3) generating new trait sizes using the static allometries. The percentage of simulations generating equal or greater bimodality than the data represents the likelihood that the bimodality can be explained by the allometry alone. Our method offers a null model linking sigmoid allometries and bimodal distributions, providing a test for mechanisms that accentuate trait bimodality beyond what the trait allometry generates.
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Data availability
All data, codes, and simulated data generated are available at the GitHub page of one of the authors (A.V.P.; https://github.com/alexandrepalaoro/simulations-bimodality).
References
Ameijeiras-Alonso J, Crujeiras RM, Rodríguez-Casal A (2018) Mode testing, critical bandwidth and excess mass. Test 28(3):900–919. https://doi.org/10.1007/s11749-018-0611-5
Bateson W, Brindley HH (1892) 3. On some cases of variation in secondary sexual characters, statistically examined. Proc Zool Soc Lond 60(4):585–596. https://doi.org/10.1111/j.1096-3642.1892.tb01785.x
Buzatto BA, Machado G (2014) Male dimorphism and alternative reproductive tactics in harvestmen (Arachnida: Opiliones). Behav Process 109:2–13. https://doi.org/10.1016/j.beproc.2014.06.008
Buzatto BA, Tomkins JL, Simmons LW, Machado G (2014) Correlated evolution of sexual dimorphism and male dimorphism in a clade of neotropical harvestmen. Evolution 68(6):1671–1686
Eberhard WG, Gutierrez EE (1991) Male dimorphisms in beetles and earwigs and the question of developmental constraints. Evolution 45(1):18–28
Elzhov TV, Mullen KM, Spiess A, Bolker B (2023) minpack.lm: R interface to the Levenberg-Marquardt nonlinear least-squares algorithm found in MINPACK, plus support for bounds. R package version 1.2-4. doi:https://CRAN.R-project.org/package=minpack.lm
Emlen DJ (1996) Artificial selection on horn length body size allometry in the horned beetle Onthophagus Acuminatus (Coleoptera: Scarabaeidae). Evolution 50(3):1219–1230
Emlen DJ (1997) Alternative reproductive tactics and male dimorphism in the horned beetle Onthophagus Acuminatus. Behav Ecol Sociobiol 41:355–341
Emlen DJ (2008) The roles of genes and the environment in the expression and evolution of alternative tactics. In: Oliveira RF, Taborsky M, Brockmann HJ (eds) Alternative Reproductive tactics: an Integrative Approach. Cambridge University Press, Cambridge, pp 85–108
Emlen DJ, Nijhout HF (2000) The development and evolution of exaggerated morphologies in insects. Annu Rev Entomol 45:661–708
Freeman JB, Dale R (2013) Assessing bimodality to detect the presence of a dual cognitive process. Beh Res Meth 45(1):83–97. https://doi.org/10.3758/s13428-012-0225-x
Gadgil M (1972) Male dimorphism as a consequence of sexual selection. Am Nat 106(951):574–580
Hissmann K (1990) Strategies of mate finding in the European field cricket (Gryllus-campestris) at different population-densities - a field-study. Ecol Entomol 15(3):281–291. https://doi.org/10.1111/j.1365-2311.1990.tb00810.x
Kahm M, Hasenbrink G, Lichtenberg-Frate H, Ludwig J, Kschischo M (2010) Grofit: fitting biological growth curves with R. J Stat Softw 33(7):1–21. http://www.jstatsoft.org/v33/i07/
Kang Y-J, Noh Y (2019) Development of Hartigan’s dip statistic with bimodality coefficient to assess multimodality of distributions. Math Probl Eng 2019:1–17. https://doi.org/10.1155/2019/4819475
Knell RJ (2009) On the analysis of non-linear allometries. Ecol Entomol 34(1):1–11. https://doi.org/10.1111/j.1365-2311.2008.01022.x
Kotiaho JS, Tomkins JL (2001) The discrimination of alternative male morphologies. Behav Ecol 12(5):553–557
Machado G, Buzatto BA, Samia DSM (2021) It is not always about body size: evidence of Rensch’s rule in a male weapon. Biol Lett 17(6):20210234. https://doi.org/10.1098/rsbl.2021.0234
Nijhout HF (1999) Control mechanisms of polyphenic development in insects. Bioscience 49(3):181–192
Packard GC (2021a) Is allometric variation in the cephalic horn on male rhinoceros beetles discontinuously dimorphic? Evol Biol 48(2):233–245. https://doi.org/10.1007/s11692-021-09533-1
Packard GC (2021b) When perception isn’t reality: allometric variation in the exaggerated mandibles of male stag beetles (Coleoptera: Lucanidae). Biol J Linn Soc 134:760–772
Packard GC (2022) Bimodal frequency distribution as a criterion for dimorphism in male rhinoceros beetles (Coleoptera: Scarabaeidae: Dynastinae). Biol J Linn Soc 136(2):384–392. https://doi.org/10.1093/biolinnean/blac028
Packard GC (2023) Frequency distributions, dimorphisms, and allometric variation in size of the weapon on male harvestmen (Arthropoda, Arachnida, Opiliones). Evol Ecol 37(4): 749-760
Palaoro AV, García-Hernández S, Buzatto BA, Machado G (2022) Function predicts the allometry of contest-related traits, but not sexual or male dimorphism in the amazonian tusked harvestman. Evol Ecol 36:605–630. https://doi.org/10.1007/s10682-022-10152-1
Prates MO, Cabral CR, Lachos VH (2013) Mixsmsn: fitting finite mixture of scale mixture of skew-normal distributions. J Stat Softw 54(12):1–20
Rowland JM, Qualls CR (2005) Likelihood models for discriminating alternative phenotypes in morphologically dimorphic species. Evol Ecol Res 7(3):421–434
Rowland JM, Qualls CR, Beaudoin-Ollivier L (2005) Discrimination of alternative male phenotypes in Scapanes australis (Boisduval) (Coleoptera: Scarabaeidae: Dynastinae). Aust J Entomol 44:22–28. https://doi.org/10.1111/j.1440-6055.2005.00440.x
Shuster SM (2008) The expression of crustacean mating strategies. In: Oliveira RF, Taborsky M, Brockmann HJ (eds) Alternative Reproductive tactics: an Integrative Approach. Cambridge University Press), Cambridge, pp 224–250
Taborsky M, Oliveira RF, Brockmann HJ (2008) The evolution of alternative reproductive tactics: concepts and questions. In: Oliveira RF, Taborsky M, Brockmann HJ (eds) Alternative Reproductive tactics: an Integrative Approach. Cambridge University Press, Cambridge), pp 1–21
Tomkins JL, Hazel WN (2007) The status of the conditional evolutionarily stable strategy. Trends Ecol Evol 22(10):522–528. https://doi.org/10.1016/j.tree.2007.09.002
Tomkins JL, Moczek AP (2009) Patterns of threshold evolution in polyphenic insects under different developmental models. Evolution 63(2):459–468. https://doi.org/10.1111/j.1558-5646.2008.00563.x
Zatz C, Werneck RM, Macias-Ordonez R, Machado G (2011) Alternative mating tactics in dimorphic males of the harvestman Longiperna Concolor (Arachnida: Opiliones). Behav Ecol Sociobiol 65(5):995–1005
Acknowledgements
We thank Gary Packard for triggering this interesting debate on the topic of male dimorphism and static allometry, and also for revising an early version of the manuscript. We also thank an anonymous reviewer for comments and Matthew Symonds and Emma Sherratt for considering this submission for Evolutionary Ecology.
Funding
GM is supported by research grants from Fundação de Apoio à Pesquisa do Estado de São Paulo (FAPESP 2021/00915-5) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq 306550/2014–8).
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B.A.B, A.V.P and G.M. conceived the project, B.A.B and A.V.P performed the analyses, B.A.B, A.V.P and G.M. interpreted the results and wrote the manuscript.
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We have authored four of the five studies that originated the data reanalysed by Packard (2023), and we therefore support the conclusions of those original studies. We declare no further conflicts of interest.
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Buzatto, B.A., Machado, G. & Palaoro, A.V. Sigmoid allometries generate male dimorphism in secondary sexual traits: a comment on Packard (2023). Evol Ecol (2024). https://doi.org/10.1007/s10682-024-10303-6
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DOI: https://doi.org/10.1007/s10682-024-10303-6