## ABSTRACT

The distance a small insect moves through the air during a jump is limited by the launch velocity at take-off and by air resistance. The launch velocity is limited by the length of the jumping legs and the maximum power that the jump apparatus can provide for pushing against the ground. The effect of air resistance is determined by the insect mass-to-area ratio. Both limitations are highly dependent on body size, making high jumps a challenge for smaller insects. We studied both effects in the tiny Encyrtid wasp *Anagyrus pseudococci.* Males are smaller than females (mean body length 1.2 and 1.8 mm, respectively), but both sexes take off in a powerful jump. Using high-speed cameras, we analyzed the relationship between take-off kinematics and distance traveled through the air. We show that the velocity, acceleration and mass-specific power when leaving the ground places *A. pseudococci* among the most prominent jumpers of the insect world. However, the absolute distance moved through the air is modest compared with other jumping insects, as a result of air resistance acting on the small body. A biomechanical model suggests that air resistance reduces the jump distance of these insects by 49% compared with jumping in the absence of air resistance. The effect of air resistance is more pronounced in the smaller males, resulting in a segregation of the jumping performance between sexes. The limiting effect of air resistance is inversely proportional to body mass, seriously constraining jumping as a form of moving through the air in these and other small insects.

## INTRODUCTION

Jumping presents a ubiquitous mechanism for moving through the air without flying. It can serve the purpose of passing over obstacles (Fleagle, 1976; Kohlsdorf and Navas, 2007), escaping from predators (Burrows and Dorosenko, 2014; Suter and Gruenwald, 2000), capturing prey (Jackson and Pollard, 1996), righting the body (Evans, 1972; Frantsevich, 2004; Ribak and Weihs, 2011) and even ‘hitchhiking’ on a larger organism (phoresy) as a means of dispersal (Fatouros and Huigens, 2012; Houck and O'Connor, 1991). In addition, many flying animals jump into the air as part of the flight initiation (take-off) process (Burrows and Dorosenko, 2017; Card and Dickinson, 2008; Earls, 2000; Heppner and Anderson, 1985; Kutsch and Fuchs, 2000; Ribak et al., 2016). The energy required to move the body in the air during a jump is generated at take-off as the jumper pushes against the ground. After leaving the ground, the jumper must work against gravity and air resistance, but the energy invested in jumping cannot be replenished. Thus, the height and distance jumped are limited by the energy invested at take-off.

Smaller jumpers encounter a problem in providing the high mechanical power needed for take-off using muscle contraction (Bennet-Clark, 1977). The smaller the jumper, the shorter its jumping appendages, providing a shorter distance within which to accelerate the center of mass as the legs push against the ground. Thus, to achieve the same take-off velocity as a jumper with longer legs, the smaller jumper needs to accelerate its body at a higher rate, requiring greater force generation within a shorter time interval. This results in an increase in the power demand of the muscles powering the jump. The power needed by some small arthropods to jump can vastly exceed the capacity of normal skeletal muscles (reviewed by Alexander, 1995). The problem is circumvented in those small insects that have evolved elastic energy storage and release mechanisms in order to catapult their body into the air. Various insects, such as froghoppers (Burrows, 2006), snow fleas (Burrows, 2011), fleas (Bennet-Clark and Lucey, 1967; Sutton and Burrows, 2011), flea beetles (Brackenbury and Wang, 1995; Nadein and Betz, 2016) and click-beetles (Evans, 1973), among others, store the work of slow muscle contraction in elastic materials within the body. These serve as biological springs that enable release of the stored energy within a very short time interval (Gronenberg, 1996; Patek et al., 2011). The release of elastic energy is not time constrained, unlike muscle contraction, and thus enables provision of the high power needed to launch an insect into the air to a height of many body lengths (Bennet-Clark, 1977; Pennycuick, 1992).

Once in the air, air resistance may be negligible in larger animals, and the jump performance (distance and maximum height) can be predicted from the launch angle and speed at take-off using simple ballistic projectile equations. However, for small creatures, the restrictive effect of air resistance on the aerial trajectory becomes increasingly more significant as the body size of the jumper decreases (Bennet-Clark, 1977, 1979). When the body size of the jumper is scaled down, the ratio between surface area and mass increases. The kinetic energy of the jumper is proportional to body mass and the square power of velocity, while air resistance is proportional to area and the square power of velocity. Thus, as the area increases relative to mass, the influence of air resistance over the jump trajectory increases. In addition, the flow of air over the body depends on the non-dimensional Reynolds number (Vogel, 1994). In very small jumpers, the lower Reynolds number (insect diameter times speed divided by the kinematic viscosity of air; see Eqn 9 below) results in a higher drag coefficient for similarly shaped animals differing in size. This again makes air resistance more influential (Vogel, 2005).

Some of the tiniest extant insects belong to the group of parasitoid wasps. This superfamily is one of the most diverse and largest (over 200,000 species described to date) insect groups, comprising about 75% of all known Hymenopteran species (Pennacchio and Strand, 2006). Parasitoid wasps have an important ecological and agricultural role in biological control over host insect populations (DeBach and Rosen, 1991). With a body length of less than 1 mm in some species, small parasitoid wasps provide an interesting research model with which to examine the constraints associated with body miniaturization. Jumps are routinely executed as the flight initiation mechanism of several parasitoid wasps (Burrows and Dorosenko, 2017), including the *Anagyrus pseudococci* species (Fig. 1) studied here.

*Anagyrus pseudococci* (Encyrtidae) is a small (body length <2.5 mm; Table 1) parasitoid wasp. Although not the smallest species, these wasps are members of the superfamily Chalcidoidea, many of which are known to be agile jumpers. Their typical tibial spur (extending from the tibial-tarsus joint) and enlarged musculature in the mesothorax were suggested to be anatomical adaptations for jumping (Riek, 1970). The sexes vary in body size (females are larger) as well as morphology (Rosen and Rossler, 1966; Fig. 1). Gibson (1986) studied the mesothoracic skeletomusculature of the Eupelmidae (another family within the superfamily) and found that the sexes differ in the muscle and skeletal structures used for jumping. Despite the economic importance of *A. pseudococci* in the biological control of mealybug pests, little is known about the take-off kinematics or energetics of the jump in general, and about inter-sex variation in jumping performance in particular.

Here, we used high-speed cameras to film male and female *A. pseudococci* while jumping, and analyzed the jumping performance and the effect of air resistance on the jump and on the inter-sex differences in jumping performance. We hypothesized that because of their small size, the jumping of these wasps would be substantially restricted by air resistance, with the smaller males either jumping to lower heights than the larger females or compensating for the increased effect of air resistance. To test this hypothesis and determine which of the two options is correct, we compared the theoretically expected jumping height in the absence of air resistance (calculated based on the take-off kinematics) with the actual observed distance moved through the air.

## MATERIALS AND METHODS

### Insects

Pupal stages of *Anagyrus pseudococci* (Girault 1915) were obtained from a commercial breeder (Bio-Bee Biological Systems, Sde Eliyaho, Israel). They were divided into individual plastic containers, each containing about 30 pupae, and kept at room temperature (25°C). Wasps started emerging after 4 days and were tested on the day of emergence.

### Filming

To film the jumps, a vertical glass tube (7 mm diameter) was inserted into the upper opening of the plastic containers, enabling the imagines to climb up the tube to exit the container. The upper tip of the vertical glass tube was covered with a horizontal microscope coverglass, leaving only a small opening for the parasitoids to exit, one at a time. After walking up the glass tube and exiting through the opening, the wasps typically paused for a moment and then jumped from the horizontal coverglass. The jumps were filmed with two orthogonal high-speed cameras (FASTCAM SA3 model 120K-M2, Photron, Japan), fitted with Nikon 80 mm lenses, providing views of the jump from two angles (Fig. 2). A third high-speed camera was placed 5 cm away from the jumping point, providing a close-up view to identify the sex of the jumping wasps (easily distinguishable because of the prominent sexual dimorphism; Fig. 1C). We carried out two alternative filming approaches. First, the two cameras were positioned 1 m from the jumping insects and filmed at 3000 frames s^{−1}, providing a wide enough field of view to see the entire jump trajectory. Second, the cameras were moved 50 cm towards the insects, filming at 10,000 frames s^{−1}, which allowed us to zoom-in on the take-off phase of the jump. In both approaches, combining the views from the two orthogonal cameras allowed us to extract the kinematics of the jump in 3D. Filming was carried out against a background illuminated with infrared (IR, 850 nm) floodlights, while the room in which the experiments took place was illuminated by visible light from fluorescent light bulbs on the ceiling. The two high-speed cameras were spatially calibrated by filming an object with known dimensions that was visible to both cameras (Hedrick, 2008). In the first approach, this resulted in a calibrated volume of a rectangular cuboid with dimensions 7.6×7.6×16.5 cm (L×W×H). In the second approach, the smaller field of view (3×3×3 cm) required an alternative method and we used the orthogonal positioning of the cameras to calculate motion of the insect in the *x*-, *y*- and *z*-axes directly from the two views after correcting for scale in each camera.

### Morphometric measurements

*A. pseudococci*are sexually dimorphic (Fig. 1). The body length of 12 males and 11 females was measured from images (resolution: 2048×1536 pixels) taken with a Leica EZ4 HD dissecting microscope, including a 1 mm grid in the background for scale correction. On the resulting images, we measured the distance between the tip of the head and the tip of the closed wings (

*l*

_{HT}) and between the tip of the head and the tip of the abdomen (

*l*

_{HA}). We also measured the maximum width of the body (

*l*

_{W}), which is the maximal width of the thorax along the lateral axis. The insect body volume (

*V*) can be estimated through these measurements by assuming an ellipsoid body shape: As the small size of the insects restricted direct measurement of body mass, we resorted to an estimation using the body volume (Eqn 1) and an average body density. Body density (ρ) of different insects can vary between 0.4 to 1.050 mg mm

^{−3}(Bennet-Clark and Alder, 1979). We assumed an intermediate value of body density of ρ=0.7 mg mm

^{−3}for

*A. pseudococci*. The body mass of the insect (

*m*) was calculated as body density times the body volume of our insect: The error associated with this indirect estimation of body mass is evaluated below (see Discussion)

### Kinematics of the jump

In each film frame, we used the Matlab code DLTdv5 (Hedrick, 2008) to track and obtain the 3D position of the center of the insect’s body. The velocity at leaving the ground (*v*_{0}) was found from the displacement of the body during the first 3.3 ms (10 movie frames) after leaving the ground.

where Δ*x* and Δ*y* are the horizontal translations along the *x* and *y* horizontal axes and *t* is the time elapsed (3.3 ms).

*z*-axis was vertical with the positive end pointing up. Consequently, the vertical component of

*v*

_{0}was found from the equation of linear translation with constant (gravitational) acceleration: where Δ

*z*is the vertical translation made in

*t*=3.3 ms and

**is the gravitational acceleration (9.8 m s**

*g*^{−2}).

#### Estimating the effect of air resistance

*h*) is determined by the take-off speed (

*v*

_{0}) and take-off angle (θ) according to standard ballistic equations (Kittel et al., 1973): where the subscript ‘vac’ denotes that this is the predicted jump height in a vacuum. In practice, some of the jump energy is wasted in overcoming air resistance as the insect moves through the air. We estimated the effect of drag on the aerial trajectory using basic equations for small particles. As the insects moved through the air, their body rotated about all three anatomical axes. Consequently, the frontal area projected onto the flow changed instantaneously, making precise assessment impractical. Instead, we sought a mean value. The planform area of each sex was calculated as an ellipse, with the major and minor radii being half the length and width of an average male and female. Thus, the planform area of the ellipsoid (

*A*) is: For our jumping insects, the Reynolds number (

*Re*) was defined as: where ν is the kinematic viscosity of air at 20°C (ν=15×10

^{−6}m

^{2}s

^{−1}) and

*v*is the velocity of the body.

*D*) is defined as: where ρ

_{a}is the density of dry air at 20°C (ρ=1.204 kg m

^{−3}) and

*C*

_{D}is the drag coefficient. For the latter, we used the drag coefficient for a spherical particle as a function of the Reynolds number (found in Eqn 9) as suggested by Vogel (2005): For non-spherical objects at low Reynolds numbers (

*Re*∼150),

*C*

_{D}will somewhat deviate from the values predicted by Eqn 10 depending on the shape and orientation of the object. For ellipsoids, the actual value depends on the proportions of the major and minor axes (Breach, 1961). Corrections for various prolate and oblate ellipsoids exist, but these are exact only for

*Re*<5 (Happel and Brenner, 1983). Furthermore, as the body is rotating in the air (Movie 1),

*C*

_{D}changes with the orientation of the long axis relative to the direction of movement. Here, we were not interested in the value of

*C*

_{D}per se but, rather, we sought to evaluate the effect of air resistance on the jumper. Hence, we used Eqn 11 with no correction for non-spherical shape, and the effect of errors in estimating

*C*

_{D}is evaluated below (see Discussion).

Using the take-off angle and speed (Eqns 5 and 6) and the calculated drag coefficient (Eqns 9 and 11), we simulated the trajectory of each jump by calculating the instantaneous height and horizontal distance relative to the take-off point, as well as the instantaneous direction (angle) of air-borne movement. The equations required for the simulation (see Appendix) were embedded in a custom-written Matlab code to predict the jump trajectory in air while accounting for the effect of air resistance. The predicted trajectories were compared with those observed by the high-speed cameras. A good agreement between the observed jump trajectory and that predicted by our simulation indicates that the estimated values of *C*_{D} (Eqn 11) are close enough to the real value. If the predicted jump trajectory is an overestimate or underestimate of the observed trajectory, then the estimated value of *C*_{D} is too low or too high, respectively.

In addition, for each jump we found the take-off speed and angle from Eqns 3–6 and calculated the maximal jump height (in the absence of air resistance) from Eqn 7. We then measured the actual jump height observed in the high-speed films, with the difference between the observed and expected jump height being due to the effect of air resistance on the insect.

#### Jump energetics

*t*is the duration of pushing against the ground. The mean acceleration on the ground (

*a*) was found from the take-off speed (

*v*

_{0}) and

*t*as: The power divided by our estimate of body mass gave the mass-specific power (

*p**), allowing us to compare the jump energetics between the larger females and smaller males.

### Statistics

Statistical tests were performed to compare the jumping performance between males and females and between the predicted and observed jump heights. We used *t*-tests to compare jump kinematics parameters between males and females. The drag coefficients, calculated as the mean for each jump, did not have a normal distribution and therefore the difference in drag coefficient between the sexes was evaluated using a Mann–Whitney *U*-test. The jump heights of males and females were also compared using ANCOVA with the take-off speed as a covariate (see Results). All tests were performed using Statistica (StatSoft Inc.) Unless noted otherwise, all results are reported as means±s.d.

## RESULTS

Males are smaller than females and consequently have lower *Re* and higher *C*_{D} (Table 1). The mean jump kinematics of the two sexes in addition to the mechanical power (Eqn 12), calculated based on the observed kinematics, are given in Table 2.

While both sexes (females: *n*=44; males: *n*=26) left the ground at similar take-off angles (females at 67.3±10.1 deg and males at 63.1±10.4 deg, *t*-test, *P*=0.10, Table 2), the mean take-off speed of females (2.37±0.41 m s^{−1}) was significantly (*t*-test, *P*=0.003) greater (by 18%) than that of males (2.02±0.52 m s^{−1}). Consequently, mean female jump height was 1.54-fold the mean jump height of males (12.2±3.9 and 7.9±3.1 cm, respectively; *t*-test, *P*<0.001).

Sequences of images taken by the high-speed cameras during take-off (e.g. Fig. 3) revealed that the insects pushed against the ground with their middle legs, which were the last legs to leave the ground. The entire period of pushing lasted less than 1 ms (Table 2). In the air, the wings opened and started flapping only after the insect had reached the peak of the ballistic flight trajectory (maximal height). The insects left the ground rotating about all axes (Movie 1).

ANCOVA on observed jump height with the vertical take-off speed as a covariate revealed that jump height corrected for vertical take-off speed was still higher in females than in males (ANCOVA, *P*<0.001). A significant interaction between sex and the vertical take-off speed (ANCOVA, *P*<0.001) revealed that the inter-sex difference in jump height was larger at higher take-off speeds (Fig. 4).

The observed jump trajectories (Fig. 5A) show a difference in jumping performance between males and females. Fig. 5B shows two of the jumps (one jump by a male and one by a female) in three ways: first, as tracked directly from the high-speed video; second, as simulated for the case of no air resistance (see also agreement with Eqn 7); and third, simulated as described in the Appendix for the inclusion of air resistance. It is evident that the simulations in air predict the actual jump height (peak of the trajectory) quite well (mean±s.d. error was: 7.5±5.5%), and that air resistance leads to jump heights that are half of the mean predicted jump heights in a vacuum (25.1±10 cm for females and 17.1±8.3 for males).

We used the velocity at take-off and the duration of pushing against the ground, as measured directly in the films, to determine the mechanical power output required for the jump (Eqn 12). The mean jump power and body mass-specific power were significantly greater in females than in males (*t*-tests, *P*<0.003 in both cases, females: *p*=0.47±0.15 mW and *p**=3.1±1 kW kg^{−1}; males: *p*=0.16±0.07 mW and *p**=2.3±1 kW kg^{−1}).

## DISCUSSION

Bennet-Clark and Alder (1979) defined jump efficiency as the ratio of actual (observed) jump height to predicted jump height in a vacuum. They showed that this efficiency is mostly affected by the ratio between surface area and mass of the jumper. According to our measurements, jumping efficiency of *A. pseudococci* was 0.51± 0.07 and 0.51±0.13 for females and males, respectively. The area-to-mass ratio of our insects was 7.65 and 5.15 for males and females, respectively (based on the data in Table 1). These ratios predict an efficiency of ∼0.6 for *A. pseudococci* according to Bennet-Clark and Alder (1979, see their fig. 4). The reasons for the lower efficiencies found here are likely 2-fold. (1) the calculation of efficiency in the previous work was based on a constant drag coefficient (*C*_{D}=1), whereas air resistance was higher in our experiments, considering the change in *C*_{D} with Reynolds number. The mean values of the drag coefficient per jump were 27% higher in males (2.15±0.31 and 1.69±0.14 for males and females, respectively, Mann–Whitney *U*-test, *P*<0.001). Thus, the higher air resistance for the same speed led to poorer jump efficiency in our insects. (2) In addition, the jump efficiency found by Bennet-Clark and Alder (1979) was calculated for vertical jumps, in which all the take-off speed is vertical speed. In contrast, our wasps had jump angles of ∼60 deg on average (Table 2), implying that the vertical take-off speed is only about 86% of the total (3D) take-off speed. The fact that the total take-off speed is higher implies higher drag on the body for the same vertical speed (Eqn 10), resulting in greater attenuation of the jump energy and lower values of jump efficiency. Hence, our study demonstrates that the problem of air resistance in small insects that do not jump vertically is more severe than suggested in the seminal study of Bennet-Clark and Alder (1979).

Male and female *A. pseudococci* are sexually dimorphic, with male body length approximately 56.5% shorter than that of females and calculated male body mass 2.26-fold smaller (Table 1). This dimorphism has implications for the jumping performance of males. First, males have shorter legs (Table 1) and thus need to launch their body at higher accelerations in order to reach the same take-off speed as that of females. Second, once in the air, the effect of drag is more pronounced in males because of their higher area-to-mass ratio. This dimorphism affects the jump height in males and females even when they leave the ground at the same speed and take-off angle (Fig. 5C). While the predicted jump height in a vacuum is the same for the two insects, the simulated jump height in the air is lower for males simply because of the increased effect of air resistance on the smaller body size. In practice, however, the inter-sex difference in jump height is larger as a result of females leaving the ground at higher speeds compared with males (Table 2). This also resulted in females demonstrating a higher power demand when jumping, while the mass-specific power of the larger females was also higher. Thus, male *A. pseudococci* do not seem to compensate for their smaller size in order to reach a similar jump height to females. We examined whether this is due to the higher drag coefficient, making higher jumps less efficient for males. Even when exerting the same mass-specific power, females reach a higher jump height, with the increase in height gain compared with males increasing with mass-specific power (Fig. 6A). In order to reach similar jump heights to females, males need to increase the mass-specific power exerted during a jump (which is due to the larger effect of air resistance) (Fig. 6B). There is a consistent increase of male-to-female power ratio as jump height increases. Hence, the difference in the non-linear increase in mass-specific power needed for higher jumps sets a physiological–biomechanical barrier between males and females. Correspondingly, ∼75% of the male jumps were to heights lower than 10 cm, while 75% of the female jumps were to heights higher than 10 cm, where the inter-sex ratio of the mass-specific power is 1.4. Hence, the increase in power dictated by the small body size and higher drag coefficient of males results in a functional segregation of jumping performance between the dimorphic males and females. *Anagyrus* is a member of the Chalcidoidea superfamily, in which sexual dimorphism is common in many species. Gibson (1986) reported that male and female eupelminid wasps (Chalcidoidea) utilize different anatomical structures for jumping. If the same is true for *A. pseudococci*, this could indicate a divergence dictated by the biomechanical constraints on jumping in smaller males.

We used basic ballistics equations to predict the trajectory of the insects in the air and in a vacuum. These simplified equations do not account for body rotation and the fact that the body is not a sphere. Nevertheless, we found a reasonable agreement between the observed and predicted jump trajectories (Figs 5B and 7). However, the simulations matched the observed jumps well only up to the maximum height of the jump. This is because once the insect reached the peak of the trajectory it opened its wings and flapped them, transitioning into active flight. As a result, its motion deviated from projectile dynamics from this point on. Because uncertainties in both body mass and drag coefficient estimates affect the jump outcome (trajectory) predicted by the model, we tested the sensitivity of our predicted jump height to variation in these values. This was done by recalculating simulated jump height, with the body density changed to 1 and 0.4 mg mm^{−3}, i.e. the maximum and minimum body densities reported for insects by Bennet-Clark and Alder (1979), and comparing the outcome (predicted jump height) with the actual jump height measured from the films, using linear regressions (Fig. 7A,B). For both males and females, all linear regressions were significant (*P*<0.001 in all cases) and the *R*^{2} value was higher than 0.92 in all cases. Our original estimate of body density for females gave a slope of 1.03 while the higher and lower mass estimates gave a slope of 0.84 and 1.46, respectively. Thus, the assumed body density for females is fairly accurate. In males, the slopes for our estimate and the higher and lower body density were 1.1, 0.9 and 1.62, respectively. Thus, we may have slightly underestimated male body density. Similarly, we recalculated jump height (Fig. 7C,D) with the drag coefficient ±33% of the estimate found using Eqn 10. For females and males, all linear regressions were significant (*P*<0.001 in all cases) and the *R*^{2} was higher than 0.92 for females and males. The slopes for our estimate and the higher and lower drag coefficient for females were 1.03, 1.22 and 0.78, respectively. For males the slopes were 1.1, 1.32 and 0.83, respectively, suggesting a slight overestimation. This sensitivity analysis demonstrates that while the exact values predicted by the model depend on the estimates used for body mass and *C*_{D}, the trend in inter-sex difference resulting from the greater effect of air resistance on the smaller males remains consistent and pronounced.

The short duration of pushing against the ground prior to take-off (<1 ms) and exceptionally high mass-specific power exerted by *A. pseudococci* during the jump (Table 2) suggest the presence of a power-amplification mechanism. Such mechanisms for jumping are common in many small insects (Table 2). Elastic energy storage has been suggested for jumping wasps based on musculoskeletal anatomy (Gibson, 1986), but to the best of our knowledge our report is the first quantitative measurement of jumping performance for a wasp using legged power amplification. Take-off jumps have been reported in hymenopteran species before (Burrows and Dorosenko, 2017) but the reported jumps were moderate compared with those of *A. pseudococci*. One such example is that of *Pteromalus puparum* (Burrows and Dorosenko, 2017), which has a similar body length (2.7±0.01 mm) and mass (1.0±0.10 mg) to *A. pseudococci*. The mass-specific power exerted by these wasps during a jump was only 60 W kg^{−1}, i.e. roughly 50-fold lower than the mass-specific power exerted by our female *A. pseudococci*.

Compared with other insects using power amplification for jumping, both male and female *A. pseudococci* achieve modest jump heights. Their acceleration to the take-off speed and their jump height relative to body length, however, are among the highest reported for jumping insects (Table 2). The froghopper *Philaenus spumarius* is considered one of the most prominent jumpers in the insect world (Burrows, 2003). Table 2 reveals that our wasps require similar values of mass-specific power to those of the froghopper in order to reach similar accelerations and heights relative to body length. Remarkably, the mean body mass and length of the froghopper is 180-fold and 5.2-fold, respectively, higher than the mean body mass and length of male *A. pseudococci.* Thus, while the two insects achieve a similar take-off performance, the air resistance is much more substantial for the wasps, resulting in a decrease in their jumping performance. For example, if we use the data given in Table 2 and attempt to predict the jump efficiency of the larger froghopper using the take-off speed of the highest jump and the mean jump angle and substituting into Eqn 7, we obtain a value of ∼90% jump efficiency for the froghopper. The difference from the 51% efficiency of *A. pseudococci* is entirely due to the lower *A*/*m* ratio of the froghopper. The acceleration of *A. pseudococci* at take-off (∼260 times gravity in females) is much lower than that in click-beetles (which do not use their legs for jumping) but only ∼10% lower than the acceleration of the froghoppers (Table 2). Thus, despite (or perhaps because of) its smaller size, *A. pseudococci* is a prominent jumper that nevertheless demonstrates the constraints of moving through air on a small creature.

### APPENDIX

## Acknowledgements

We thank Shimon Steinberg from Bio-Bee Biological Systems for providing the insects used in this study. Eyal Dafni provided invaluable assistance with the experiments. The late Prof. Dan Gerling contributed many useful insights that led to the conceptualization of the study.

## Footnotes

**Author contributions**

Conceptualization: T.U., G.R.; Methodology: T.U., G.R.; Software: T.U., G.R.; Validation: T.U., G.R.; Formal analysis: T.U.; Investigation: T.U., G.R.; Resources: G.R.; Data curation: T.U., G.R.; Writing - original draft: T.U., G.R.; Writing - review & editing: T.U., G.R.; Visualization: T.U.; Supervision: G.R.; Project administration: G.R.; Funding acquisition: G.R.

**Funding**

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

**Data availability**

Data used in the study and code to simulate jumps are available for download from: https://galribak.weebly.com/codes--multimedia.html.

## References

**Competing interests**

The authors declare no competing or financial interests.