Male stag beetles have evolved extremely large mandibles in a wide range of extraordinary shapes. These mandibles function as weaponry in pugnacious fights for females. The robust mandibles of Cyclommatus metallifer are as long as their own body and their enlarged head houses massive, hypertrophied musculature. Owing to this disproportional weaponry, trade-offs exist with terrestrial locomotion: running is unstable and approximately 40% more costly. Therefore, flying is most probably essential to cover larger distances towards females and nesting sites. We hypothesized that weight, size and shape of the weaponry will affect flight performance. Our computational fluid dynamics simulations of steady-state models (without membrane wings) reveal that male stag beetles must deliver 26% more mechanical work to fly with their heavy weaponry. This extra work is almost entirely required to carry the additional weight of the massive armature. The size and shape of the mandibles have only negligible influence on flight performance (less than 0.1%). This indicates that the evolution of stag beetle weaponry is constrained by its excessive weight, not by the size or shape of the mandibles and head as such. This most probably paved the way for the wide diversity of extraordinary mandible morphologies that characterize the stag beetle family.
In sexual selection theory, the differential costs of sexual traits are essential to balance their benefits and to ensure honest signalling. However, it has been hard to prove these costs experimentally [1,2]. Even in species with very impressive ornaments and armaments, the costs are often relatively small. For example, peacock trains (6.9% of body mass) do not decrease take-off performance, and rhinoceros beetle horns (1.5% of body mass) do not influence flight distance, travel speed or aerodynamic costs [3–5]. The detection of such costs is even harder when animals have also evolved compensatory traits . For example, malachite sunbirds have increased their wingspan to compensate for the negative effects of their long tail (0.3% of body mass) on flight cost and manoeuvrability , and swordtail fishes have a larger heart ventricle to compensate for the decreased swimming endurance because of their long tail . The costs of bearing weapons become more evident when the armaments are heavier. Walking around with hermit crab chelipeds (30% of body mass), fiddler crab claws (50% of body mass) or stag beetle mandibles and their musculature (18% of body mass) is energetically very costly [8–10]. Stag beetle males need their robust mandibles and the associated massive muscles to bite extremely forcefully (three times as forcefully as scaled female bites) in their aggressive battles  (figure 1). As a result, the heavy armature not only increases the energy cost of running, but Cyclommatus metallifer stag beetles are also statically unstable twice every stride . Therefore, flying is most probably essential to cover larger distances towards females and nesting sites. The importance of flight over running has also been observed for the European Lucanus cervus stag beetles . It should therefore be questioned whether or not the weaponry impairs the aerial performance of male stag beetles. If so, this would be an important factor in the evolution of stag beetle armature.
In this paper, we investigate the influence of stag beetle weaponry on their cost of flight, by simulating the aerodynamic forces and moments on a flying stag beetle body with computational fluid dynamics (CFD). Given the high armature mass, we hypothesize that the associated increase in the cost of flight will be significant, as opposed to that of the lightweight horns of rhinoceros beetles. In addition to calculating the effect of the weight of the armature, we will quantify the energetic and aerodynamic consequences of its size and shape, as these characteristics determine the flow of air around the head during flight. Armature weight, size and shape are obviously inter-related. Theoretically, however, these aspects can be decoupled from each other in the case of structural changes (e.g. cuticle thickness). Size and shape (and not the weight as such) determine the aerodynamic performance of the morphology. Therefore, from an evolutionary point of view, it is important to consider the effects of armature size and shape on the net mechanical flight cost separately. We therefore test the aerodynamic and energetic consequences of different mandible weights, sizes and shapes explicitly, by comparing CFD models with modified armature.
2. Material and methods
2.1. Micro-computed tomography imaging
We obtained C. metallifer stag beetles from a commercial dealer (Kingdom of Beetle, Taipei, Taiwan). The anterior body part of the male and female specimens was scanned by the Centre for X-ray Tomography of Ghent University (voxel sizes of, respectively, 38 µm and 13 µm). For the posterior body parts, we used the Skyscan 1172 high-resolution micro-CT scanner (Bruker micro-CT, Kontich, Belgium; voxel size of 13 µm for both the male and the female specimens). The same Skyscan micro-CT scanner was used for the limbs (voxel size of 27 µm for both sexes), and the mandibles of four other stag beetle species (voxel size of 13 µm for Lamprima adolphinae, Dorcus bucephalus and Prismognathus davidis; 11 µm for Nigidius obesus).
2.2. Three-dimensional beetle model
Using the micro-CT scans, we created surface models of each of the body parts (head, thorax, abdomen, mandibles, limbs, elytra). This was performed in the three-dimensional image processing software Amira (v. 5.4.4; VSG Systems, Mérignac, France), in which we annotated the voxels that belong to the body using automatic thresholds (based on greyscale values) and manual corrections in the three orthogonal views. To obtain information on the natural flight position, we made video recordings (125 frames s−1, JVC GZ-V515; JVC Kenwood Corporation, Kanagawa, Japan) of flying male stag beetles in the laboratory. We positioned the triangulated three-dimensional surface models of the body parts relative to each other and the global frame of reference (Capture v. 11; Geomagic, Cary, NC; figure 2). The hard wing covers (elytra) of dead specimens were manipulated to find their ‘locked position’ as adopted in flight [13,14]. The entire beetle body, yet without the membrane wings, was represented as one single rigid body for which force and moment balances were calculated (similar to the approach of Taylor & Thomas  for the desert locust and Sum & Xiong  for the bumblebee). As we consider only the body in our simulations, the aerodynamic interactions between the body and the membrane wings are neglected. For example, interactions of the flapping wings with vortices that are shed by the body are not considered. However, because numerical Navier–Stokes simulations have shown that the aerodynamic interactions between the body and the wings of insects are negligible, their aerodynamic characteristics can adequately be analysed separately . By performing CFD simulations (see below), the net forces (i.e. as averaged over a wing beat cycle) and therefore also the external work (i.e. whole body level) per unit distance that need to be delivered by the wings can be calculated. This enables the straightforward comparisons of the effect of the mandibles aimed at in this paper.
2.3. Computational fluid dynamic simulations
We constructed a virtual wind tunnel around the beetle model in Designmodeller (v. 14.5; Ansys, Canonsburg, PA). The boundaries of the virtual wind tunnel left sufficient space around this rigid body to prevent artefacts. The volume between these boundaries and the beetle model was subsequently filled with tetrahedrons. To enhance the accuracy, this mesh became finer the closer it was to the beetle model (figure 3). The required number of mesh elements were tested in a convergence analysis (see below).
In Fluent (v. 14.5; Ansys, Canonsburg, PA), a velocity inlet was created in the boundary in front of the beetle model, with a horizontal flow directed towards the anterior side of the stag beetle model. The velocity was 0.57 m s−1, corresponding to the highest flight velocity observed. The boundaries that are parallel with the flow direction were modelled as moving walls with the same velocity as the air. A pressure outlet was defined at the exit of the virtual wind tunnel.
The Reynolds number (Re) is where ρ is the density of air at 20°C, ν is the flow velocity, L is the length of the beetle along the flow direction and μ is the dynamic viscosity of air at 20°C. Based on this intermediate Re, we assumed a laminar boundary layer, although the wake may already show turbulent flow patterns . Therefore, a transition shear-stress transport flow model was used . The Reynolds-averaged equations of a fluid flow were solved numerically through the finite-volume method. We used a node-based Green–Gauss gradient treatment, because it achieves higher accuracy than cell-based gradient treatments in unstructured tetrahedral meshes (Ansys user manual). The simulations ran until convergence of the scaled flow equation residuals was reached, usually after approximately 3000 iterations. The total drag and lift forces on the rigid body were computed by Fluent. These forces apply at the body centre of mass, are directed, respectively, horizontally and vertically and take both pressure and viscous force components into account.
2.4. Convergence analysis
The number of tetrahedral mesh elements in our simulation is a compromise between computational power and accuracy. We compared the aerodynamic forces and moments of 28 models of between 300 000 and 10 million mesh elements. For the mesh with 2.7 million elements, used for the simulation experiments, the relative aerodynamic differences with the finest mesh model output was only 2.2%.
We simulated a sphere with the Re of the beetle under the same settings as the beetle simulations. The simulation predicted a drag coefficient of 0.49, which is nearly identical to the experimentally determined value of 0.47 . Furthermore, a recent study using almost identical model settings proved accurate in calculating the forces and moments by the flow around the more complex biological shapes of boxfish carapaces .
2.6. Force balance
The CFD simulations provide the aerodynamic forces and moments that apply on the steadily and horizontally flying rigid body. The gravitational force acts at its centre of mass. The equilibrating external forces and moments needed to maintain this steady state can derive only from the wings and must be considered as the net effect (in terms of forces and moment transfer) from the wings to the body as averaged over a wing beat cycle. These net forces and moments apply on the rigid body almost vertically above the experimentally determined position of the body centre of mass . Horizontal and vertical forces (wing, aerodynamic and gravity) must cancel each other out to proceed in a steady horizontal manner (figure 2). Note that the moments of the forces (wing and aerodynamic), as well as potential direct torque transfer from the wings on the body, must be in equilibrium to maintain the average orientation of the body over a cycle. The moment balance is, however, redundant for the present analysis, hence it is not considered further. The cycle-averaged wing forces derive from the net work done by the flight muscles. As such, these forces can be considered a direct measure for the net mechanical energy requirements per unit distance covered (net mechanical cost of flight). It should be noted that the present simulations pretend by no means to reflect the instantaneous unsteady features of the dynamics of insect flight. These simulations deliberately aim at capturing just the basic costs of the beetle's flight (void of all intricate confounding factors) useful for a solid comparison.
2.7. Simulation experiments
We compared the complete beetle model with other models to assess the influence of the mandible's weight, size and shape on the aerodynamic and wing forces:
— Weight: complete model versus a model without mandibles and a model of a non-dimorph conspecific. The non-dimorph state is represented by the scaled female morphology, because the male and female morphology is very alike in stag beetle species without sexual dimorphism [10,22] (figures 1 and 4a,c,d). The non-dimorph model was scaled in such a way that the posterior body part (meso- and metathorax + abdomen) was as large as in the complete model . For both the without mandibles and non-dimorph model, the total mass was reduced to conform to our weight measurements (on fresh material with an analytical microbalance; MT5; Mettler Toledo, Columbus, OH; precision: 5 µg; table 1), and the position of the centre of mass was adapted according to the altered model volume. The differences in the aerodynamic forces between the complete model and the two other models reveal the influence on the aerodynamic drag and lift of the large mandibles (for the without mandibles model) or the large mandibles and bite muscles (for the non-dimorph model).
— Size: complete model versus a model with short mandibles (isometrically scaled to half the original length; figure 4b) and the model without mandibles (i.e. infinitely small mandibles). Because we want to examine the size effect separately from the weight effect, we simulated these three models with an identical total body mass and position of the body centre of mass. Hence, the without mandibles model in the size comparison differs from the without mandibles model in the weight comparison. In nature, the male mandible size varies strongly between C. metallifer individuals . We use isometrically scaled mandibles for the short mandibles model, to exclude the additional shape effects that would have occurred if we would have used the naturally short mandibles of a conspecific individual.
— Shape: complete model versus four models with mandibles of different stag beetle species (figure 4e–h). In these models, we replaced the original mandibles with those of other species (L. adolphinae, N. obesus, Dorcus bucephalus, P. davidis; figure 4e–h). We scaled the mandibles to the same cuticle volume as the original mandibles, to arrive at a similar armature weight (i.e. assuming the same mass density of the mandible cuticle). We used armature weight as a scaling factor, rather than the frontal surface area, because the latter already partly eliminates the shape effect. All models in the shape comparison had the same body mass and location of the body centre of mass as the complete model, in order to exclude the weight effects.
We calculated the frontal area of the models, which is defined as the projected area of the beetle (at the body angle of attack used in the model) on a plane perpendicular to the flow direction. Combining this frontal area (A) with the drag force (FD, as calculated by CFD), we calculated the coefficient of drag (CD) of the entire beetle body using the formula of aerodynamic drag: FD = 1/2 CD ρν2A. To arrive at coefficients of drag of the mandibles only, we subtracted the drag force and frontal area of the models without mandibles from those of the model with mandibles.
The mandibles of the male C. metallifer individual that was used to build the model weighed 8.7% of its body mass (table 1). The complete sexual dimorphism of its head (taking the scaled weight of the female head into account), weighed 23% of its body mass (table 1). This is close to the value that we calculated for a population of 10 male individuals (18%; table 1). Male C. metallifer stag beetles fly at a body angle of 58° (±14° s.d.). We recorded a maximal velocity of 0.57 m s−1 (on average 0.46 m s−1 ± 0.13 m s−1 s.d.) which is relatively slow compared with similarly sized rhinoceros beetles [3,4]. We implemented these characteristics in a CFD model, to calculate the aerodynamic forces and moments on a steadily and horizontally flying male stag beetle body (without the wings).
The simulations show how air flows over and around the body and the mandibles (figures 5 and 6). The flow separates approximately when passing the widest points on both sides of the mandibles, causing a wide low-pressure region behind the mandibles. The same is true for the rest of the body. Hence, both are poorly streamlined and the beetle has a high CD (table 2). This causes a high (pressure) drag force on the beetle. Averaged over a wing beat cycle, the summed effects of the aerodynamic and gravitational forces on the body must be balanced by thrust and lift that are produced by the wings (figure 2).
3.1. Weaponry weight
The gravitational force (1.3 × 10−2 N) is several orders of magnitude larger than the aerodynamic forces on the stag beetle body (1.2 × 10−4 N drag and 6.1 × 10−5 N lift; table 2). This suggests that the weight of the heavy weaponry (mandibles + head) has a significant impact on the required membrane wing forces, hence also on the net work done by the flight muscles. As such, the membrane wing forces can be considered a direct measure for the net mechanical energy requirements per unit distance covered (net mechanical cost of flight). We evaluate this weight effect by comparing the results of the CFD simulations of the complete three-dimensional beetle model (figure 4a), a model that lacks the mandibles (without mandibles model; figure 4c), and a model of a non-dimorph conspecific (size-normalized non-dimorph model; figure 4d). The differences in the aerodynamic forces between the complete model and the two additional models can be attributed to the sexual dimorphism (mandibles and, in the case of the non-dimorph model, the accompanying muscles). The mandibles create some lift, but have a negative effect on drag. The same is true for the wide male head (table 2). The mandibles (11% of body weight) have a pronounced negative effect on the mechanical cost of flight (8.4% more costly than the without mandibles model). When we also take the effect of the enlarged head into account (in total 27% of body weight, compared with the non-dimorph model), the mechanical cost of the sexual dimorphism increases to 26%.
3.2. Weaponry size
To investigate the pure size effect of the mandibles, we compare three geometrically different models (figure 4a–c; complete, short mandibles and without mandibles), yet with an identical total body mass. The complete model experiences more lift, but also more drag, than the short mandibles model. We find a similar, but more pronounced, difference when using the without mandibles model that mimics infinitely small mandibles (table 2). However, the aerodynamic changes owing to the altered mandible size do not affect the net mechanical flight cost at all (less than 0.1% difference from the complete model; table 2).
3.3. Weaponry shape
To assess the influence of the shape or ‘ornamentation’ of the weaponry, we use four additional models in which the original mandibles are replaced by those of other stag beetle species (figure 4e–h). By scaling the mandibles in such a way that cuticle volume remains identical, we exclude weight differences. The mandible shape influences the aerodynamic forces. For example, the large and ornamented mandibles of Lamprima adolphinae cause a high frontal area and CD (table 2). Hence, the model experiences more drag, and, combined with a decreased lift force, this results in the lowest absolute lift-to-drag ratio magnitude of all models (0.37; table 2). However, it is clear that the shape effect is trivial compared with the weight effect of the weaponry and its musculature (less than 0.1% difference from the complete model; table 2).
4.1. Performance consequences of weaponry weight
In many species, elaborate ornaments and armaments incur no, or only subtle, costs . This is, for example, the case for flying with rhinoceros horns (1.5% of body weight), hummingbird tails (0.4% of body weight) and peacock trains (6.9% of body weight) [3,5,24]. Species for which substantial costs of ornaments were observed often compensate for these costs, for example, by increasing power generation with a larger wingspan (e.g. sunbird tails, 0.3% of body weight; earwig forceps) and/or flight muscles (e.g. stalk-eyed flies, 4.92% of body weight) [2,6,25]. Stag beetle mandibles are more extreme in terms of armament weight. Because fighting males must bite forcefully, they have not only robust mandibles but also a massive head to house the large bite muscles [11,26]. In Lucanus maculifemoratus  and C. metallifer stag beetles, this results in a head that weighs, respectively, 28% and 18% of the total male body weight. Moreover, nutritional constraints cause stag beetles with large mandibles to develop relatively small wings . Therefore, they do not compensate for the flight cost of their heavy armature by an increased wing span. Consequently, our CFD models highlight an evolutionary constraint: natural selection on flight cost hampers the sexual selection towards heavier armaments in male stag beetles. This is in addition to the negative effect of armature weight on the cost and stability of male terrestrial locomotion .
4.2. Evolution of weaponry size and shape
The size and shape of stag beetle weaponry determine the dynamics of the air flow over the flying beetle body. None of the models show signs of a delayed flow separation at any position along their mandibles, and all models have large coefficients of drag (CD >1; table 2). These CDs are comparable to those of an upright cyclist or a circular cylinder (with its axis normal to the flow at a Re of about 103) [18,20]. Hence, the mandibles are poorly streamlined. However, the aerodynamic forces are very small compared with the gravitational force. This relative difference depends on the flight velocity, but, even if the beetles flew a lot faster, the conclusions would still hold true: at a hypothetical forward velocity of 2 m s−1 (3.5 times as fast as in reality), the maximal effect of size and shape would only be 1%. Hence, the aerodynamic effects of the weaponry size and shape are swamped by the weight effect. Therefore, the strong sexual selection for large mandibles and bite muscles has (contra-intuitively) released the constraints on mandible sizes and shapes. Consequently, the absence of a selective pressure to enhance the lift properties and to streamline the mandibles probably provided the freedom to evolve the wide variety of elaborate mandible morphologies that is characteristic for the stag beetle family Lucanidae  (figure 4e–h). Also for other animal taxa (e.g. rhinoceros horns and avian tails), it was argued that the low cost of these armaments/ornaments causes their remarkable diversification owing to more labile evolution [3,24]. However, the origin of the low armament/ornament cost differs: the selective pressure on stag beetle mandible morphology is minimal because of the dominant effect of weight on the net flight cost, whereas for rhinoceros horns and avian tails this is the case because of a relative large wake from the body compared with the armament/ornament [3,24].
We used flow simulations to compare the effects of weaponry weight, size and shape on the net additional wing force that male stag beetles have to deliver in order to maintain a steady, horizontal flight. Even though these three characteristics are intuitively inter-related (e.g. larger structures are often heavier), they can change independently in case of structural changes (e.g. a larger structure with a thinner cuticle). The weight of the armament determines the gravitational force, and its size and shape determine the aerodynamic forces. We showed that, because of the robust mandibles and their hypertrophied muscles, stag beetle flight is heavily dependent on armature weight. Relative to this weight effect, the impact of armature size and shape is negligible. Such a decoupling between morphology (size, shape) and weight of armaments and ornaments may well be a more general characteristic of flying animals. This was for example suggested, but not explicitly tested, for rhinoceros beetles . For the stag beetle family, the low cost of the weaponry morphology probably made it liable to evolutionary changes, which led to its extraordinary diversity in mandible sizes and shapes.
This study was supported by a BOF grant (ID BOF UA 2011-445-a) of the Research Council of the University of Antwerp. The micro-CT reconstructions of the anterior body parts were provided by the Centre for X-Ray Tomography, Ghent University, Belgium. The SkyScan 1172 high-resolution micro-CT scanner, located at the VUB facilities, was funded by the Hercules Foundation (grant no. UABR/ 11/004).
The authors thank Ms Josie Meaney for proofreading of the manuscript.
- Received March 12, 2015.
- Accepted March 26, 2015.
- © 2015 The Author(s) Published by the Royal Society. All rights reserved.