Figures and data
The studied hoverfly species shown with their phylogenetic relationships. Phylogenetic positions of the studied species in a larger hoverfly phylogeny (91 species) are shown in the bottom left corner. Sample size as number of individuals for each species is in parentheses. Mean values (± standard deviation) for body mass are presented, along with a photography of a representative individual and a wing from each species. Phylogeny was obtained from (Wong et al., 2023). Photographs were taken by the second author.
Quantifying the in-flight wingbeat kinematics and wing morphology of hoverflies. (A) Hoverflies were released in an octagon-shaped flight arena. We recorded stereoscopic high-speed videos of the flying hoverflies using three synchronised high-speed video cameras; from the videos we reconstructed the three-dimensional body and wingbeat kinematics. Infrared light panels positioned as the bottom of the arena enabled high contrast between the flying insect and the background. (B) Conventional wing angles were measured at each time step (t=0.0004 s) in the body reference frame. ϕ, wing stroke angle within the stroke plane; η, wing deviation angle out of the stroke plane; θ, wing rotation angle along the spanwise axis; U, air velocity vector relative to the wing; α, angle-of-attack of the wing. (C) Wing morphological parameters including their definitions: wingspan R, wing surface area S, radial position along the span r, local wing chord c at distance r, and the second-moment-of-area S2.
Relationship between wingbeat kinematics and body mass among hoverfly species. (A-E) Temporal dynamics of the wingbeat kinematics throughout the cycle of all digitised wingbeats. Separate wingbeats are colour coded by body mass (see legend on top), and the black lines show the average wingbeat kinematics for all wingbeats combined. (A-C) Temporal dynamics of the three conventional wingbeat kinematics angle (stroke, deviation, and rotation angle, respectively). (D-E) Angular speed and angle-of-attack of the wings throughout the wingbeat cycle, derived from the stroke and rotation angle, respectively. (F-I) Derived Wingbeat kinematics parameters for each studied species versus the body mass of that species. The kinematics parameters are mean angular wing speed, stroke amplitude, angle-of-attack at mid stroke, and wingbeat frequency. Each data point shows mean ± standard error. None of the wingbeat kinematic parameters were significantly associated with body mass.
Results from reduced major axis regression of hovering-flight wingbeat kinematic parameters against body mass. Test were performed using mean values per species (n=8). See figures 2 and 3F-I for definitions of primary kinematics parameter and visualisations of the data, respectively.
Results from reduced major axis regression of wing morphology parameters against body mass (n=44 individuals among eight species). See figures 2 and 4 for definitions of primary morphology parameter and visualizations of the data and regressions, respectively.
Wing morphology parameters versus body mass for all studied species, including isometric scaling and the observed reduced major axis regression. (A-D) body weight (abscissa) versus on the ordinate the second-moment-of-area, wingspan, mean chord and normalised second-moment-of-area, respectively. Each data point shows results for a different individual, and is color-coded according to species (see top of A). The expected slopes for isometric scaling are indicated by dashed grey lines, and the best fitting reduced major axis regressions are indicated by continuous black lines (see top of B). All fitted regression slopes are significantly lower than expected under isometry, suggesting negative allometric scaling of wing size and shape with respect to body mass.
Changes in wing shape and size associated with body mass variation and according the geometric morphometrics analysis. (A) normalised second-moment-of-area (S2) vs. the primary Principal Component (PC1) of the geometric morphometrics analysis on the wing outlines (left), and weight-normalised wing outlines (with coordinates X*=X/m1/3) color-coded with body mass (see legend) (right). (B) body mass (m) versus the primary Principal Component (PC1) (left), and weight-normalised wing outlines and wingspan (R*) versus body mass (right). Small transparent data points and large opaque data points show values per individual and per species, respectively. All data points and wing outlines in B are color-coded according to species (see right of B). (A) Variation in the normalised second-moment-of-area (S2) were captured well by PC1 (see also Fig. S2). (B) In larger species, wing surface area was located more proximally (lower values on PC 1) than in smaller species, in which wing area tend to be located more distally (higher value on PC 1). The left and right gritted wing shapes on the PC1 axis show the hypothetical wings for PC1=-0.08 and PC1=0.06, respectively. PC1 explains 65.24% of the variations in wing shape (see also figure. S2). (A-B) Combined with the changes in wing shape (left), weight-normalised wing size and wingspan (R*) tend to be larger in smaller species (right).
Aerodynamic forces produced by hovering hoverflies, as estimated using Computational Fluid Dynamic (CFD) simulations. (A-B) For our simulations, we used the species-specific wing shapes and sizes (figure 5), and average wingbeat kinematics (A) and the mean wingbeat frequency (B) across all eight studied hoverfly species. (C) The resulting temporal dynamic of vertical forces throughout the wingbeat cycle, coloured by species (see legend above B). (D) The wingbeat-average vertical force versus body mass, for all simulated hoverfly species operating at both the average wingbeat frequency for all species and their species-specific frequency (square and round data points, respectively). Trendlines for weight support (F=mg), and results of the average wingbeat frequency and species-specific frequency simulations are shown as grey dashed, black dashed and black solid lines, respectively. With the species-specific wingbeat frequency (shown in panel B) hoverflies are closer to producing weight support than for the average wingbeat frequency simulations.
Relationships between wing morphology and vertical force production during simulated hovering flight. (A-D) Wingbeat-average vertical force (abscissa) vs. on the ordinate the dimensional and normalised second-moment-of-area, wingspan, and mean wing chord, respectively. Data points are results for different species, as color-coded according to the legend above panel E). The expected slopes for isometric scaling are indicated by dashed grey lines, and the best fitting reduced major axis regressions are indicated by continuous black lines (see top of A). All fitted regression slopes are significantly lower than expected under isometry, suggesting negative allometric scaling. (E) The relative contributions of different aspects of wing morphology (R, 
Ancestral state estimates for body size plotted onto a phylogeny of hoverflies. Ancestral states were estimated and plotted along the tree branches following method described in Revell (2013). Thorax width data (a proxy of body size) were obtained from Gilbert (1985) and the phylogeny from Wong et al. (2023). n=29 species.
Result of geometric morphometrics analysis on the wing outlines of eight hoverfly species. Data points show results for different individuals, coloured by species (see legend on top). (A) The two first principal components (PC1 and PC2) of the geometric morphometrics analysis, together with the associated shape changes and percentage of the variation explained by the principal components. Shown wing shapes represent the extreme values associated with each PC axis. (B) PC1 and PC2 (first and second row, respectively) plotted against body mass, highlighting that the change in wing shape carried on PC1 is associated with variations in body mass.
Temporal dynamic of the vertical aerodynamic forces produced during the wingbeat of the eight studied hoverfly species, estimated using Computation Fluid Dynamics (CFD) simulations. Data for the eight species are color-coded according to the legend on top. All simulations were run with the mean wingbeat kinematics of all hoverflies combined (Fig 6A), but with species-specific wing morphology (see legend on top). The aerodynamic forces in the different panels are scaled and normalized using four different methods: (A) aerodynamic forces in mN as directly estimated using the CFD simulations, and where all wings operate at the mean wingbeat frequency of all hoverflies 
Number of female and male individuals in the studied species.
Phylogenetic signal computed on morphological and flight traits on the eight studied species.
Results from multiple regressions testing the effect of body dynamics on the wingbeat kinematics parameters.
