RESEARCH ARTICLE

The effect of air resistance on the jump performance of a smallparasitoid wasp Anagyrus pseudococci (Encyrtidae)Tomer Urca1 and Gal Ribak12

ABSTRACTThe distance a small insect moves through the air during a jump islimited by the launch velocity at take-off and by air resistance Thelaunch velocity is limited by the length of the jumping legs and themaximum power that the jump apparatus can provide for pushingagainst the ground The effect of air resistance is determined by theinsect mass-to-area ratio Both limitations are highly dependent onbody size making high jumps a challenge for smaller insects Westudied both effects in the tiny Encyrtid wasp Anagyrus pseudococciMales are smaller than females (mean body length 12 and 18 mmrespectively) but both sexes take off in a powerful jump Using high-speed cameras we analyzed the relationship between take-offkinematics and distance traveled through the air We show that thevelocity acceleration and mass-specific power when leaving theground places A pseudococci among the most prominent jumpers ofthe insect world However the absolute distance moved through theair is modest compared with other jumping insects as a result of airresistance acting on the small body A biomechanical model suggeststhat air resistance reduces the jump distance of these insects by 49compared with jumping in the absence of air resistance The effect ofair resistance is more pronounced in the smaller males resulting in asegregation of the jumping performance between sexes The limitingeffect of air resistance is inversely proportional to body massseriously constraining jumping as a form of moving through the air inthese and other small insects

KEY WORDS Sexual dimorphism Locomotion ParasitoidsMiniaturization Drag Power Ballistics Elastic energy storage

INTRODUCTIONJumping presents a ubiquitous mechanism for moving through theair without flying It can serve the purpose of passing over obstacles(Fleagle 1976 Kohlsdorf and Navas 2007) escaping frompredators (Burrows and Dorosenko 2014 Suter and Gruenwald2000) capturing prey (Jackson and Pollard 1996) righting thebody (Evans 1972 Frantsevich 2004 Ribak andWeihs 2011) andeven lsquohitchhikingrsquo on a larger organism (phoresy) as a means ofdispersal (Fatouros and Huigens 2012 Houck and OrsquoConnor1991) In addition many flying animals jump into the air as part ofthe flight initiation (take-off ) process (Burrows and Dorosenko2017 Card and Dickinson 2008 Earls 2000 Heppner andAnderson 1985 Kutsch and Fuchs 2000 Ribak et al 2016) The

energy required to move the body in the air during a jump isgenerated at take-off as the jumper pushes against the ground Afterleaving the ground the jumper must work against gravity and airresistance but the energy invested in jumping cannot bereplenished Thus the height and distance jumped are limited bythe energy invested at take-off

Smaller jumpers encounter a problem in providing the highmechanical power needed for take-off using muscle contraction(Bennet-Clark 1977) The smaller the jumper the shorter itsjumping appendages providing a shorter distance within which toaccelerate the center of mass as the legs push against the groundThus to achieve the same take-off velocity as a jumper with longerlegs the smaller jumper needs to accelerate its body at a higher raterequiring greater force generation within a shorter time intervalThis results in an increase in the power demand of the musclespowering the jump The power needed by some small arthropods tojump can vastly exceed the capacity of normal skeletal muscles(reviewed by Alexander 1995) The problem is circumvented inthose small insects that have evolved elastic energy storage andrelease mechanisms in order to catapult their body into the airVarious insects such as froghoppers (Burrows 2006) snow fleas(Burrows 2011) fleas (Bennet-Clark and Lucey 1967 Sutton andBurrows 2011) flea beetles (Brackenbury andWang 1995 Nadeinand Betz 2016) and click-beetles (Evans 1973) among othersstore the work of slow muscle contraction in elastic materials withinthe body These serve as biological springs that enable release ofthe stored energy within a very short time interval (Gronenberg1996 Patek et al 2011) The release of elastic energy is nottime constrained unlike muscle contraction and thus enablesprovision of the high power needed to launch an insect into the airto a height of many body lengths (Bennet-Clark 1977 Pennycuick1992)

Once in the air air resistance may be negligible in larger animalsand the jump performance (distance and maximum height) can bepredicted from the launch angle and speed at take-off using simpleballistic projectile equations However for small creatures therestrictive effect of air resistance on the aerial trajectory becomesincreasingly more significant as the body size of the jumperdecreases (Bennet-Clark 1977 1979) When the body size of thejumper is scaled down the ratio between surface area and massincreases The kinetic energy of the jumper is proportional to bodymass and the square power of velocity while air resistance isproportional to area and the square power of velocity Thus as thearea increases relative to mass the influence of air resistance overthe jump trajectory increases In addition the flow of air over thebody depends on the non-dimensional Reynolds number (Vogel1994) In very small jumpers the lower Reynolds number (insectdiameter times speed divided by the kinematic viscosity of air seeEqn 9 below) results in a higher drag coefficient for similarly shapedanimals differing in size This again makes air resistance moreinfluential (Vogel 2005)Received 17 January 2018 Accepted 15 February 2018

1School of Zoology Faculty of Life Sciences Tel Aviv University Tel Aviv 6997801Israel 2The Steinhardt Museum of Natural History Israel National Center forBiodiversity Studies Tel Aviv 6997801 Israel

Author for correspondence (gribakposttauacil)

GR 0000-0002-6267-5471

1

copy 2018 Published by The Company of Biologists Ltd | Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

Some of the tiniest extant insects belong to the group of parasitoidwasps This superfamily is one of the most diverse and largest (over200000 species described to date) insect groups comprising about75 of all known Hymenopteran species (Pennacchio and Strand2006) Parasitoid wasps have an important ecological and agriculturalrole in biological control over host insect populations (DeBach andRosen 1991) With a body length of less than 1 mm in some speciessmall parasitoid wasps provide an interesting research model withwhich to examine the constraints associated with bodyminiaturizationJumps are routinely executed as the flight initiation mechanism ofseveral parasitoid wasps (Burrows and Dorosenko 2017) includingthe Anagyrus pseudococci species (Fig 1) studied hereAnagyrus pseudococci (Encyrtidae) is a small (body length

lt25 mm Table 1) parasitoid wasp Although not the smallestspecies these wasps are members of the superfamily Chalcidoideamany of which are known to be agile jumpers Their typical tibialspur (extending from the tibial-tarsus joint) and enlargedmusculature in the mesothorax were suggested to be anatomicaladaptations for jumping (Riek 1970) The sexes vary in body size(females are larger) as well as morphology (Rosen and Rossler1966 Fig 1) Gibson (1986) studied the mesothoracicskeletomusculature of the Eupelmidae (another family within thesuperfamily) and found that the sexes differ in the muscle andskeletal structures used for jumping Despite the economicimportance of A pseudococci in the biological control ofmealybug pests little is known about the take-off kinematics orenergetics of the jump in general and about inter-sex variation injumping performance in particularHere we used high-speed cameras to film male and female

A pseudococci while jumping and analyzed the jumpingperformance and the effect of air resistance on the jump and onthe inter-sex differences in jumping performance We hypothesizedthat because of their small size the jumping of thesewasps would besubstantially restricted by air resistance with the smaller maleseither jumping to lower heights than the larger females orcompensating for the increased effect of air resistance To test thishypothesis and determine which of the two options is correct wecompared the theoretically expected jumping height in the absenceof air resistance (calculated based on the take-off kinematics) withthe actual observed distance moved through the air

MATERIALS AND METHODSInsectsPupal stages of Anagyrus pseudococci (Girault 1915) were obtainedfrom a commercial breeder (Bio-Bee Biological Systems SdeEliyaho Israel) They were divided into individual plasticcontainers each containing about 30 pupae and kept at roomtemperature (25degC) Wasps started emerging after 4 days and weretested on the day of emergence

FilmingTo film the jumps a vertical glass tube (7 mm diameter) wasinserted into the upper opening of the plastic containers enabling

the imagines to climb up the tube to exit the container The upper tipof the vertical glass tube was covered with a horizontal microscopecoverglass leaving only a small opening for the parasitoids to exitone at a time After walking up the glass tube and exiting through theopening the wasps typically paused for a moment and then jumpedfrom the horizontal coverglass The jumps were filmed with twoorthogonal high-speed cameras (FASTCAM SA3 model 120K-M2Photron Japan) fitted with Nikon 80 mm lenses providing viewsof the jump from two angles (Fig 2) A third high-speed camerawasplaced 5 cm away from the jumping point providing a close-upview to identify the sex of the jumping wasps (easily distinguishablebecause of the prominent sexual dimorphism Fig 1C) We carriedout two alternative filming approaches First the two cameraswere positioned 1 m from the jumping insects and filmed at3000 frames sminus1 providing a wide enough field of view to see the

Table 1 Average morphometric measurements and calculated values of male and female A pseudococci

Leg length (mm) lHA (mm) lHT (mm) lW (mm) m (μg) Re CD

Male 128plusmn0074 (11) 12plusmn017 (12) 162plusmn012 (12) 04plusmn004 (12) 67 109 215plusmn031 (26)Female 163plusmn01 (9) 18plusmn02 (11) 21plusmn015 (11) 047plusmn0063 (11) 152 158 169plusmn014 (44)

Data are meansplusmnsd with N in parentheseslHA length from head tip to tip of abdomen lHT length from head tip to wing tip lW body width m calculated body mass (Eqns 1 and 2) Re Reynolds number(Eqn 8) calculated based onmean velocity at take-off andmean (planform area)12CD drag coefficient (Eqn 10) calculated asmeanCD for each simulated jump

A

B

C

Fig 1 Sexual dimorphism in Anagyrus pseudococci (A) Female(B) Male (C) Both sexes as seen by the third (zoomed-in) high-speed camera(see Materials and methods and Fig 2) The female is on the left and the maleis on the right

2

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entire jump trajectory Second the cameras were moved 50 cmtowards the insects filming at 10000 frames sminus1 which allowed usto zoom-in on the take-off phase of the jump In both approachescombining the views from the two orthogonal cameras allowed us toextract the kinematics of the jump in 3D Filming was carried outagainst a background illuminated with infrared (IR 850 nm)floodlights while the room in which the experiments took placewas illuminated by visible light from fluorescent light bulbs on theceiling The two high-speed cameras were spatially calibrated byfilming an object with known dimensions that was visible to bothcameras (Hedrick 2008) In the first approach this resulted in acalibrated volume of a rectangular cuboid with dimensions76times76times165 cm (LtimesWtimesH) In the second approach the smallerfield of view (3times3times3 cm) required an alternative method and weused the orthogonal positioning of the cameras to calculate motionof the insect in the x- y- and z-axes directly from the two views aftercorrecting for scale in each camera

Morphometric measurementsMale and female A pseudococci are sexually dimorphic (Fig 1)The body length of 12 males and 11 females was measured fromimages (resolution 2048times1536 pixels) taken with a Leica EZ4 HDdissecting microscope including a 1 mm grid in the background forscale correction On the resulting images we measured the distancebetween the tip of the head and the tip of the closed wings (lHT) andbetween the tip of the head and the tip of the abdomen (lHA) Wealso measured the maximum width of the body (lW) which is themaximal width of the thorax along the lateral axis The insect bodyvolume (V) can be estimated through these measurements byassuming an ellipsoid body shape

V frac14 4

3p lHA

2 lW4

2

eth1THORN

As the small size of the insects restricted direct measurement ofbody mass we resorted to an estimation using the body volume(Eqn 1) and an average body density Body density (ρ) of differentinsects can vary between 04 to 1050 mg mmminus3 (Bennet-Clark andAlder 1979) We assumed an intermediate value of body density of

ρ=07 mg mmminus3 for A pseudococci The body mass of the insect(m) was calculated as body density times the body volume of ourinsect

m frac14 r V eth2THORNThe error associated with this indirect estimation of body mass isevaluated below (see Discussion)

Kinematics of the jumpIn each film frame we used the Matlab code DLTdv5 (Hedrick2008) to track and obtain the 3D position of the center of the insectrsquosbody The velocity at leaving the ground (v0) was found from thedisplacement of the body during the first 33 ms (10 movie frames)after leaving the ground

The horizontal component of v0 was calculated from

v0ethhTHORN frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2 thorn Dy2

pt

eth3THORN

where Δx and Δy are the horizontal translations along the x and yhorizontal axes and t is the time elapsed (33 ms)

In the 3D coordinate system the z-axis was vertical with thepositive end pointing up Consequently the vertical component ofv0 was found from the equation of linear translation with constant(gravitational) acceleration

v0ethzTHORN frac14 Dz

tthorn gt

2 eth4THORN

where Δz is the vertical translation made in t=33 ms and g is thegravitational acceleration (98 m sminus2)

The take-off angle (θ0) in the plane of the jump was found fromthe horizontal and vertical components of the take-off velocity

u0 frac14 tan1 v0ethzTHORNv0ethhTHORN

eth5THORN

and the magnitude of the three-dimensional take-off velocity is

v0 frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv20ethhTHORN thorn v20ethzTHORN

q eth6THORN

Estimating the effect of air resistanceIn the absence of air resistance the kinetic energy is converted topotential energy and the maximal height of the jump (h) isdetermined by the take-off speed (v0) and take-off angle (θ)according to standard ballistic equations (Kittel et al 1973)

hvac frac14 v20sin2u0

2g eth7THORN

where the subscript lsquovacrsquo denotes that this is the predicted jumpheight in a vacuum In practice some of the jump energy is wastedin overcoming air resistance as the insect moves through the air Weestimated the effect of drag on the aerial trajectory using basicequations for small particles As the insects moved through the airtheir body rotated about all three anatomical axes Consequently thefrontal area projected onto the flow changed instantaneouslymaking precise assessment impractical Instead we sought a meanvalue The planform area of each sex was calculated as an ellipsewith the major and minor radii being half the length and width of anaverage male and female Thus the planform area of the ellipsoid

1 m

1 m

Fig 2 Camera setup for recording jumps Two orthogonal cameras wereset 1 or 05 m away from the take-off point in order to capture the entire jumpand a close-up of the take-off phase of the jump respectively The take-offpoint was 12 cm above the table surface and was filmed by a camera placed5 cm from the take-off point in order to identify the sex of the jumping waspDiffused infrared (IR) lights were placed behind a screen in order to providebackground illumination for the cameras without attracting the jumping insects

3

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

(A) is

A frac14 p lHT lW4

eth8THORN

For our jumping insects the Reynolds number (Re) was defined as

Re frac14 v ffiffiffiffiffiffiffiffiffiA=p

pn

eth9THORN

where ν is the kinematic viscosity of air at 20degC (ν=15times10minus6 m2 sminus1)and v is the velocity of the bodyThe drag due to movement through air (D) is defined as

D frac14 ra A CDv2

2 eth10THORN

where ρa is the density of dry air at 20degC (ρ=1204 kgmminus3) andCD isthe drag coefficient For the latter we used the drag coefficient for aspherical particle as a function of the Reynolds number (found inEqn 9) as suggested by Vogel (2005)

CD frac14 24

Rethorn 6

1thorn Re1=2thorn 04 eth11THORN

For non-spherical objects at low Reynolds numbers (Re sim150) CD

will somewhat deviate from the values predicted by Eqn 10depending on the shape and orientation of the object For ellipsoidsthe actual value depends on the proportions of the major and minoraxes (Breach 1961) Corrections for various prolate and oblateellipsoids exist but these are exact only for Relt5 (Happel andBrenner 1983) Furthermore as the body is rotating in the air(Movie 1) CD changes with the orientation of the long axis relativeto the direction of movement Here we were not interested in thevalue of CD per se but rather we sought to evaluate the effect of airresistance on the jumper Hence we used Eqn 11 with no correctionfor non-spherical shape and the effect of errors in estimating CD isevaluated 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 thetrajectory of each jump by calculating the instantaneous height andhorizontal distance relative to the take-off point as well as theinstantaneous direction (angle) of air-borne movement Theequations required for the simulation (see Appendix) wereembedded in a custom-written Matlab code to predict the jumptrajectory in air while accounting for the effect of air resistance Thepredicted trajectories were compared with those observed by thehigh-speed cameras A good agreement between the observed jumptrajectory and that predicted by our simulation indicates that theestimated values of CD (Eqn 11) are close enough to the real valueIf the predicted jump trajectory is an overestimate or underestimateof the observed trajectory then the estimated value of CD is too lowor too high respectivelyIn addition for each jump we found the take-off speed and angle

from Eqns 3ndash6 and calculated the maximal jump height (in theabsence of air resistance) from Eqn 7 We then measured the actualjump height observed in the high-speed films with the differencebetween the observed and expected jump height being due to theeffect of air resistance on the insect Ta

ble2

Com

parativ

eda

taof

Ap

seud

ococ

cian

dse

veralo

fthe

topinse

ctjumpe

rs

Bod

ymas

sBod

yleng

thTim

eto

take

-off

Tak

e-off

velocity

Tak

e-off

angle

Acceleration

arelativeto

grav

ityJu

mp

height

Jumphe

ight

body

leng

thKinetic

energy

Pow

erMas

s-sp

ecific

power

Sym

bolformula

mt

vθ

a=vt

ag

Ek=05m

v2p=

Ekt

p=pm

Units

mg

mm

ms

msminus

1de

gm

sminus2

g=98m

sminus2

mm

mm

μJmW

Wkg

minus1

Ana

gyrus

pseu

doco

cci

Mea

n

015

2179

plusmn02

(N=11

)093

(N=11

)237

plusmn041

(N=44

)67

3plusmn10

1(N

=44

)25

53plusmn43

6(N

=44

)26

0plusmn44

(N=44

)12

2plusmn39

(N=44

)68

2plusmn22

8(N

=44

)044

plusmn014

(N=44

)047

plusmn015

(N=44

)31

16plusmn10

06(N

=44

)Bes

t297

813

3189

323

195

1088

067

072

4729

Mea

n

006

7118

plusmn017

(N=12

)093

(N=11

)202

plusmn052

(N=26

)63

1plusmn10

4(N

=26

)21

54plusmn56

6(N

=26

)21

9plusmn58

(N=26

)79

plusmn31

(N=26

)67

3plusmn26

2(N

=26

)015

plusmn006

(N=26

)016

plusmn007

(N=26

)23

26plusmn10

27(N

=26

)Bes

t289

661

3116

316

153

1295

028

030

4514

1Frogh

oppe

r(N

=34

)Mea

n12

3plusmn074

61plusmn

008

128plusmn

010

468plusmn20

2800

286

428plusmn

261

7016

4848

3902

Philaen

ussp

umarius

Bes

t087

547

5400

550

700

11475

155

155

12602

2Le

afho

pper

(N=43

)Mea

n18

4plusmn13

85plusmn

022

44plusmn

018

25plusmn

009

371plusmn440

568

5811

613

65

5813

7065

Aph

rode

smak

arov

iBes

t275

29

1055

108

156

1835

7728

1522

3Hed

geho

gfle

a(N

=10

)Mea

n07

18

14

13

3996

098

3318

306

043

6143

Archa

eops

yllus

erinac

eiBes

t1

212

19

1600

160

687

344

18

15

1500

4Clickbe

etle

Athou

sha

emorrhoida

lisMea

n40

11064

24

3750

382

300

273

115

180

4492

1Burrows(200

6)2Burrows(200

7)3Suttonan

dBurrows(201

1)4Eva

ns(197

2)Ifa

vailable

data

arepres

entedforthemea

nas

wella

sthebe

st(m

axim

um)jump

4

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

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iology

entire jump trajectory Second the cameras were moved 50 cmtowards the insects filming at 10000 frames sminus1 which allowed usto zoom-in on the take-off phase of the jump In both approachescombining the views from the two orthogonal cameras allowed us toextract the kinematics of the jump in 3D Filming was carried outagainst a background illuminated with infrared (IR 850 nm)floodlights while the room in which the experiments took placewas illuminated by visible light from fluorescent light bulbs on theceiling The two high-speed cameras were spatially calibrated byfilming an object with known dimensions that was visible to bothcameras (Hedrick 2008) In the first approach this resulted in acalibrated volume of a rectangular cuboid with dimensions76times76times165 cm (LtimesWtimesH) In the second approach the smallerfield of view (3times3times3 cm) required an alternative method and weused the orthogonal positioning of the cameras to calculate motionof the insect in the x- y- and z-axes directly from the two views aftercorrecting for scale in each camera

Morphometric measurementsMale and female A pseudococci are sexually dimorphic (Fig 1)The body length of 12 males and 11 females was measured fromimages (resolution 2048times1536 pixels) taken with a Leica EZ4 HDdissecting microscope including a 1 mm grid in the background forscale correction On the resulting images we measured the distancebetween the tip of the head and the tip of the closed wings (lHT) andbetween the tip of the head and the tip of the abdomen (lHA) Wealso measured the maximum width of the body (lW) which is themaximal width of the thorax along the lateral axis The insect bodyvolume (V) can be estimated through these measurements byassuming an ellipsoid body shape

V frac14 4

3p lHA

2 lW4

2

eth1THORN

As the small size of the insects restricted direct measurement ofbody mass we resorted to an estimation using the body volume(Eqn 1) and an average body density Body density (ρ) of differentinsects can vary between 04 to 1050 mg mmminus3 (Bennet-Clark andAlder 1979) We assumed an intermediate value of body density of

ρ=07 mg mmminus3 for A pseudococci The body mass of the insect(m) was calculated as body density times the body volume of ourinsect

m frac14 r V eth2THORNThe error associated with this indirect estimation of body mass isevaluated below (see Discussion)

Kinematics of the jumpIn each film frame we used the Matlab code DLTdv5 (Hedrick2008) to track and obtain the 3D position of the center of the insectrsquosbody The velocity at leaving the ground (v0) was found from thedisplacement of the body during the first 33 ms (10 movie frames)after leaving the ground

The horizontal component of v0 was calculated from

v0ethhTHORN frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2 thorn Dy2

pt

eth3THORN

where Δx and Δy are the horizontal translations along the x and yhorizontal axes and t is the time elapsed (33 ms)

In the 3D coordinate system the z-axis was vertical with thepositive end pointing up Consequently the vertical component ofv0 was found from the equation of linear translation with constant(gravitational) acceleration

v0ethzTHORN frac14 Dz

tthorn gt

2 eth4THORN

where Δz is the vertical translation made in t=33 ms and g is thegravitational acceleration (98 m sminus2)

The take-off angle (θ0) in the plane of the jump was found fromthe horizontal and vertical components of the take-off velocity

u0 frac14 tan1 v0ethzTHORNv0ethhTHORN

eth5THORN

and the magnitude of the three-dimensional take-off velocity is

v0 frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv20ethhTHORN thorn v20ethzTHORN

q eth6THORN

Estimating the effect of air resistanceIn the absence of air resistance the kinetic energy is converted topotential energy and the maximal height of the jump (h) isdetermined by the take-off speed (v0) and take-off angle (θ)according to standard ballistic equations (Kittel et al 1973)

hvac frac14 v20sin2u0

2g eth7THORN

where the subscript lsquovacrsquo denotes that this is the predicted jumpheight in a vacuum In practice some of the jump energy is wastedin overcoming air resistance as the insect moves through the air Weestimated the effect of drag on the aerial trajectory using basicequations for small particles As the insects moved through the airtheir body rotated about all three anatomical axes Consequently thefrontal area projected onto the flow changed instantaneouslymaking precise assessment impractical Instead we sought a meanvalue The planform area of each sex was calculated as an ellipsewith the major and minor radii being half the length and width of anaverage male and female Thus the planform area of the ellipsoid

1 m

1 m

Fig 2 Camera setup for recording jumps Two orthogonal cameras wereset 1 or 05 m away from the take-off point in order to capture the entire jumpand a close-up of the take-off phase of the jump respectively The take-offpoint was 12 cm above the table surface and was filmed by a camera placed5 cm from the take-off point in order to identify the sex of the jumping waspDiffused infrared (IR) lights were placed behind a screen in order to providebackground illumination for the cameras without attracting the jumping insects

3

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

(A) is

A frac14 p lHT lW4

eth8THORN

For our jumping insects the Reynolds number (Re) was defined as

Re frac14 v ffiffiffiffiffiffiffiffiffiA=p

pn

eth9THORN

where ν is the kinematic viscosity of air at 20degC (ν=15times10minus6 m2 sminus1)and v is the velocity of the bodyThe drag due to movement through air (D) is defined as

D frac14 ra A CDv2

2 eth10THORN

where ρa is the density of dry air at 20degC (ρ=1204 kgmminus3) andCD isthe drag coefficient For the latter we used the drag coefficient for aspherical particle as a function of the Reynolds number (found inEqn 9) as suggested by Vogel (2005)

CD frac14 24

Rethorn 6

1thorn Re1=2thorn 04 eth11THORN

For non-spherical objects at low Reynolds numbers (Re sim150) CD

will somewhat deviate from the values predicted by Eqn 10depending on the shape and orientation of the object For ellipsoidsthe actual value depends on the proportions of the major and minoraxes (Breach 1961) Corrections for various prolate and oblateellipsoids exist but these are exact only for Relt5 (Happel andBrenner 1983) Furthermore as the body is rotating in the air(Movie 1) CD changes with the orientation of the long axis relativeto the direction of movement Here we were not interested in thevalue of CD per se but rather we sought to evaluate the effect of airresistance on the jumper Hence we used Eqn 11 with no correctionfor non-spherical shape and the effect of errors in estimating CD isevaluated 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 thetrajectory of each jump by calculating the instantaneous height andhorizontal distance relative to the take-off point as well as theinstantaneous direction (angle) of air-borne movement Theequations required for the simulation (see Appendix) wereembedded in a custom-written Matlab code to predict the jumptrajectory in air while accounting for the effect of air resistance Thepredicted trajectories were compared with those observed by thehigh-speed cameras A good agreement between the observed jumptrajectory and that predicted by our simulation indicates that theestimated values of CD (Eqn 11) are close enough to the real valueIf the predicted jump trajectory is an overestimate or underestimateof the observed trajectory then the estimated value of CD is too lowor too high respectivelyIn addition for each jump we found the take-off speed and angle

from Eqns 3ndash6 and calculated the maximal jump height (in theabsence of air resistance) from Eqn 7 We then measured the actualjump height observed in the high-speed films with the differencebetween the observed and expected jump height being due to theeffect of air resistance on the insect Ta

ble2

Com

parativ

eda

taof

Ap

seud

ococ

cian

dse

veralo

fthe

topinse

ctjumpe

rs

Bod

ymas

sBod

yleng

thTim

eto

take

-off

Tak

e-off

velocity

Tak

e-off

angle

Acceleration

arelativeto

grav

ityJu

mp

height

Jumphe

ight

body

leng

thKinetic

energy

Pow

erMas

s-sp

ecific

power

Sym

bolformula

mt

vθ

a=vt

ag

Ek=05m

v2p=

Ekt

p=pm

Units

mg

mm

ms

msminus

1de

gm

sminus2

g=98m

sminus2

mm

mm

μJmW

Wkg

minus1

Ana

gyrus

pseu

doco

cci

Mea

n

015

2179

plusmn02

(N=11

)093

(N=11

)237

plusmn041

(N=44

)67

3plusmn10

1(N

=44

)25

53plusmn43

6(N

=44

)26

0plusmn44

(N=44

)12

2plusmn39

(N=44

)68

2plusmn22

8(N

=44

)044

plusmn014

(N=44

)047

plusmn015

(N=44

)31

16plusmn10

06(N

=44

)Bes

t297

813

3189

323

195

1088

067

072

4729

Mea

n

006

7118

plusmn017

(N=12

)093

(N=11

)202

plusmn052

(N=26

)63

1plusmn10

4(N

=26

)21

54plusmn56

6(N

=26

)21

9plusmn58

(N=26

)79

plusmn31

(N=26

)67

3plusmn26

2(N

=26

)015

plusmn006

(N=26

)016

plusmn007

(N=26

)23

26plusmn10

27(N

=26

)Bes

t289

661

3116

316

153

1295

028

030

4514

1Frogh

oppe

r(N

=34

)Mea

n12

3plusmn074

61plusmn

008

128plusmn

010

468plusmn20

2800

286

428plusmn

261

7016

4848

3902

Philaen

ussp

umarius

Bes

t087

547

5400

550

700

11475

155

155

12602

2Le

afho

pper

(N=43

)Mea

n18

4plusmn13

85plusmn

022

44plusmn

018

25plusmn

009

371plusmn440

568

5811

613

65

5813

7065

Aph

rode

smak

arov

iBes

t275

29

1055

108

156

1835

7728

1522

3Hed

geho

gfle

a(N

=10

)Mea

n07

18

14

13

3996

098

3318

306

043

6143

Archa

eops

yllus

erinac

eiBes

t1

212

19

1600

160

687

344

18

15

1500

4Clickbe

etle

Athou

sha

emorrhoida

lisMea

n40

11064

24

3750

382

300

273

115

180

4492

1Burrows(200

6)2Burrows(200

7)3Suttonan

dBurrows(201

1)4Eva

ns(197

2)Ifa

vailable

data

arepres

entedforthemea

nas

wella

sthebe

st(m

axim

um)jump

4

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(A) is

A frac14 p lHT lW4

eth8THORN

For our jumping insects the Reynolds number (Re) was defined as

Re frac14 v ffiffiffiffiffiffiffiffiffiA=p

pn

eth9THORN

where ν is the kinematic viscosity of air at 20degC (ν=15times10minus6 m2 sminus1)and v is the velocity of the bodyThe drag due to movement through air (D) is defined as

D frac14 ra A CDv2

2 eth10THORN

where ρa is the density of dry air at 20degC (ρ=1204 kgmminus3) andCD isthe drag coefficient For the latter we used the drag coefficient for aspherical particle as a function of the Reynolds number (found inEqn 9) as suggested by Vogel (2005)

CD frac14 24

Rethorn 6

1thorn Re1=2thorn 04 eth11THORN

For non-spherical objects at low Reynolds numbers (Re sim150) CD

will somewhat deviate from the values predicted by Eqn 10depending on the shape and orientation of the object For ellipsoidsthe actual value depends on the proportions of the major and minoraxes (Breach 1961) Corrections for various prolate and oblateellipsoids exist but these are exact only for Relt5 (Happel andBrenner 1983) Furthermore as the body is rotating in the air(Movie 1) CD changes with the orientation of the long axis relativeto the direction of movement Here we were not interested in thevalue of CD per se but rather we sought to evaluate the effect of airresistance on the jumper Hence we used Eqn 11 with no correctionfor non-spherical shape and the effect of errors in estimating CD isevaluated 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 thetrajectory of each jump by calculating the instantaneous height andhorizontal distance relative to the take-off point as well as theinstantaneous direction (angle) of air-borne movement Theequations required for the simulation (see Appendix) wereembedded in a custom-written Matlab code to predict the jumptrajectory in air while accounting for the effect of air resistance Thepredicted trajectories were compared with those observed by thehigh-speed cameras A good agreement between the observed jumptrajectory and that predicted by our simulation indicates that theestimated values of CD (Eqn 11) are close enough to the real valueIf the predicted jump trajectory is an overestimate or underestimateof the observed trajectory then the estimated value of CD is too lowor too high respectivelyIn addition for each jump we found the take-off speed and angle

from Eqns 3ndash6 and calculated the maximal jump height (in theabsence of air resistance) from Eqn 7 We then measured the actualjump height observed in the high-speed films with the differencebetween the observed and expected jump height being due to theeffect of air resistance on the insect Ta

ble2

Com

parativ

eda

taof

Ap

seud

ococ

cian

dse

veralo

fthe

topinse

ctjumpe

rs

Bod

ymas

sBod

yleng

thTim

eto

take

-off

Tak

e-off

velocity

Tak

e-off

angle

Acceleration

arelativeto

grav

ityJu

mp

height

Jumphe

ight

body

leng

thKinetic

energy

Pow

erMas

s-sp

ecific

power

Sym

bolformula

mt

vθ

a=vt

ag

Ek=05m

v2p=

Ekt

p=pm

Units

mg

mm

ms

msminus

1de

gm

sminus2

g=98m

sminus2

mm

mm

μJmW

Wkg

minus1

Ana

gyrus

pseu

doco

cci

Mea

n

015

2179

plusmn02

(N=11

)093

(N=11

)237

plusmn041

(N=44

)67

3plusmn10

1(N

=44

)25

53plusmn43

6(N

=44

)26

0plusmn44

(N=44

)12

2plusmn39

(N=44

)68

2plusmn22

8(N

=44

)044

plusmn014

(N=44

)047

plusmn015

(N=44

)31

16plusmn10

06(N

=44

)Bes

t297

813

3189

323

195

1088

067

072

4729

Mea

n

006

7118

plusmn017

(N=12

)093

(N=11

)202

plusmn052

(N=26

)63

1plusmn10

4(N

=26

)21

54plusmn56

6(N

=26

)21

9plusmn58

(N=26

)79

plusmn31

(N=26

)67

3plusmn26

2(N

=26

)015

plusmn006

(N=26

)016

plusmn007

(N=26

)23

26plusmn10

27(N

=26

)Bes

t289

661

3116

316

153

1295

028

030

4514

1Frogh

oppe

r(N

=34

)Mea

n12

3plusmn074

61plusmn

008

128plusmn

010

468plusmn20

2800

286

428plusmn

261

7016

4848

3902

Philaen

ussp

umarius

Bes

t087

547

5400

550

700

11475

155

155

12602

2Le

afho

pper

(N=43

)Mea

n18

4plusmn13

85plusmn

022

44plusmn

018

25plusmn

009

371plusmn440

568

5811

613

65

5813

7065

Aph

rode

smak

arov

iBes

t275

29

1055

108

156

1835

7728

1522

3Hed

geho

gfle

a(N

=10

)Mea

n07

18

14

13

3996

098

3318

306

043

6143

Archa

eops

yllus

erinac

eiBes

t1

212

19

1600

160

687

344

18

15

1500

4Clickbe

etle

Athou

sha

emorrhoida

lisMea

n40

11064

24

3750

382

300

273

115

180

4492

1Burrows(200

6)2Burrows(200

7)3Suttonan

dBurrows(201

1)4Eva

ns(197

2)Ifa

vailable

data

arepres

entedforthemea

nas

wella

sthebe

st(m

axim

um)jump

4

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Jump energeticsThe mechanical power needed for the jump was calculated from thekinetic energy at take-off (Ek) as

p frac14 Ek

tfrac14 1

2mv20=t eth12THORN

where t is the duration of pushing against the ground The meanacceleration on the ground (a) was found from the take-off speed(v0) and t as

a frac14 v0=t eth13THORNThe power divided by our estimate of body mass gave the mass-specific power ( p) allowing us to compare the jump energeticsbetween the larger females and smaller males

StatisticsStatistical tests were performed to compare the jumpingperformance between males and females and between thepredicted and observed jump heights We used t-tests to comparejump kinematics parameters between males and females The dragcoefficients calculated as the mean for each jump did not have anormal distribution and therefore the difference in drag coefficientbetween the sexes was evaluated using a MannndashWhitney U-testThe jump heights of males and females were also compared usingANCOVA with the take-off speed as a covariate (see Results) Alltests were performed using Statistica (StatSoft Inc) Unless notedotherwise all results are reported as meansplusmnsd

RESULTSMales are smaller than females and consequently have lower Re andhigher CD (Table 1) The mean jump kinematics of the two sexes inaddition to the mechanical power (Eqn 12) calculated based on theobserved kinematics are given in Table 2While both sexes (females n=44 males n=26) left the ground at

similar take-off angles (females at 673plusmn101 deg and males at 631plusmn

104 deg t-test P=010 Table 2) the mean take-off speed of females(237plusmn041 m sminus1) was significantly (t-test P=0003) greater (by18) than that of males (202plusmn052 m sminus1) Consequently meanfemale jump height was 154-fold the mean jump height of males(122plusmn39 and 79plusmn31 cm respectively t-test Plt0001)

Sequences of images taken by the high-speed cameras duringtake-off (eg Fig 3) revealed that the insects pushed againstthe ground with their middle legs which were the last legs to leavethe ground The entire period of pushing lasted less than 1 ms(Table 2) In the air thewings opened and started flapping only afterthe 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-offspeed as a covariate revealed that jump height corrected for verticaltake-off speed was still higher in females than in males (ANCOVAPlt0001) A significant interaction between sex and the verticaltake-off speed (ANCOVA Plt0001) revealed that the inter-sexdifference in jump height was larger at higher take-off speeds(Fig 4)

The observed jump trajectories (Fig 5A) show a difference injumping performance between males and females Fig 5B showstwo of the jumps (one jump by a male and one by a female) inthree ways first as tracked directly from the high-speed videosecond as simulated for the case of no air resistance (see alsoagreement with Eqn 7) and third simulated as described in theAppendix for the inclusion of air resistance It is evident thatthe simulations in air predict the actual jump height (peak of thetrajectory) quite well (meanplusmnsd error was 75plusmn55) and thatair resistance leads to jump heights that are half of the meanpredicted jump heights in a vacuum (251plusmn10 cm for females and171plusmn83 for males)

We used the velocity at take-off and the duration of pushing againstthe ground as measured directly in the films to determine themechanical power output required for the jump (Eqn 12) The meanjump power and body mass-specific power were significantly greater

ndash231 msA

B

ndash163 ms ndash08 ms ndash07 ms ndash06 ms ndash05 ms ndash04 ms

ndash03 ms

1 mm

1 mm

ndash07 ms ndash06 ms ndash05 ms ndash04 ms ndash03 ms ndash02 ms ndash01 ms 0 msTake-off

+01 ms

ndash02 ms ndash01 ms 0 msTake-off

+01 ms +02 ms +03 ms

Fig 3 Take-off sequence of A pseudococci filmed at 10000 frames sminus1 (A) Back view The insect (male) leaves the ground at time 0 Before take-off theantennae and wings are tucked in towards the body (minus231 ms) The body is propelled upwards using the middle legs In this particular jump the take-offangle was 87 deg and the take-off velocity was 246 m sminus1 (B) Side view of a different individual (also male) The wings are tucked in towards the bodybefore take-off (minus06 ms) The body is pushed against the ground via the middle legs (yellow triangles at minus02 ms) while the forelegs and hindlegs are off theground or remain idle The take-off angle of this jump was 84 deg and the take-off velocity was 23 m sminus1

5

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in females than in males (t-tests Plt0003 in both cases femalesp=047plusmn015 mWand p=31plusmn1 kW kgminus1 males p=016plusmn007 mWand p=23plusmn1 kW kgminus1)

DISCUSSIONBennet-Clark and Alder (1979) defined jump efficiency as the ratioof actual (observed) jump height to predicted jump height in avacuum They showed that this efficiency is mostly affected by theratio between surface area and mass of the jumper According to ourmeasurements jumping efficiency of A pseudococci was 051plusmn007 and 051plusmn013 for females and males respectively The area-to-mass ratio of our insects was 765 and 515 for males and femalesrespectively (based on the data in Table 1) These ratios predict anefficiency of sim06 for A pseudococci according to Bennet-Clarkand Alder (1979 see their fig 4) The reasons for the lowerefficiencies found here are likely 2-fold (1) the calculation ofefficiency in the previous work was based on a constant dragcoefficient (CD=1) whereas air resistance was higher in ourexperiments considering the change in CD with Reynolds numberThe mean values of the drag coefficient per jump were 27 higherin males (215plusmn031 and 169plusmn014 for males and femalesrespectively MannndashWhitney U-test Plt0001) Thus the higherair resistance for the same speed led to poorer jump efficiency in ourinsects (2) In addition the jump efficiency found by Bennet-Clarkand Alder (1979) was calculated for vertical jumps in which all thetake-off speed is vertical speed In contrast our wasps had jumpangles of sim60 deg on average (Table 2) implying that the verticaltake-off speed is only about 86 of the total (3D) take-off speedThe fact that the total take-off speed is higher implies higher drag onthe body for the same vertical speed (Eqn 10) resulting in greaterattenuation of the jump energy and lower values of jump efficiency

20 A B

C

30

25

20

15

10

5

00 10 20 30

No drag

Male FemaleMales Females

FemaleMale

40 50

0 10 20Distance (cm)

30 40

25

20

15

10

5

0

18

16

14

12

10

Hei

ght (

cm)

Hei

ght (

cm)

8

6

4

2

00 5 10

Distance (cm)15

Fig 5 Comparison of measured and simulated jump trajectories in males and females (A) The measured jump trajectories of 44 females and 26 males(B) Examples of a single jump by a male (593 deg angle and 232 m sminus1 velocity) and by a female (66 deg angle and 252 m sminus1 velocity) The trajectoriesmeasured directly from the video are depicted by solid lines The simulated trajectory in air is depicted by dashed lines The predicted trajectory in a vacuum(dotted lines) underlines the effect of air resistance on the jump trajectories which lowers the jump height by sim50 (C) Inter-sex differences in jumpingperformance due to air resistance Simulated jump trajectory in air of an average male and female In both simulations take-off velocity was set to 22 m sminus1 andthe jump angle was set to 65 deg (ie in the simulations only body mass and size differ between the sexes) Under these conditions in a vacuum the two insectsshould display the same jump trajectory (gray dashed line) illustrating the greater effect of air resistance on the smaller males

y=34494x15528

R2=09467

y=34341x14208

R2=09662

0

5

10

15

20

25

0 05 1 15 2 25 3 35

Obs

erve

d ju

mp

heig

ht (c

m)

Vertical take-off velocity (m sndash1)

FemalesMales

Fig 4 Observed jump height as a function of vertical take-off velocity inmales and females The trend lines for males and females are statisticallysignificant with females jumping significantly higher than males (ANCOVAPlt0001)

6

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Hence our study demonstrates that the problem of air resistance insmall insects that do not jump vertically is more severe thansuggested in the seminal study of Bennet-Clark and Alder (1979)Male and female A pseudococci are sexually dimorphic with

male body length approximately 565 shorter than that of femalesand calculated male body mass 226-fold smaller (Table 1) Thisdimorphism has implications for the jumping performance of malesFirst males have shorter legs (Table 1) and thus need to launch theirbody at higher accelerations in order to reach the same take-offspeed as that of females Second once in the air the effect of drag ismore pronounced in males because of their higher area-to-massratio This dimorphism affects the jump height in males and femaleseven when they leave the ground at the same speed and take-offangle (Fig 5C) While the predicted jump height in a vacuum is thesame for the two insects the simulated jump height in the air islower for males simply because of the increased effect of airresistance on the smaller body size In practice however the inter-sex difference in jump height is larger as a result of females leavingthe ground at higher speeds compared with males (Table 2) Thisalso resulted in females demonstrating a higher power demand whenjumping while the mass-specific power of the larger females wasalso higher Thus male A pseudococci do not seem to compensatefor their smaller size in order to reach a similar jump height tofemales We examined whether this is due to the higher dragcoefficient making higher jumps less efficient for males Evenwhen exerting the same mass-specific power females reach a higherjump height with the increase in height gain compared with malesincreasing with mass-specific power (Fig 6A) In order to reachsimilar jump heights to females males need to increase the mass-specific power exerted during a jump (which is due to the largereffect of air resistance) (Fig 6B) There is a consistent increase ofmale-to-female power ratio as jump height increases Hence thedifference in the non-linear increase in mass-specific power neededfor higher jumps sets a physiologicalndashbiomechanical barrierbetween males and females Correspondingly sim75 of the malejumps were to heights lower than 10 cm while 75 of the femalejumps were to heights higher than 10 cm where the inter-sex ratio ofthe mass-specific power is 14 Hence the increase in power dictatedby the small body size and higher drag coefficient of males results ina functional segregation of jumping performance between thedimorphic males and females Anagyrus is a member of theChalcidoidea superfamily in which sexual dimorphism is commonin many species Gibson (1986) reported that male and femaleeupelminid wasps (Chalcidoidea) utilize different anatomicalstructures for jumping If the same is true for A pseudococci thiscould indicate a divergence dictated by the biomechanicalconstraints on jumping in smaller malesWe used basic ballistics equations to predict the trajectory of the

insects in the air and in a vacuum These simplified equations do notaccount for body rotation and the fact that the body is not a sphereNevertheless we found a reasonable agreement between theobserved and predicted jump trajectories (Figs 5B and 7)However the simulations matched the observed jumps well onlyup to the maximum height of the jump This is because once theinsect reached the peak of the trajectory it opened its wings andflapped them transitioning into active flight As a result its motiondeviated from projectile dynamics from this point on Becauseuncertainties in both body mass and drag coefficient estimates affectthe jump outcome (trajectory) predicted by the model we tested thesensitivity of our predicted jump height to variation in these valuesThis was done by recalculating simulated jump height with thebody density changed to 1 and 04 mg mmminus3 ie the maximum and

minimum body densities reported for insects by Bennet-Clark andAlder (1979) and comparing the outcome (predicted jump height)with the actual jump height measured from the films using linearregressions (Fig 7AB) For both males and females all linearregressions were significant (Plt0001 in all cases) and the R2 valuewas higher than 092 in all cases Our original estimate of bodydensity for females gave a slope of 103 while the higher and lowermass estimates gave a slope of 084 and 146 respectively Thus theassumed body density for females is fairly accurate In males theslopes for our estimate and the higher and lower body density were11 09 and 162 respectively Thus we may have slightlyunderestimated male body density Similarly we recalculated jumpheight (Fig 7CD) with the drag coefficient plusmn33 of the estimatefound using Eqn 10 For females and males all linear regressionswere significant (Plt0001 in all cases) and the R2 was higher than092 for females and males The slopes for our estimate and thehigher and lower drag coefficient for females were 103 122 and078 respectively For males the slopes were 11 132 and 083respectively suggesting a slight overestimation This sensitivityanalysis demonstrates that while the exact values predicted by the

0

5

10

15

20

25

0 2 4 6 8 10

Hei

ght (

cm)

p (kW kgndash1)

FemalesMales

A

1

11

12

13

14

15

16

17

0 2 4 6 8 10 12 14 16

Mal

e p

fem

ale p

Jump height (cm)

B

Fig 6 Jump height as a function of mass-specific power (p)(A) Simulated jump heights of males (blue) and females (orange) In allsimulations the take-off anglewas 65 deg and the duration of pushing against theground remained constant at 1 ms The take-off speed was increased from 0 to4 m sminus1 to increase the power of the jumpWe then found the jumpheight for eachsimulation as a function of the body mass-specific power In order to achievesimilar heightsmalesmust exertamass-specific power larger than that of females(eg to reacha jumpheightof12 cmdashedgray linemaleshave toexertamass-specific power of 44 kW kgminus1 ie 15 times the mass-specific power exerted byfemales 29 kW kgminus1) (B) Data from A presented as the ratio of male to femalemass-specific power as a function of simulated jump height The observed meanjump heights for males and females are denoted by the blue and orange circlesrespectively Shaded blue and orange rectangles indicate the interquartile range(IQR 25ndash75) of observed jump heights in males and females respectively

7

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model depend on the estimates used for body mass andCD the trendin inter-sex difference resulting from the greater effect of airresistance on the smaller males remains consistent and pronouncedThe short duration of pushing against the ground prior to take-off

(lt1 ms) and exceptionally high mass-specific power exerted byA pseudococci during the jump (Table 2) suggest the presence of apower-amplification mechanism Such mechanisms for jumping arecommon in many small insects (Table 2) Elastic energy storage hasbeen suggested for jumping wasps based on musculoskeletalanatomy (Gibson 1986) but to the best of our knowledge our reportis the first quantitative measurement of jumping performance for awasp using legged power amplification Take-off jumps have been

reported in hymenopteran species before (Burrows and Dorosenko2017) but the reported jumps were moderate compared with those ofA pseudococci One such example is that of Pteromalus puparum(Burrows and Dorosenko 2017) which has a similar body length(27plusmn001 mm) and mass (10plusmn010 mg) to A pseudococci Themass-specific power exerted by these wasps during a jump was only60 W kgminus1 ie roughly 50-fold lower than the mass-specific powerexerted by our female A pseudococci

Compared with other insects using power amplification forjumping both male and female A pseudococci achieve modestjump heights Their acceleration to the take-off speed and their jumpheight relative to body length however are among the highest

18 A B

C D

16

25

20

y=11134xndash0468R2=09346

y=16216xndash11291R2=09267

y=09027xndash00875R2=09388

y=13188xndash07685R2=0931

y=08311x+00653R2=09403

y=1495xndash18817R2=09311

y=1032xndash06089R2=09295

y=08413x+00981R2=09279

y=11134xndash0468R2=09346

y=12194xndash11842R2=09304

y=1032xndash06089R2=09295

y=07775x+03707R2=0927

15

10

5

0

25

20

15

10

5

0

14

12

10

Obs

erve

d ju

mp

heig

ht (c

m)

8

6

4

2

00 0 5 10 15 20 252 4 6

0 5 10 15 20 25

8 10 12 14 16

0 2 4 6Simulated jump height (cm)

8 10 12 14 16 18

18

16

14

12

10

8

6

4

2

0

Fig 7 Sensitivity analysis to evaluate the effect of variation in body density and drag coefficient on the predicted jumpheight in air from the simulationAll panels show plots of the maximummeasured height versus predicted heights using our estimates (blue) and higher (gray) and lower (orange) values for bodydensity (AB) or the drag coefficient (CD) (AC) Jumps by males (BD) jumps by females

8

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reported for jumping insects (Table 2) The froghopper Philaenusspumarius is considered one of the most prominent jumpers in theinsect world (Burrows 2003) Table 2 reveals that our wasps requiresimilar values of mass-specific power to those of the froghopper inorder to reach similar accelerations and heights relative to bodylength Remarkably the mean body mass and length of thefroghopper is 180-fold and 52-fold respectively higher than themean bodymass and length of male A pseudococci Thus while thetwo insects achieve a similar take-off performance the air resistanceis much more substantial for the wasps resulting in a decrease intheir jumping performance For example if we use the data given inTable 2 and attempt to predict the jump efficiency of the largerfroghopper using the take-off speed of the highest jump and themean jump angle and substituting into Eqn 7 we obtain a value ofsim90 jump efficiency for the froghopper The difference from the51 efficiency of A pseudococci is entirely due to the lower Amratio of the froghopper The acceleration of A pseudococci at take-off (sim260 times gravity in females) is much lower than that in click-beetles (which do not use their legs for jumping) but only sim10lower than the acceleration of the froghoppers (Table 2) Thusdespite (or perhaps because of ) its smaller size A pseudococci is aprominent jumper that nevertheless demonstrates the constraints ofmoving through air on a small creature

APPENDIXThe instantaneous height above the take-off point of a projectile iscalculated numerically from the speed and air resistance according to

hethtTHORN frac14 hethtdtTHORN thorn vzethtTHORNdt ethA1THORNwhere dt is the time increment (step) in the simulation and vz(t) is theinstantaneous vertical speed calculated as

vzethtTHORN frac14 vzethtdtTHORN thorn azethtTHORNdt ethA2THORNwhere az(t) is the instantaneous vertical acceleration calculated as

azethtTHORN frac14 g ethDethtTHORN=mTHORN sin uethtTHORN ethA3THORNSimilarly the instantaneous horizontal distance (x) moved

through the air is

xethtTHORN frac14 xethtdtTHORN thorn vxethtTHORNdt ethA4THORNwhere vx(t) is the instantaneous horizontal speed calculated as

vxethtTHORN frac14 vxethtdtTHORN thorn axethtTHORNdt ethA5THORNwhere ax(t) is the instantaneous horizontal acceleration found from

axethtTHORN frac14 ethDethtTHORN=mTHORN cos uethtTHORN ethA6THORNThe instantaneous angle θ(t) in Eqns A3 and A6 is

uethtTHORN frac14 tan1 vzethtTHORNvxethtTHORN

ethA7THORN

AcknowledgementsWe thank Shimon Steinberg from Bio-Bee Biological Systems for providing theinsects used in this study Eyal Dafni provided invaluable assistance with theexperiments The late Prof Dan Gerling contributed many useful insights that led tothe conceptualization of the study

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization TU GR Methodology TU GR Software TU GRValidation TU GR Formal analysis TU Investigation TU GR Resources

GR Data curation TU GR Writing - original draft TU GR Writing - review ampediting TU GR Visualization TU Supervision GR Project administrationGR Funding acquisition GR

FundingThis research received no specific grant from any funding agency in the publiccommercial or not-for-profit sectors

Data availabilityData used in the study and code to simulate jumps are available for download fromhttpsgalribakweeblycomcodesndashmultimediahtml

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb177600supplemental

ReferencesAlexander R M (1995) Leg design and jumping technique for humans other

vertebrates and insects Philos Trans R Soc Lond B Biol Sci 347 235-248Bennet-Clark H C (1977) Scale effects in jumping animals In Scale Effects in

Animal Locomotion (ed T J Pedley) pp 185-201 London Academic PressBennet-Clark H C and Alder G M (1979) The effect of air resistance on the

jumping performance of insects J Exp Biol 82 105-121Bennet-Clark H C and Lucey E C A (1967) The jump of the flea a study of the

energetics and a model of the mechanism J Exp Biol 47 59-76Brackenbury J andWang R (1995) Ballistics and visual targeting in flea-beetles

(Alticinae) J Exp Biol 198 1931-1942Breach D R (1961) Slow flow past ellipsoids of revolution J Fluid Mech 10

306-314Burrows M (2003) Froghopper insects leap to new heights Nature 424 509Burrows M (2006) Jumping performance of froghopper insects J Exp Biol 209

4607-4621Burrows M (2007) Kinematics of jumping in leafhopper insects (Hemiptera

Auchenorrhyncha Cicadellidae) J Exp Biol 210 3578-3589Burrows M (2011) Jumping mechanisms and performance of snow fleas

(Mecoptera Boreidae) J Exp Biol 214 2362-2374Burrows M and Dorosenko M (2014) Jumping mechanisms in lacewings

(Neuroptera Chrysopidae and Hemerobiidae) J Exp Biol 217 4252-4261Burrows M and Dorosenko M (2017) Take-off mechanisms in parasitoid wasps

J Exp Biol 220 3812-3825Card G and Dickinson M H (2008) Performance trade-offs in the flight initiation

of Drosophila J Exp Biol 211 341-353DeBach P and Rosen D (1991) Biological Control by Natural Enemies 2nd edn

New York Cambridge University PressEarls K D (2000) Kinematics and mechanics of ground take-off in the starling

Sturnis Vulgaris and the Quail Coturnix coturnix J Exp Biol 203 725-739Evans M E G (1972) The jump of the click beetle (Coleoptera Elateridae)- a

preliminary study J Zool 167 319-336Evans M E G (1973) The jump of the click beetle (Coleoptera Elateridae)-

energetics and mechanics J Zool 169 181-194FatourosNE andHuigensME (2012) Phoresy in the field natural occurrenceof

Trichogramma egg parasitoids on butterflies and moths BioControl 57 493-502Fleagle J G (1976) Locomotion and posture of the malayan siamang and

implications for hominoid evolution Folia Primatol 26 245-269Frantsevich L (2004) Righting kinematics in beetles (Insecta Coleoptera)

Arthropod Struct Dev 33 221-235Gibson G A P (1986) Mesothoracic skeletomusculature and mechanics of flight

and jumping in Eupelminae (Hymenoptera Chalcidoidea Eupelmidae) CanEntomol 118 691-728

Gronenberg W (1996) Fast actions in small animals springs and clickmechanisms J Comp Physiol A 178 727-734

Happel J and Brenner H (1983) Low Reynolds Number HydrodynamicsMartinus Nijhoff Publishers The Hague

Hedrick T L (2008) Software techniques for two- and three-dimensional kinematicmeasurements of biological and biomimetic systemsBioinspir Biomim 3 34001

Heppner F H and Anderson J G T (1985) Leg thrust important in flight take-offin the pigeon J Exp Biol 114 285-288

Houck M A andOConnor B M (1991) Ecological and evolutionary significanceof Phoresy in the Astigmata Annu Rev Entomol 36 611-636

Jackson R R and Pollard S D (1996) Predatory behavior of jumping spidersAnnu Rev Entomol 41 287-308

Kittel C Knight W D Ruderman M A Helmholz A C and Moyer B J(1973) Mechanics New York McGraw-Hill

Kohlsdorf T and Navas C A (2007) Evolution of jumping capacity inTropidurinae lizards does habitat complexity influence obstacle-crossingability Biol J Linn Soc 91 393-402

Kutsch W and Fuchs U (2000) Locust flight initiation a comparison of normaland artificial release Physiol Entomol 25 370-382

9

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Hence our study demonstrates that the problem of air resistance insmall insects that do not jump vertically is more severe thansuggested in the seminal study of Bennet-Clark and Alder (1979)Male and female A pseudococci are sexually dimorphic with

male body length approximately 565 shorter than that of femalesand calculated male body mass 226-fold smaller (Table 1) Thisdimorphism has implications for the jumping performance of malesFirst males have shorter legs (Table 1) and thus need to launch theirbody at higher accelerations in order to reach the same take-offspeed as that of females Second once in the air the effect of drag ismore pronounced in males because of their higher area-to-massratio This dimorphism affects the jump height in males and femaleseven when they leave the ground at the same speed and take-offangle (Fig 5C) While the predicted jump height in a vacuum is thesame for the two insects the simulated jump height in the air islower for males simply because of the increased effect of airresistance on the smaller body size In practice however the inter-sex difference in jump height is larger as a result of females leavingthe ground at higher speeds compared with males (Table 2) Thisalso resulted in females demonstrating a higher power demand whenjumping while the mass-specific power of the larger females wasalso higher Thus male A pseudococci do not seem to compensatefor their smaller size in order to reach a similar jump height tofemales We examined whether this is due to the higher dragcoefficient making higher jumps less efficient for males Evenwhen exerting the same mass-specific power females reach a higherjump height with the increase in height gain compared with malesincreasing with mass-specific power (Fig 6A) In order to reachsimilar jump heights to females males need to increase the mass-specific power exerted during a jump (which is due to the largereffect of air resistance) (Fig 6B) There is a consistent increase ofmale-to-female power ratio as jump height increases Hence thedifference in the non-linear increase in mass-specific power neededfor higher jumps sets a physiologicalndashbiomechanical barrierbetween males and females Correspondingly sim75 of the malejumps were to heights lower than 10 cm while 75 of the femalejumps were to heights higher than 10 cm where the inter-sex ratio ofthe mass-specific power is 14 Hence the increase in power dictatedby the small body size and higher drag coefficient of males results ina functional segregation of jumping performance between thedimorphic males and females Anagyrus is a member of theChalcidoidea superfamily in which sexual dimorphism is commonin many species Gibson (1986) reported that male and femaleeupelminid wasps (Chalcidoidea) utilize different anatomicalstructures for jumping If the same is true for A pseudococci thiscould indicate a divergence dictated by the biomechanicalconstraints on jumping in smaller malesWe used basic ballistics equations to predict the trajectory of the

insects in the air and in a vacuum These simplified equations do notaccount for body rotation and the fact that the body is not a sphereNevertheless we found a reasonable agreement between theobserved and predicted jump trajectories (Figs 5B and 7)However the simulations matched the observed jumps well onlyup to the maximum height of the jump This is because once theinsect reached the peak of the trajectory it opened its wings andflapped them transitioning into active flight As a result its motiondeviated from projectile dynamics from this point on Becauseuncertainties in both body mass and drag coefficient estimates affectthe jump outcome (trajectory) predicted by the model we tested thesensitivity of our predicted jump height to variation in these valuesThis was done by recalculating simulated jump height with thebody density changed to 1 and 04 mg mmminus3 ie the maximum and

minimum body densities reported for insects by Bennet-Clark andAlder (1979) and comparing the outcome (predicted jump height)with the actual jump height measured from the films using linearregressions (Fig 7AB) For both males and females all linearregressions were significant (Plt0001 in all cases) and the R2 valuewas higher than 092 in all cases Our original estimate of bodydensity for females gave a slope of 103 while the higher and lowermass estimates gave a slope of 084 and 146 respectively Thus theassumed body density for females is fairly accurate In males theslopes for our estimate and the higher and lower body density were11 09 and 162 respectively Thus we may have slightlyunderestimated male body density Similarly we recalculated jumpheight (Fig 7CD) with the drag coefficient plusmn33 of the estimatefound using Eqn 10 For females and males all linear regressionswere significant (Plt0001 in all cases) and the R2 was higher than092 for females and males The slopes for our estimate and thehigher and lower drag coefficient for females were 103 122 and078 respectively For males the slopes were 11 132 and 083respectively suggesting a slight overestimation This sensitivityanalysis demonstrates that while the exact values predicted by the

0

5

10

15

20

25

0 2 4 6 8 10

Hei

ght (

cm)

p (kW kgndash1)

FemalesMales

A

1

11

12

13

14

15

16

17

0 2 4 6 8 10 12 14 16

Mal

e p

fem

ale p

Jump height (cm)

B

Fig 6 Jump height as a function of mass-specific power (p)(A) Simulated jump heights of males (blue) and females (orange) In allsimulations the take-off anglewas 65 deg and the duration of pushing against theground remained constant at 1 ms The take-off speed was increased from 0 to4 m sminus1 to increase the power of the jumpWe then found the jumpheight for eachsimulation as a function of the body mass-specific power In order to achievesimilar heightsmalesmust exertamass-specific power larger than that of females(eg to reacha jumpheightof12 cmdashedgray linemaleshave toexertamass-specific power of 44 kW kgminus1 ie 15 times the mass-specific power exerted byfemales 29 kW kgminus1) (B) Data from A presented as the ratio of male to femalemass-specific power as a function of simulated jump height The observed meanjump heights for males and females are denoted by the blue and orange circlesrespectively Shaded blue and orange rectangles indicate the interquartile range(IQR 25ndash75) of observed jump heights in males and females respectively

7

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

model depend on the estimates used for body mass andCD the trendin inter-sex difference resulting from the greater effect of airresistance on the smaller males remains consistent and pronouncedThe short duration of pushing against the ground prior to take-off

(lt1 ms) and exceptionally high mass-specific power exerted byA pseudococci during the jump (Table 2) suggest the presence of apower-amplification mechanism Such mechanisms for jumping arecommon in many small insects (Table 2) Elastic energy storage hasbeen suggested for jumping wasps based on musculoskeletalanatomy (Gibson 1986) but to the best of our knowledge our reportis the first quantitative measurement of jumping performance for awasp using legged power amplification Take-off jumps have been

reported in hymenopteran species before (Burrows and Dorosenko2017) but the reported jumps were moderate compared with those ofA pseudococci One such example is that of Pteromalus puparum(Burrows and Dorosenko 2017) which has a similar body length(27plusmn001 mm) and mass (10plusmn010 mg) to A pseudococci Themass-specific power exerted by these wasps during a jump was only60 W kgminus1 ie roughly 50-fold lower than the mass-specific powerexerted by our female A pseudococci

Compared with other insects using power amplification forjumping both male and female A pseudococci achieve modestjump heights Their acceleration to the take-off speed and their jumpheight relative to body length however are among the highest

18 A B

C D

16

25

20

y=11134xndash0468R2=09346

y=16216xndash11291R2=09267

y=09027xndash00875R2=09388

y=13188xndash07685R2=0931

y=08311x+00653R2=09403

y=1495xndash18817R2=09311

y=1032xndash06089R2=09295

y=08413x+00981R2=09279

y=11134xndash0468R2=09346

y=12194xndash11842R2=09304

y=1032xndash06089R2=09295

y=07775x+03707R2=0927

15

10

5

0

25

20

15

10

5

0

14

12

10

Obs

erve

d ju

mp

heig

ht (c

m)

8

6

4

2

00 0 5 10 15 20 252 4 6

0 5 10 15 20 25

8 10 12 14 16

0 2 4 6Simulated jump height (cm)

8 10 12 14 16 18

18

16

14

12

10

8

6

4

2

0

Fig 7 Sensitivity analysis to evaluate the effect of variation in body density and drag coefficient on the predicted jumpheight in air from the simulationAll panels show plots of the maximummeasured height versus predicted heights using our estimates (blue) and higher (gray) and lower (orange) values for bodydensity (AB) or the drag coefficient (CD) (AC) Jumps by males (BD) jumps by females

8

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

reported for jumping insects (Table 2) The froghopper Philaenusspumarius is considered one of the most prominent jumpers in theinsect world (Burrows 2003) Table 2 reveals that our wasps requiresimilar values of mass-specific power to those of the froghopper inorder to reach similar accelerations and heights relative to bodylength Remarkably the mean body mass and length of thefroghopper is 180-fold and 52-fold respectively higher than themean bodymass and length of male A pseudococci Thus while thetwo insects achieve a similar take-off performance the air resistanceis much more substantial for the wasps resulting in a decrease intheir jumping performance For example if we use the data given inTable 2 and attempt to predict the jump efficiency of the largerfroghopper using the take-off speed of the highest jump and themean jump angle and substituting into Eqn 7 we obtain a value ofsim90 jump efficiency for the froghopper The difference from the51 efficiency of A pseudococci is entirely due to the lower Amratio of the froghopper The acceleration of A pseudococci at take-off (sim260 times gravity in females) is much lower than that in click-beetles (which do not use their legs for jumping) but only sim10lower than the acceleration of the froghoppers (Table 2) Thusdespite (or perhaps because of ) its smaller size A pseudococci is aprominent jumper that nevertheless demonstrates the constraints ofmoving through air on a small creature

APPENDIXThe instantaneous height above the take-off point of a projectile iscalculated numerically from the speed and air resistance according to

hethtTHORN frac14 hethtdtTHORN thorn vzethtTHORNdt ethA1THORNwhere dt is the time increment (step) in the simulation and vz(t) is theinstantaneous vertical speed calculated as

vzethtTHORN frac14 vzethtdtTHORN thorn azethtTHORNdt ethA2THORNwhere az(t) is the instantaneous vertical acceleration calculated as

azethtTHORN frac14 g ethDethtTHORN=mTHORN sin uethtTHORN ethA3THORNSimilarly the instantaneous horizontal distance (x) moved

through the air is

xethtTHORN frac14 xethtdtTHORN thorn vxethtTHORNdt ethA4THORNwhere vx(t) is the instantaneous horizontal speed calculated as

vxethtTHORN frac14 vxethtdtTHORN thorn axethtTHORNdt ethA5THORNwhere ax(t) is the instantaneous horizontal acceleration found from

axethtTHORN frac14 ethDethtTHORN=mTHORN cos uethtTHORN ethA6THORNThe instantaneous angle θ(t) in Eqns A3 and A6 is

uethtTHORN frac14 tan1 vzethtTHORNvxethtTHORN

ethA7THORN

AcknowledgementsWe thank Shimon Steinberg from Bio-Bee Biological Systems for providing theinsects used in this study Eyal Dafni provided invaluable assistance with theexperiments The late Prof Dan Gerling contributed many useful insights that led tothe conceptualization of the study

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization TU GR Methodology TU GR Software TU GRValidation TU GR Formal analysis TU Investigation TU GR Resources

GR Data curation TU GR Writing - original draft TU GR Writing - review ampediting TU GR Visualization TU Supervision GR Project administrationGR Funding acquisition GR

FundingThis research received no specific grant from any funding agency in the publiccommercial or not-for-profit sectors

Data availabilityData used in the study and code to simulate jumps are available for download fromhttpsgalribakweeblycomcodesndashmultimediahtml

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb177600supplemental

ReferencesAlexander R M (1995) Leg design and jumping technique for humans other

vertebrates and insects Philos Trans R Soc Lond B Biol Sci 347 235-248Bennet-Clark H C (1977) Scale effects in jumping animals In Scale Effects in

Animal Locomotion (ed T J Pedley) pp 185-201 London Academic PressBennet-Clark H C and Alder G M (1979) The effect of air resistance on the

jumping performance of insects J Exp Biol 82 105-121Bennet-Clark H C and Lucey E C A (1967) The jump of the flea a study of the

energetics and a model of the mechanism J Exp Biol 47 59-76Brackenbury J andWang R (1995) Ballistics and visual targeting in flea-beetles

(Alticinae) J Exp Biol 198 1931-1942Breach D R (1961) Slow flow past ellipsoids of revolution J Fluid Mech 10

306-314Burrows M (2003) Froghopper insects leap to new heights Nature 424 509Burrows M (2006) Jumping performance of froghopper insects J Exp Biol 209

4607-4621Burrows M (2007) Kinematics of jumping in leafhopper insects (Hemiptera

Auchenorrhyncha Cicadellidae) J Exp Biol 210 3578-3589Burrows M (2011) Jumping mechanisms and performance of snow fleas

(Mecoptera Boreidae) J Exp Biol 214 2362-2374Burrows M and Dorosenko M (2014) Jumping mechanisms in lacewings

(Neuroptera Chrysopidae and Hemerobiidae) J Exp Biol 217 4252-4261Burrows M and Dorosenko M (2017) Take-off mechanisms in parasitoid wasps

J Exp Biol 220 3812-3825Card G and Dickinson M H (2008) Performance trade-offs in the flight initiation

of Drosophila J Exp Biol 211 341-353DeBach P and Rosen D (1991) Biological Control by Natural Enemies 2nd edn

New York Cambridge University PressEarls K D (2000) Kinematics and mechanics of ground take-off in the starling

Sturnis Vulgaris and the Quail Coturnix coturnix J Exp Biol 203 725-739Evans M E G (1972) The jump of the click beetle (Coleoptera Elateridae)- a

preliminary study J Zool 167 319-336Evans M E G (1973) The jump of the click beetle (Coleoptera Elateridae)-

energetics and mechanics J Zool 169 181-194FatourosNE andHuigensME (2012) Phoresy in the field natural occurrenceof

Trichogramma egg parasitoids on butterflies and moths BioControl 57 493-502Fleagle J G (1976) Locomotion and posture of the malayan siamang and

implications for hominoid evolution Folia Primatol 26 245-269Frantsevich L (2004) Righting kinematics in beetles (Insecta Coleoptera)

Arthropod Struct Dev 33 221-235Gibson G A P (1986) Mesothoracic skeletomusculature and mechanics of flight

and jumping in Eupelminae (Hymenoptera Chalcidoidea Eupelmidae) CanEntomol 118 691-728

Gronenberg W (1996) Fast actions in small animals springs and clickmechanisms J Comp Physiol A 178 727-734

Happel J and Brenner H (1983) Low Reynolds Number HydrodynamicsMartinus Nijhoff Publishers The Hague

Hedrick T L (2008) Software techniques for two- and three-dimensional kinematicmeasurements of biological and biomimetic systemsBioinspir Biomim 3 34001

Heppner F H and Anderson J G T (1985) Leg thrust important in flight take-offin the pigeon J Exp Biol 114 285-288

Houck M A andOConnor B M (1991) Ecological and evolutionary significanceof Phoresy in the Astigmata Annu Rev Entomol 36 611-636

Jackson R R and Pollard S D (1996) Predatory behavior of jumping spidersAnnu Rev Entomol 41 287-308

Kittel C Knight W D Ruderman M A Helmholz A C and Moyer B J(1973) Mechanics New York McGraw-Hill

Kohlsdorf T and Navas C A (2007) Evolution of jumping capacity inTropidurinae lizards does habitat complexity influence obstacle-crossingability Biol J Linn Soc 91 393-402

Kutsch W and Fuchs U (2000) Locust flight initiation a comparison of normaland artificial release Physiol Entomol 25 370-382

9

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

model depend on the estimates used for body mass andCD the trendin inter-sex difference resulting from the greater effect of airresistance on the smaller males remains consistent and pronouncedThe short duration of pushing against the ground prior to take-off

(lt1 ms) and exceptionally high mass-specific power exerted byA pseudococci during the jump (Table 2) suggest the presence of apower-amplification mechanism Such mechanisms for jumping arecommon in many small insects (Table 2) Elastic energy storage hasbeen suggested for jumping wasps based on musculoskeletalanatomy (Gibson 1986) but to the best of our knowledge our reportis the first quantitative measurement of jumping performance for awasp using legged power amplification Take-off jumps have been

reported in hymenopteran species before (Burrows and Dorosenko2017) but the reported jumps were moderate compared with those ofA pseudococci One such example is that of Pteromalus puparum(Burrows and Dorosenko 2017) which has a similar body length(27plusmn001 mm) and mass (10plusmn010 mg) to A pseudococci Themass-specific power exerted by these wasps during a jump was only60 W kgminus1 ie roughly 50-fold lower than the mass-specific powerexerted by our female A pseudococci

Compared with other insects using power amplification forjumping both male and female A pseudococci achieve modestjump heights Their acceleration to the take-off speed and their jumpheight relative to body length however are among the highest

18 A B

C D

16

25

20

y=11134xndash0468R2=09346

y=16216xndash11291R2=09267

y=09027xndash00875R2=09388

y=13188xndash07685R2=0931

y=08311x+00653R2=09403

y=1495xndash18817R2=09311

y=1032xndash06089R2=09295

y=08413x+00981R2=09279

y=11134xndash0468R2=09346

y=12194xndash11842R2=09304

y=1032xndash06089R2=09295

y=07775x+03707R2=0927

15

10

5

0

25

20

15

10

5

0

14

12

10

Obs

erve

d ju

mp

heig

ht (c

m)

8

6

4

2

00 0 5 10 15 20 252 4 6

0 5 10 15 20 25

8 10 12 14 16

0 2 4 6Simulated jump height (cm)

8 10 12 14 16 18

18

16

14

12

10

8

6

4

2

0

Fig 7 Sensitivity analysis to evaluate the effect of variation in body density and drag coefficient on the predicted jumpheight in air from the simulationAll panels show plots of the maximummeasured height versus predicted heights using our estimates (blue) and higher (gray) and lower (orange) values for bodydensity (AB) or the drag coefficient (CD) (AC) Jumps by males (BD) jumps by females

8

RESEARCH ARTICLE Journal of Experimental Biology (2018) 221 jeb177600 doi101242jeb177600

Journal

ofEx

perim

entalB

iology

reported for jumping insects (Table 2) The froghopper Philaenusspumarius is considered one of the most prominent jumpers in theinsect world (Burrows 2003) Table 2 reveals that our wasps requiresimilar values of mass-specific power to those of the froghopper inorder to reach similar accelerations and heights relative to bodylength Remarkably the mean body mass and length of thefroghopper is 180-fold and 52-fold respectively higher than themean bodymass and length of male A pseudococci Thus while thetwo insects achieve a similar take-off performance the air resistanceis much more substantial for the wasps resulting in a decrease intheir jumping performance For example if we use the data given inTable 2 and attempt to predict the jump efficiency of the largerfroghopper using the take-off speed of the highest jump and themean jump angle and substituting into Eqn 7 we obtain a value ofsim90 jump efficiency for the froghopper The difference from the51 efficiency of A pseudococci is entirely due to the lower Amratio of the froghopper The acceleration of A pseudococci at take-off (sim260 times gravity in females) is much lower than that in click-beetles (which do not use their legs for jumping) but only sim10lower than the acceleration of the froghoppers (Table 2) Thusdespite (or perhaps because of ) its smaller size A pseudococci is aprominent jumper that nevertheless demonstrates the constraints ofmoving through air on a small creature

APPENDIXThe instantaneous height above the take-off point of a projectile iscalculated numerically from the speed and air resistance according to

hethtTHORN frac14 hethtdtTHORN thorn vzethtTHORNdt ethA1THORNwhere dt is the time increment (step) in the simulation and vz(t) is theinstantaneous vertical speed calculated as

vzethtTHORN frac14 vzethtdtTHORN thorn azethtTHORNdt ethA2THORNwhere az(t) is the instantaneous vertical acceleration calculated as

azethtTHORN frac14 g ethDethtTHORN=mTHORN sin uethtTHORN ethA3THORNSimilarly the instantaneous horizontal distance (x) moved

through the air is

xethtTHORN frac14 xethtdtTHORN thorn vxethtTHORNdt ethA4THORNwhere vx(t) is the instantaneous horizontal speed calculated as

vxethtTHORN frac14 vxethtdtTHORN thorn axethtTHORNdt ethA5THORNwhere ax(t) is the instantaneous horizontal acceleration found from

axethtTHORN frac14 ethDethtTHORN=mTHORN cos uethtTHORN ethA6THORNThe instantaneous angle θ(t) in Eqns A3 and A6 is

uethtTHORN frac14 tan1 vzethtTHORNvxethtTHORN

ethA7THORN

AcknowledgementsWe thank Shimon Steinberg from Bio-Bee Biological Systems for providing theinsects used in this study Eyal Dafni provided invaluable assistance with theexperiments The late Prof Dan Gerling contributed many useful insights that led tothe conceptualization of the study

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization TU GR Methodology TU GR Software TU GRValidation TU GR Formal analysis TU Investigation TU GR Resources

GR Data curation TU GR Writing - original draft TU GR Writing - review ampediting TU GR Visualization TU Supervision GR Project administrationGR Funding acquisition GR

FundingThis research received no specific grant from any funding agency in the publiccommercial or not-for-profit sectors

Data availabilityData used in the study and code to simulate jumps are available for download fromhttpsgalribakweeblycomcodesndashmultimediahtml

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb177600supplemental

ReferencesAlexander R M (1995) Leg design and jumping technique for humans other

vertebrates and insects Philos Trans R Soc Lond B Biol Sci 347 235-248Bennet-Clark H C (1977) Scale effects in jumping animals In Scale Effects in

Animal Locomotion (ed T J Pedley) pp 185-201 London Academic PressBennet-Clark H C and Alder G M (1979) The effect of air resistance on the

jumping performance of insects J Exp Biol 82 105-121Bennet-Clark H C and Lucey E C A (1967) The jump of the flea a study of the

energetics and a model of the mechanism J Exp Biol 47 59-76Brackenbury J andWang R (1995) Ballistics and visual targeting in flea-beetles

(Alticinae) J Exp Biol 198 1931-1942Breach D R (1961) Slow flow past ellipsoids of revolution J Fluid Mech 10

306-314Burrows M (2003) Froghopper insects leap to new heights Nature 424 509Burrows M (2006) Jumping performance of froghopper insects J Exp Biol 209

4607-4621Burrows M (2007) Kinematics of jumping in leafhopper insects (Hemiptera

Auchenorrhyncha Cicadellidae) J Exp Biol 210 3578-3589Burrows M (2011) Jumping mechanisms and performance of snow fleas

(Mecoptera Boreidae) J Exp Biol 214 2362-2374Burrows M and Dorosenko M (2014) Jumping mechanisms in lacewings

(Neuroptera Chrysopidae and Hemerobiidae) J Exp Biol 217 4252-4261Burrows M and Dorosenko M (2017) Take-off mechanisms in parasitoid wasps

J Exp Biol 220 3812-3825Card G and Dickinson M H (2008) Performance trade-offs in the flight initiation

of Drosophila J Exp Biol 211 341-353DeBach P and Rosen D (1991) Biological Control by Natural Enemies 2nd edn

New York Cambridge University PressEarls K D (2000) Kinematics and mechanics of ground take-off in the starling

Sturnis Vulgaris and the Quail Coturnix coturnix J Exp Biol 203 725-739Evans M E G (1972) The jump of the click beetle (Coleoptera Elateridae)- a

preliminary study J Zool 167 319-336Evans M E G (1973) The jump of the click beetle (Coleoptera Elateridae)-

energetics and mechanics J Zool 169 181-194FatourosNE andHuigensME (2012) Phoresy in the field natural occurrenceof

Trichogramma egg parasitoids on butterflies and moths BioControl 57 493-502Fleagle J G (1976) Locomotion and posture of the malayan siamang and

implications for hominoid evolution Folia Primatol 26 245-269Frantsevich L (2004) Righting kinematics in beetles (Insecta Coleoptera)

Arthropod Struct Dev 33 221-235Gibson G A P (1986) Mesothoracic skeletomusculature and mechanics of flight

and jumping in Eupelminae (Hymenoptera Chalcidoidea Eupelmidae) CanEntomol 118 691-728

Gronenberg W (1996) Fast actions in small animals springs and clickmechanisms J Comp Physiol A 178 727-734

Happel J and Brenner H (1983) Low Reynolds Number HydrodynamicsMartinus Nijhoff Publishers The Hague

Hedrick T L (2008) Software techniques for two- and three-dimensional kinematicmeasurements of biological and biomimetic systemsBioinspir Biomim 3 34001

Heppner F H and Anderson J G T (1985) Leg thrust important in flight take-offin the pigeon J Exp Biol 114 285-288

Houck M A andOConnor B M (1991) Ecological and evolutionary significanceof Phoresy in the Astigmata Annu Rev Entomol 36 611-636

Jackson R R and Pollard S D (1996) Predatory behavior of jumping spidersAnnu Rev Entomol 41 287-308

Kittel C Knight W D Ruderman M A Helmholz A C and Moyer B J(1973) Mechanics New York McGraw-Hill

Kohlsdorf T and Navas C A (2007) Evolution of jumping capacity inTropidurinae lizards does habitat complexity influence obstacle-crossingability Biol J Linn Soc 91 393-402

Kutsch W and Fuchs U (2000) Locust flight initiation a comparison of normaland artificial release Physiol Entomol 25 370-382

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reported for jumping insects (Table 2) The froghopper Philaenusspumarius is considered one of the most prominent jumpers in theinsect world (Burrows 2003) Table 2 reveals that our wasps requiresimilar values of mass-specific power to those of the froghopper inorder to reach similar accelerations and heights relative to bodylength Remarkably the mean body mass and length of thefroghopper is 180-fold and 52-fold respectively higher than themean bodymass and length of male A pseudococci Thus while thetwo insects achieve a similar take-off performance the air resistanceis much more substantial for the wasps resulting in a decrease intheir jumping performance For example if we use the data given inTable 2 and attempt to predict the jump efficiency of the largerfroghopper using the take-off speed of the highest jump and themean jump angle and substituting into Eqn 7 we obtain a value ofsim90 jump efficiency for the froghopper The difference from the51 efficiency of A pseudococci is entirely due to the lower Amratio of the froghopper The acceleration of A pseudococci at take-off (sim260 times gravity in females) is much lower than that in click-beetles (which do not use their legs for jumping) but only sim10lower than the acceleration of the froghoppers (Table 2) Thusdespite (or perhaps because of ) its smaller size A pseudococci is aprominent jumper that nevertheless demonstrates the constraints ofmoving through air on a small creature

APPENDIXThe instantaneous height above the take-off point of a projectile iscalculated numerically from the speed and air resistance according to

hethtTHORN frac14 hethtdtTHORN thorn vzethtTHORNdt ethA1THORNwhere dt is the time increment (step) in the simulation and vz(t) is theinstantaneous vertical speed calculated as

vzethtTHORN frac14 vzethtdtTHORN thorn azethtTHORNdt ethA2THORNwhere az(t) is the instantaneous vertical acceleration calculated as

azethtTHORN frac14 g ethDethtTHORN=mTHORN sin uethtTHORN ethA3THORNSimilarly the instantaneous horizontal distance (x) moved

through the air is

xethtTHORN frac14 xethtdtTHORN thorn vxethtTHORNdt ethA4THORNwhere vx(t) is the instantaneous horizontal speed calculated as

vxethtTHORN frac14 vxethtdtTHORN thorn axethtTHORNdt ethA5THORNwhere ax(t) is the instantaneous horizontal acceleration found from

axethtTHORN frac14 ethDethtTHORN=mTHORN cos uethtTHORN ethA6THORNThe instantaneous angle θ(t) in Eqns A3 and A6 is

uethtTHORN frac14 tan1 vzethtTHORNvxethtTHORN

ethA7THORN

AcknowledgementsWe thank Shimon Steinberg from Bio-Bee Biological Systems for providing theinsects used in this study Eyal Dafni provided invaluable assistance with theexperiments The late Prof Dan Gerling contributed many useful insights that led tothe conceptualization of the study

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization TU GR Methodology TU GR Software TU GRValidation TU GR Formal analysis TU Investigation TU GR Resources

GR Data curation TU GR Writing - original draft TU GR Writing - review ampediting TU GR Visualization TU Supervision GR Project administrationGR Funding acquisition GR

FundingThis research received no specific grant from any funding agency in the publiccommercial or not-for-profit sectors

Data availabilityData used in the study and code to simulate jumps are available for download fromhttpsgalribakweeblycomcodesndashmultimediahtml

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb177600supplemental

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