The studied hoverfly species shown with their phylogenetic relationships.

Sample size as number of individuals for each species is in parentheses. Species marked with coloured circles are displayed on the right, and coloured circles with black outlines are of species for which both morphology and flight kinematics were quantified. Body mass is presented as mean values and standard deviation. Phylogeny was obtained from Wong et al. (2023).

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.

Wingbeat kinematics during hovering flight of the eight studied hoverfly species.

(A-E) Temporal dynamics of the wingbeat kinematics throughout the wingbeat 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; see Figure 2 for definitions). (D-E) Angular speed and angle-of-attack of the wings throughout the wingbeat cycle, derived from wing stroke, deviation and rotation angles. (F-I) Derived wingbeat-average wingbeat kinematics parameters for each studied species versus the body mass of that species. The kinematics parameters are wing stroke amplitude Aϕ, wingbeat frequency f, wingbeat-average angular wing speed ω), and mean angle-of-attack at mid wing stroke)α, respectively. Each data point shows mean ± standard error per species, and is colour-coded according to the legend on the top. None of the wingbeat-average wingbeat kinematic parameters were significantly associated with body mass (Table 1). Horizontal dashed lines show the mean parameter values, as expected under kinematic similarity.

Results of phylogenetic general least square (PGLS) regressions of the log10-transformed morphological and wingbeat kinematic parameters from the aerodynamic model (Eqns 1,2) relative to log10-transformed body mass.

Estimated scaling factors with 95% confidence intervals that exclude the scaling factor for isometry are indicated in bold and with a star, suggesting allometric scaling of that metric (see last column). The preceding two columns show the scaling factor for geometric or kinematic similarity, and the scaling factor for maintaining weight support across sizes via allometric changes of the specific parameter, assuming all other parameters scale under geometric and kinematic similarity.

The scaling of wing morphology with body mass for all 28 studied hoverfly species (A-D), and how allometric scaling of wing morphology contributes to maintaining weight support across the hoverfly size range (E,F).

(A-D) body mass (abscissa) versus on the ordinate the second-moment-of-area S2, wingspan R, mean chord 𝑐̅, and normalized second-moment-of-area 𝑆2∗, respectively. Each data point shows a species average value, where data points of species for which flight was studied are colour-coded (see top row), and species with only quantified morphology are shown in grey (see the corresponding individual-specific data in Figure S3). The black, red and grey trendlines show the best fitting line from a PGLS regression, isometric scaling, and the expected scaling for metric-specific maintenance of weight support, respectively (see legend above A). All fitted PGLS regression slopes, except for wing chord, are significantly lower than expected under isometry, suggesting negative allometric scaling of wing size and shape with respect to body mass (see also Table 1). (E,F) The relative contributions of different wing morphology metrics to maintaining weight-support across the studied range of hoverfly sizes, expressed by the relative allometric scaling factor a*. (E) the relative allometric scaling factor for the second-moment-of-area S2, and (F) the relative allometric scaling factor for the separate morphological components of S2: wingspan R, mean chord 𝑐̅, and normalized second-moment-of-area 𝑆2∗.

Changes in wing shape and size associated with body mass variation.

Each data point shows a species-average value (see the individual-specific data in Figure S5), and are colour-coded for species of which both flight and morphology were studied (see right column of B), and in grey for species with only morphology quantified. (A) Normalised second-moment-of-area (S2*) versus the primary Principal Component (PC1) of the phylogenetic Principal Component Analysis (phyloPCA) performed on the mean wing shape coordinates of all 28 studied hoverfly species. On the right, we show the 28 wing outlines colour-coded and normalised with body mass (with coordinates X∗=X/m1/3). (B) body mass (m) versus PC1 (left), and mass-normalised wing outlines versus body mass for the eight species used in our flight experiments (right). The left and right gritted wing shapes on the PC1 axis show the theoretical wing shapes at maximum and minimum value of PC1. PC1 explains 65.24% of the variations in wing shape (see also Figures S4 and S5). (A-B) The combined results show that in larger species, wing surface area was located more proximally (lower PC1 and S*2 values) than in smaller species, in which wing area tend to be located more distally (higher PC1 and S*2 values). Combined with the changes in wing shape (left), weight-normalised wingspan (R∗) and mean chord (𝑐̅) 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 (top row), but average wingbeat kinematics and frequency (𝑓̅) across all eight studied hoverfly species (A,B). (C) The resulting temporal dynamic of vertical forces throughout the wingbeat cycle, coloured by species (see legend on top). (D) The wingbeat-average vertical force versus body mass, for all simulated hoverfly species operating at the average wingbeat frequency for all species 𝑓̅ (square data points and dashed trend line), and forces scaled to the species-specific wingbeat frequency f (round data points in B and D, and solid trend line). The trendline for weight support (F=mg) is shown with a grey dashed line. With the species-specific wingbeat frequency f (solid trend line and circles in B and D) hoverflies are closer to producing weight support than for the simulations at the average wingbeat frequency 𝑓̅ (dashed lines in B and D).

The relative contribution of allometric variations in wing morphology and wingbeat kinematics to CFD-derived vertical aerodynamic force production, and how this contributes to maintaining in-hovering weight-support across the sampled range of hoverfly sizes.

(A-C, D-F) Wingbeat-average vertical force at species-specific wingbeat frequency f (abscissa) vs. on the ordinate various wing morphology and kinematics traits. In each panel, data points are results for different species, colour-coded according to the legend on the top. The solid and dashed black lines show the significant and non-significant OLS regression fits, respectively. The dashed red and grey lines show the expected slopes for scaling under morphological and kinematic similarity and for 100% metric-specific weight-support, respectively. (G,H) The relative contributions of allometric variations in wing morphology and kinematics to maintaining weight-support across the studied range of hoverfly sizes, expressed by the relative allometric scaling factor a* (Eqn 3). Contributions based on significant and non-significant OLS regressions are shown in dark and light grey, respectively. (A) The wingbeat-average vertical force scales linearly with the product of second-moment-of-area and wingbeat-average angular speed squared (Eqn 1: 𝐹∼𝑆2 ω̄2), resulting in weight-support across all sizes. (B-C) Separating this product into its main components (B and C, respectively) shows that the second-moment-of-area and wingbeat frequency contribute 81% and 22% to maintaining weight support across sizes, respectively (D). (E-H) Parting the contribution of second-moment-of-area into its components (D-F) shows that allometric variations in wingspan, mean chord and nondimensional second-moment-of-area contribute 55%, 19% and 5% to maintaining weight support across sizes, respectively (H).

Results of ordinary least square (OLS) regression between the log10-transformed vertical aerodynamic force estimated from CFD and log10-transformed morphological and kinematics parameters, for all eight hoverfly species studied using CFD.

OLS regressions for metrics that scale significantly with aerodynamic force magnitude are indicated with a star, and have P-values in bold (P<0.05). For each tested parameter, we estimated its relative contribution to maintaining weight support across sizes using the relative allometric scaling factor a* (Eqn 3), based on the estimated scaling factor, the scaling factor for geometric or kinematic similarity, and the expected scaling for maintaining metric-specific weight support.

List of symbols and abbreviations

The body mass of museum specimens was approximated using the relationship between thorax width and the fresh mass in wild caught specimens.

(A) A cubic polynomial regression model was fitted to the data from wild caught specimens. (B) The equation of the fitted model was used to estimated body mass from thorax width in museum specimens.

Derived wingbeat-average wing kinematics parameters for each studied species versus the body mass of that species.

The kinematics parameters are (A) wingbeat frequency f, (B) wing stroke amplitude Aϕ (C) wingbeat-average angular wing speed ω), and (D) mean angle-of-attack at mid wing stroke α), respectively. Each data point shows mean ± standard error per species. None of the wingbeat kinematic parameters were significantly associated with body mass, as shown by the non-significant dashed black trend lines from PGLS regressions. The corresponding scaling based on kinematic similarity is shown with red dashed lines, and the expected scaling for maintaining weight support across sizes, when assuming that all other parameters scale under kinematic similarity are shown in dotted grey lines (see Table 1 for details).

Wing morphology parameters versus body mass for all studied specimens.

(A-D) body mass (abscissa) versus on the ordinate the second-moment-of-area S2, wingspan R, mean chord 𝑐̅, and normalized second-moment-of-area 𝑆2∗, respectively. Each circle shows data for a different individual. Species for which both flight and morphology were studied are colour-coded (see top row), whereas species with only quantified morphology are shown in grey.

Result of geometric morphometrics analysis on the wing outlines of 28 hoverfly species.

Species for which both flight and morphology were studied are colour-coded (see top row), whereas species with only quantified morphology are shown in grey. (A) The first two principal components (PC1 and PC2) from the phylogenetic principal component analysis (phyloPCA) on the geometric morphometrics data, along with the associated shape changes and the percentage of variation explained by these 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.

Result of geometric morphometrics analysis on the wing outlines of 28 hoverfly species.

Each data point shows a different individual value. Species for which both flight and morphology were studied are colour-coded (see top row), whereas species with only quantified morphology are shown in grey. (A) The two first principal components (PC1 and PC2) of the phylogenetic Principal Component Analysis (phyloPCA), 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 colour-coded according to the legend on the top. All simulations were run with the mean wingbeat kinematics of all hoverflies combined (Fig 6A,B), 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 as directly estimated using the CFD simulations, and where all wings operate at the mean wingbeat frequency of all hoverflies (𝑓̅); (B) aerodynamic forces as produced by each hoverfly beating its wings at the species-specific wingbeat frequency f; (C) aerodynamic forces normalized with its wingbeat-average force, 𝐹∗ = 𝐹/𝐹I; (D) aerodynamic forces produced using the species-specific wingbeat frequency, and normalized with the mean weight of the specific hoverfly species F/mg.

Results of ANOVAs testing the effect of sex and species on wing morphology parameters.

Phylogenetic signal computed on morphological and flight traits. Species number in the morphology and flight dataset is 28 and 8, respectively.

Results from multiple regressions testing correlations between the wingbeat kinematics parameters and body kinematics, expressed by flight speed and climb angle.

Results of phylogenetic generalized least square (PGLS) regressions of additional wingbeat kinematic parameters against body mass.