Scaling bat wingbeat frequency and amplitude

Chiroptera cover both tropical and temperate regions and reach high latitudes. One of the main constraints on their geographical radiation is their energy balance. The daily cycle of energy expenditure in bats is dominated by the cost of foraging flight, which is a function of their aerodynamics and, hence, their wingbeat frequency and amplitude across the range of normal flight speeds. These data are only available for a few species(Norberg et al., 1993; Aldridge, 1986; Van Den Berg and Rayner, 1995; Britton et al., 1997; Carpenter, 1986; Norberg, 1976). To improve our understanding of bat energetics, a general model is required that is scaled to a readily available parameter such as species mass and that encompasses bats from an array of climatic zones and with a range of foraging strategies.

In mammals, the duration of a muscle's contraction is adapted to its function, and the contraction performance of the muscle is affected by the resistance that it works against (Guyton and Hall, 1996). If there were gross variations in the speed of operation of the muscles driving the wingbeat of bats with different phylogenetic relationships, foraging strategies or microhabitats, then we would expect the relationships between wingbeat frequency(f_w) and airframe variables (such as mass, wing area and wing span) to be complex.

In this study, we measure wingbeat frequency and amplitude across a range of flight speeds for 23 species representing all families of insectivorous,frugivorous and carnivorous bats that occur in tropical and temperate regions of Western Australia. We then propose a general model linking these variables to various airframe attributes and flight speed.

The 23 species of Australian bat assessed in this study are listed below,together with authorities and synonyms in cited references. Table 1 gives morphological parameters and Table 2,foraging niche, climatic range and phylogeny, for each species.

Chalinolobus gouldii Grey; Chalinolobus morio Grey; Chalinolobus nigrogriseus Gould; Hipposideros ater,Templeton; Macroderma gigas Dobson; Mormopterus planicepsPeters; Miniopterus schreibersii Kuhl; Nyctophilus arnhemensis Johnson; Nyctophilus geoffroyi Leach; Nyctophilus gouldi Tomes; Nyctophilus timoriensis Geoffroy; Pteropus poliocephalus Temminck; Pteropus scapulatus Peters; Rhinonycteris aurantius Grey; Scotorepens balstoni Thomas; Saccolaimus flaviventris Peters; Scotorepens greyii Grey(previously Nycticeius balstoni caprenus Troughton); Tadarida australis Grey; Taphozous georgianus Thomas; Taphozous hilli Kitchener; Vespadelus finlaysoni Kitchener, Jones and Caputi (previously Eptesicus pumilis Gray); Vespadelus regulus Thomas.

Data for two different populations of Nyctophilus timoriensis(Nt_g and Nt_s/w) are presented and treated as separate species. The arid population Nt_g inhabits the Coolgardie woodlands and has a mean mass of 11 g, whereas the mesic population Nt_s/w is endemic to the forests of southwestern Western Australia and has a mean mass of 14.2 g.

Relevant aspects of species foraging ecologies and airframe measures were collected from existing literature. Only publications using a consistent measurement technique were used. This measurement protocol, relevant formulae and a discussion of aerodynamic mechanisms and implications are provided in Bullen and McKenzie (2001). Capture and release techniques were used to collect a library of video recordings complete with flight speed measurements. Bat flight was filmed using video cameras (Sony Video8 Professional CCD-V100E in VHS format and Sony digital Beta-cam model DVW-709WSP at a shutter speed of 1/250 s), both running at 24 frames s^-1. Wingbeat frequency and amplitude values were determined from a frame-by-frame playback. Note that the Beta format video actually showed two clear images of the bats wing position when replayed via VHS because of the differences in the recording protocols of the two video standards. It gives an effective frame rate of 48 frames s^-1 for test points recorded with the DigiBeta camera. Limited data were also collected in an indoor observation chamber at high frame rates using a cine camera running at 200 frames s^-1 (Photosonics; Burbank, CA,USA; model 61-1100).

Flight speeds were measured continuously in all cases using a hand-held K-band radar gun (model TS3, Municipal Electronics, UK, calibrated for a speed range of 1-28 m s^-1). These speeds were `called' into a hand-held recorder and, if applicable, into the audio feature of the video camera while each test animal was being filmed. For each test, the angle between the gun's line of sight to the bat and the bat's line of flight was estimated by the operator and `called' into the recorder. A cosine correction was applied to the measured flight speeds to correct for this angle. Data corresponding to angles greater than 45° were ignored.

The mean angle of the wing between shoulder and tip, above or below the body axis reference dorsal plane, was estimated within ±5° for each frame in sequence. Note that this method is different from that used by Pennycuick (1996) on birds. Pennycuick (1996) estimated the angle created by the shoulder-to-wrist joint line only. A bat's hand wings reaches higher positional angles than its arm wing at the end of the stroke,so our method gives higher amplitude values. These were plotted against time(Fig. 1) and fitted with splines using Microsoft EXCEL. A minimum of three complete wingbeat cycles was required to calculate frequency, f_w, and amplitude,θ _w, reliably. At 24 frames s^-1, a family of lines can be fitted to the sequences, differing in their frequencies by a factor of 3. Given that bats with a mass of less than 50 g are known to use f_w in the range of 3-12 Hz(Carpenter, 1985; Van Den Berg and Rayner,1995), the curve with the lowest f_w was used for all species because its frequency always fell within this range. This also agrees with our own high frame rate data. The f_winformation was then deduced directly from the time histories. The spline for each low frame rate (VHS at 24 frames s^-1) test point was reviewed to obtain amplitude. The maximum and minimum amplitudes were then averaged and compared to give θ_w values for each test point. Given that the test points were all taken during periods approximating steady level flight, peaks that were clearly out of phase with the sequence were ignored in this average (see Fig. 1B). This method is expected to give maximum and minimum θ_w values slightly lower in magnitude than those obtained from high frame rate (>100 frames s^-1) cine cameras. Because of the impracticability of extensive use of high-frame-rate cine in the majority of our field experimental situations, this was not attempted, and low-frame-rate video was used to maximise data collection. See discussion below of the effect of this procedure on the results.

Sub-adults, pregnant females and animals with damaged wings, or that were visibly distressed or considered significantly underweight, were excluded. The methods used did not result in injury to or the death of the bats tested.

Four strategies were used to collect wingbeat data over a wide range of flight speeds. First, bats were flown in a flight chamber to collect low-speed data. Individual adult bats were released to fly around in a large, well-lit room (11 m long, 5 m wide and 3.2 m high). All species were able to maintain continuous level flight in this room. Although Mormopterus planiceps,Chalinolobus gouldii and Tadarida australis did not achieve their typical in-field flight speeds (see Bullen and McKenzie, 2001),they were flying 0.3-3 m s^-1 (1-10 km h^-1) above their usual minimum steady level flight speed (R. D. Bullen and N. L. McKenzie,unpublished data). Thus, they had a considerable margin of power for manoeuvring. The floor, ceiling and walls of the room were painted in shades of white or cream, which contrasted with the brown and black colours of the fur and wing membranes of the bats. This method gave excellent coverage of the lower speed range of the bats.

Second, free-air hand releases in daylight were used to collect mid- and high-speed data. The same video and speed measuring equipment was used. The bats were prone to escape after release by accelerating to high speed. Results were most readily obtained when the released bat was filmed against bright,monotonous backgrounds such as grass or sky. The initial period of 2-3 s,while bats accelerated from rest to their normal flight speed range, was excluded from data analysis. If, during the test point, the speed of the bat varied marginally (typically less than ±1 m s^-1), then the speed at the mid-point of the run was taken as the average value for that run. If the speed varied by more that ±1 m s^-1 then the run was broken into two or more test points. All readings for a species were pooled.

Third, daytime free-flying data were collected from large pteropodids as they commuted from roost to roost. Because of their size, it was possible to film the bats in flight in full daylight and to record their speed. Again,cosine corrections were applied to the measured flight speeds, as described above, to account for the off line-of-sight measurement errors.

Fourth, night-time free-flight data were also collected to supplement the first two strategies and to check whether different wingbeat values were obtained in a natural situation. Echolocation recordings were taken from free-flying bats in situations while the foraging bat could be seen, lines of flight estimated and speeds measured. An Anabat II ultrasound detector (Titley Electronics, Australia) was used with its output stored directly onto audiocassette tapes using a Sony Walkman Professional (WMD6C) tape recorder. The species identity and f_w values were then derived from the recorded call sequences using COOL EDIT 2000 (Syntrillium Software, USA). The species was identified by reference to a library of reference calls, and the f_w data were derived based on a direct correlation of the wingbeat frequency with the echolocation call rate(Lancaster et al., 1995). These sequences were not filmed and did not provide data onθ _w.

Of the 23 species represented in this study, 11 provided data over the flight speed range of 3-9 m s^-1 that is the majority of their speed range. Seven species provided data over the range less than 6 m s^-1, covering their low-speed range only, and five provided data at speeds greater than 5 m s^-1, which is their high-speed range only. Despite having scant or incomplete data sets, these last two categories were included to assess whether the generalised scaling model applied to all types of bat.

The f_w and θ_w data for each species were then plotted against flight speed (refer, for an example plot, to Fig. 2) and the plots reviewed for a general pattern.

Maximum range speed, V_mr, was calculated and used as a reference point for low-speed flight using a quasi-steady aerodynamic model that follows the method of Pennycuick(1989). The calculated values are included in Table 3. V_mode, the `mode' speed of the test data(Bullen and McKenzie, 2001) was estimated empirically from the data to represent the divide between low and high-speed flight. Means and standard deviations for f_wcorresponding to V_mr ±1.0 m s^-1 were calculated and plotted against mass. A series of forward stepwise least-squares regression curves was tested against the frequency and flight speed data (STATISTICA SoftStat). A range of relevant morphological variables,including mass, span and wing area, was assessed as independent variables. Linear, polynomial and logarithmic variants were assessed for explaining the variation. Statistically significant relationships between θ_wand the available morphological variables were also sought.

Previously published wingbeat data for a number of other species are included for comparison: Eidolon helvum (m=315 g),high-speed cine data (Carpenter,1986); Hypsignathus monstrosus (m=260 g),high-speed cine data (Carpenter,1986); Myotis dasycneme (m=20 g), stroboscopic flash data at 30 Hz (Britton et al.,1997); Noctilio leporinus (m=70 g), synchronised cameras at 20 frames s^-1(Schnitzler et al., 1994); Pipistrellus pipistrellus (m=5 g), high-speed video at 250 frames s^-1 (Thomas et al.,1990); Pteropus poliocephalus (m=700 g), manual and high-speed cine data (Carpenter,1985); Rhinolophus ferrumequinum (m=22 g),stroboscopic flash data at 100 and 200 Hz(Aldridge, 1986); Rousettus aegyptiacus (m=180 g), high-speed cine data(Carpenter, 1986).

Data on foraging ecology and airframe variables were available for Western Australian populations of 22 species and for a southeastern Australian population of Pteropus poliocephalus. Morphological variables are listed in Table 1. The foraging strategy, biogeographical and phylogenetic data for the species are given in Table 2 (terms are defined in Bullen and McKenzie, 2001). Table 3 provides calculated V_mr and measured V_mode flight speeds for each species.

An example of the f_w and θ_w data for Chalinolobus gouldii is given in Fig. 2 and a second example for f_w, Mormopterus planiceps, is given in Fig. 6. For all 11 species assessed over a wide range of flight speed, f_w initially decreased with increasing speed until a mid-range speed was reached. f_w then remained relatively constant until high speeds were achieved. This pattern can also be seen clearly in Figs 2A and 6. Regarding amplitude, Fig. 2 also illustrates a pattern common to all 11 species: θ_w increased with speed,although there was very wide scatter.

Provided bat mass is known, the wingbeat frequency for any bat can be estimated at low or high flight speed to within ±1.5 Hz.

This model is presented in Fig. 4.

Equation 3 is presented in Fig. 5 for Chalinolobus gouldii, for comparison with the data of Fig. 2.

The two outliers in Fig. 3Bhave high-speed f_w values significantly below the line represented by this equation. They are Mormopterus planiceps (this study) and Noctilio leporinus(Schnitzler et al., 1994),which have f_w values corresponding to approximately 65 %of the value predicted by Equation 2. Our empirical data on Mormopterus planiceps are presented in Fig. 6. The Mormopterus planiceps outlier is a series of low f_w points treated separately. The upper series in Fig. 6 is accurately represented by the scaling equations.

A summary of the data included in the study is given in Table 4.

Equation 4 is presented in Fig. 5 for Chalinolobus gouldii for comparison with the data of Fig. 2.

The maximum and minimum values of θ_w recorded during the study are presented in Table 5for each species. They are much higher than the extrapolations based on Equation 4. This is to be expected given that pectoral girdle anatomy clearly permits very high θ_w values to be used for extreme speeds beyond our data set as well as during manoeuvres and periods of high acceleration when extra power is required. For the reasons given in the legend to Fig. 2, low flight speed test points at amplitudes approaching the maximum and minimum values were observed (typically +40 and -80°), but were not included in the derivation of Equation 4.

We have derived a general model for calculating wingbeat frequency and amplitude of bats in their usual speed range. The f_wequations require only mass and flight speed to provide accurate estimates for a species. The θ_w equation is less accurate and requires flight speed and wing area. These equations apply to steady, level flight in the range of speeds given in Table 4, but exclude the high and low extremes of the bat's speed range. They apply to bats from a wide range of environments and with very different foraging strategies and phylogenetic affiliations.

Strong family-level phylogenetic relationships are apparent in our results. Vespertilionidae are at one end of the relationship, Emballonuridae in the middle and Pteropodidae at the opposite end. Given the high selective pressures on morphological characters associated with modes of nutrition and foraging, phylogeny in the absence of ecologically driven aerodynamic function would be highly unlikely to produce optimum aerodynamic functionality. The similarity in aerodynamic optimisations apparent in the various families of bats presented here is consistent with the assumption that the model linking morphological variables to flight speed and wingbeat kinematics is functionally based, rather than an aerodynamically trivial artefact of phylogenetic relationships (Felsenstein,1982; McKenzie et al.,1995b).

The pooled species data included in this study show f_wvalues ranging from 3 to 12 Hz. Species with f_w values up to 12 Hz probably have downstroke and upstroke muscles composed of the fast fibre type (Guyton and Hall,1996). In species with f_w values down to 3 Hz,it is possible that the slow fibre type dominates. This hypothesis is based on our observation that the larger bats with slow wingbeat frequencies are those that are known to travel long distances (Pteropus poliocephalus,refer to Churchill, 1998; Saccolaimus flaviventris, refer to Strahan, 1995; Tadarida australis, refer to Churchill,1998) or migrate on an annual cycle. Further work is recommended to confirm this suggestion.

The variation of f_w with flight speed for these 23 species shows a two-stage characteristic that is reflected by the need for a logarithmic equation (e.g. Fig. 6). At low speeds, f_w changes with flight speed, whereas at higher speeds it is nearly constant. This is consistent with the rule that only a limited range of wingbeat frequencies are available to a species (Rayner, 1985) to provide the endurance required for long sustained flights. Initially, f_w decreases as flight speed increases, until cruising speed ranges are reached (see mode speed data in Table 3). The reduction is small, approximately 2 Hz. At and above cruising speed, f_wappears to remain almost constant until the bats reach their extreme high speed (e.g. Figs 2 and 6). In contrast to bats, bird flight is more variable. Some birds show no change with speed(Tobalske and Dial, 1996),others show a more-or-less linear relationship with speed(Tobalske, 1995) and still others show a U-shaped relationship, with an initial fall in f_w with increasing speed followed by an increase of f_w at even higher speeds(Bruderer et al., 2001; Park et al., 2001). The two-stage relationship in bats differs from published bird data. This almost constant relationship between f_w and flight speed at high speed in the bats we have assessed may be due to use of their most efficient muscle contraction frequency in the flight speed region of rapidly increasing`opposing loading' and, therefore, metabolic power requirement. The opposing loading applied to the muscles is due to the rapidly increasing drag airloads.

For low-speed f_w data, the model predicts a value that lies in the centre of the scatter of the available empirical data for 19 species (Fig. 3A). In fact,empirical values are within 1 Hz (approximately 1 S.D.) of the fitted model across the speed range. Three of the remaining species show a bias when the model is compared with the data (Mormopterus planiceps, Noctilio leporinus and Nyctophilus timoriensis_s/w). The model underestimates f_w data for Nyctophilus timoriensis_s/w (this study) and overestimates Noctilio leporinus (Schnitzler et al.,1994) by approximately 1 Hz. However, our data on Nyctophilus timoriensis_s/w are scant and the apparent bias may disappear with more data. The model overestimates the frequency data for the `low range'of Mormopterus planiceps (referred to as `outliers' in the results). For high-speed f_w data, the model predicts the empirical data for 18 of 19 species, including the high f_w range of Mormopterus planiceps. The low f_w range of Mormopterus planiceps and Noctilio leporinus(Schnitzler et al., 1994) are substantially overestimated. In both species, empirical wingbeat frequencies are approximately 65 % of the value predicted by the model.

Mormopterus planiceps was unique among the 23 species assessed during this study. The empirical f_w data are arrayed in two parallel series across the full flight speed range(Fig. 6). The model predicted the main (higher) frequency series. The other series averaged 3 Hz lower and,provided that it is not a sampling artifact, would reduce the resultant airspeed past the wing during the down- and upstroke. Preliminary calculations indicate a consequent reduction in the profile power fraction of the wing of approximately 4 % and in the inertial power fraction of 67 %. Preliminary dissections by the authors also revealed that Mormopterus planicepshas a very low flight muscle mass to total mass fraction of wing down-stroke and up-stroke muscle groups (Vaughan,1970; Hermanson and Altenbach,1985), approximately 7.5 % of total mass compared with a more typical range of approximately 9-11 % for similar insectivores. Taken together, these observations suggest a particular optimisation of this tiny interceptor, in which the upper f_w series is used for acceleration to speed and for manoeuvring to intercept prey, while the lower f_w range is used for efficient cruising/commuting.

By comparison with birds, bats have a 50 % higher wingbeat frequency for a given size range. Pennycuick(1996) gives a model for bird frequency based upon mass, wing span and wing area. This model is compared with our high-speed bat data in Fig. 8.

We had empirical data on θ_w for 24 species (including Rhinolophus ferrumequinum(Aldridge, 1986)). No species departed substantially from the general model given by Equation 4. However,unlike f_w, all species showed a high level of scatter(±20°) in the raw amplitude data (e.g. Fig. 2), forming effectively two blocks that represent the low- and high-speed experimental data collection strategies (see Fig. 2). The bats were studied in free flight at all times, during which climbing,descending, accelerating and decelerating flight would require differences in lift, drag and thrust. It is possible for the bat to generate lift and thrust by changing the mean wingbeat angle of attack (α), the average airspeed over the wings (V_wing), the wingbeat frequency and/or amplitude. At low angles of attack, lift is directly proportional to the product of α and V_wing². For level flight, when lift exactly equals weight, the bat must change the flow velocity over its wings by changing forward speed, f_w and/orθ _w if it changes its mean wing α during the stroke,otherwise, it will climb or descend. In addition, to increase speed in level flight, the bat must generate more thrust by using higher α, f_w and/or θ_w to offset the increasing drag. Given that we have shown that f_w is relatively constant across the full speed range of bats, α must decrease as flight speed increases (for constant or increasing θ_w) otherwise the bat will generate excess lift and climb. To this end, wingbeat amplitude must increase to generate the increased thrust, resulting in an even higher mean V_wing value and an even lower mean value of α. Equation 4 therefore represents the steady level-flight wingbeat amplitude independent of the experimental context in which the data were collected. Note that the previous study (Aldridge,1986) published data on θ_w of bats over a narrow range of speeds (2.7-4.8 m s^-1) using a flight tunnel. These data fall within the predictions of our equation.

The difference between data recorded at 24 frames s^-1 compared with higher frame rates is given in Table 6 and can be seen in Fig. 1. At high wingbeat frequencies (>9.5 Hz), VHS video camera or a cine camera with 24 frames s^-1 and a slow shutter speed give an accurate representation of the extrema of the wing positional angle, because they occur at a repetition rate that ensures a high probability of the extrema coinciding with the relatively long open shutter/scan time period. These extrema are then used as the frame θ_w value. Similarly, 24 frames s^-1 is sufficient to capture relatively slowly moving wings at low f_w (<7 Hz) within 5° of its maximum position(see Fig. 1C). There is,however, a range of wing frequencies (approximately 7-9.5 Hz) used by bats of 20-50 g to which neither of these situations applies. For these bats, care must be taken to use only the 24 frames s^-1 frame images that clearly show a significant variation in angle from frame to frame. Frames that do not fit this criterion should not be included in averagingθ _w values for the test point. This effect is apparent in the Tadarida australis data of Table 6, which underestimate actual amplitude by 10-20°. Underestimation occurred in four of our bats: Taphozous hilli, Taphozous georgianus, Tadarida australis and Saccolaimus. flaviventris. Even so, this bias is of the same order as the overall scatter in the data collected, and the data for these species have therefore been included in the overall regression analysis. Fig. 7C shows that including these data has little effect on the regression result. Data from Fig. 1A show that this effect is reduced at frame rates of 50 frames s^-1 and is not apparent at rates beyond 100 frames s^-1.

Field observations of bats `hand-released' in daylight(Table 5) suggest that bats approach their amplitude limits of approximately 50-60° above and 80-90° below the reference dorsal plane in extreme flight conditions. This gives a theoretical maximum of 140-150° for wingbeat amplitude at the high speed extreme of the flight speed range, compared with a more typical range of 40-80°. This theoretical maximum will be influenced by the back, shoulder,elbow and wrist morphology of the various species. Given that maximum efficiency in skeletal muscle ordinarily occurs when the velocity of contraction is approximately 30% of maximum(Guyton and Hall, 1996) and that f_w is virtually constant in the bat's upper speed range, our result of a threefold increase in amplitude at extreme speeds is consistent with constant f_w and best use of muscle efficiency in the bat's normal speed range.

The relationships between f_w, θ_w and flight speed are defined by two simple equations involving mass and wing area. The same equations fitted tropical as well as temperate species, megabats and microbats, the six microbat families assessed and species with the full range of foraging ecologies. One scaling model fitted all. Its simplicity implies that a single theme underlies bat aerodynamics. This argument is not circular because no bats showed substantial departures from the model, despite differences in foraging niche, climatic range and phylogeny. In this respect at least, the kinematics of bat flight is different from that of birds.

List of symbols
 
• f_w

  Wingbeat frequency (Hz)

 
• m

  Bat mass (kg)

 
• S_REF

  Wing reference area (m²)

 
• V_mr

  Maximum range flight speed (m s^-1)

 
• V_mode

  Flight speed representing the `mode' speed of the test data (m s^-1)

 
• V_wing

  Resultant airspeed over the wing including contributions from flight speed, wingbeat frequency and amplitude (m s^-1)

 
• α

  Angle of attack (degrees)

 
• θw

  Wingbeat amplitude (degrees)

We thank C. L. Bullen, M. H. McKenzie, W. P. Muir and A. N. Start for field assistance. Daniel Searle, Sarah Neylon and Ingo Helbig of Storyteller Productions, Perth, Western Australia, assisted with the photography. Mike Searle kindly made his visual technology production facilities available to us for data reduction and analysis. The Western Australian Department of Conservation and Land Management contributed to the cost of the project. We also thank two anonymous referees for constructive comments on an earlier manuscript.