Computational protocol: The biomechanical origin of extreme wing allometry in hummingbirds

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[…] We use our allometric framework to analyse a data set obtained from individual hummingbirds sampled at different sites in Brazil, Canada, Costa Rica, Ecuador, Peru and the United States (Fig. ). We do not explicitly distinguish sexes. Some kinematic and morphological data for these species have appeared elsewhere, , –. Sample sizes in each bivariate regression in numbers of species and individuals are presented in Supplementary Table , and decisions on species placement and taxonomy are provided in Supplementary Note . All data collection was performed in compliance with respective institutional guidelines. No randomisation or blinding was performed in this study. For the results reported here, we used all available samples, but investigated the impact of data subsets, as described below.Air density was calculated from elevation using standard pressure and temperature relationships with elevation. We emphasise that in the context of this analysis, the allometry of ρ is interpreted as evidence for an association between body mass and air density (or elevation), whether due to individuals or species selecting their environment or adapting to it, and not as hummingbirds effecting changes in local air density. Given a species’ or individual’s body mass, this regression is a prediction of the environment in which it will be found. In preliminary analyses, we found that b ρ was somewhat influenced by the inclusion of the unusually large and phylogenetically distinct species Patagona gigas, and by inclusion of species with a single observation (Supplementary Fig. ). Removal of these progressively reduces the air density allometric exponent toward zero, and so the overall influence of elevation and air density on species body mass is uncertain. Nonetheless, it is notable that the exponent is similar among and within species, which could indicate a common underlying mechanism. We investigated whether independent data sets might show evidence of a correlation between body mass and elevation. We collected species mean body masses and elevational midpoints from the Handbook of the Birds of the World (HBW) and calculated mean species elevations from range maps provided by BirdLife International (BL; see ‘Methods’ for further details of mapping procedures). Mean elevations from the two sources are well correlated (Supplementary Fig. ), though with somewhat more error for low elevation species. Predictions of species maximum elevation were uncorrelated, likely because the range maps coarsely include all elevations within a contour. Elevation and body mass were examined using a phylogenetic regression implemented in MCMCglmm (see below). For all elevational parameters (minimum, mean, and maximum) in both data sets, the CIs of the slopes overlap 0 (Supplementary Fig. ).We examined whether capturing individuals at discrete sites influences results, because discrete sampling might not reflect continuous elevational distributions. We therefore sought to compare our results to independent estimations of species elevations, derived from species range maps. Our observational data are reasonably well correlated with the derived species mean elevation and the distribution of species elevations (Supplementary Fig. ).Wing morphological variables were digitised from photographs of the spread wing as described by Altshuler et al. or from wings spread on graph paper and traced in Adobe Illustrator (CCW collection). We obtained the wing area, S, and length, R, and second and third moments of area, r^2 and r^3, from these photos, and the aspect ratio was calculated as AR = 4R 2/S. There was a high degree of correlation in wing morphology and air density measurements among authors with overlapping species measurements (DLA, CCW and PSS data sets; Supplementary Fig. ), and so apparent differences between data sets appear to be attributable to species sampling.Kinematics (mean stroke amplitude and frequency) were digitised as previously described, , . The mean wing velocity at the second moment of area was calculated as the product of stroke frequency, stroke amplitude, wing length, and the second moment of area, (Ū=4fΦr^2 R = 4fΦR 2, see ‘Methods’). Our results do not differ depending on this definition of wing velocity, or the use of the wing tip velocity directly, because r^2 is not correlated with body mass (Supplementary Table ). We calculated the vertical force coefficients in flight while hummingbirds support weight (C¯ w,V) or during burst load lifting (C¯ b,V), by rearranging Eq. (). [...] All analyses were performed with R 3.2.0 to organise data and interface with JAGS 4.2). We also used the R package dplyr for data manipulation; ape, nlme and treespace , , for phylogeny manipulation, visualisation of phylogenetic uncertainty, and comparison of our parameter estimates to those obtained by maximum likelihood; and rjags and R2jags , for interfacing with JAGS.The map in Fig.  was generated in R using the packages mapplots, raster, rworldmap and sp –. The map of the Americas, and the latitudes and longitudes of the collections sites, were transformed to a Mollweide projection centred on (Lat 0, Lon −90). For clarity, we omitted collection sites with a single record, and grouped nearby sites (especially transects) in 0.5 × 0.5° cells. The map is shaded to provide elevational context for hummingbird ranges, and the elevation of individual collection sites, relative to 5000 m, is depicted in a cartoon. The phylogeny in Fig.  was drawn with the aid of the package phytools . The sample size for partial kinematics was the number of individuals with a calculated force coefficient in hovering, and the sample size for full kinematics was determined as the number of individuals with both a hovering and burst load lifting force coefficient. The sample size for morphology alone was determined as the number of individuals with weight, elevation and wing area data. […]

Pipeline specifications

Software tools Adobe Illustrator, dplyr, APE, treespace, Phytools
Applications Miscellaneous, Phylogenetics
Diseases Stroke