Scaling of wingbeat frequency with body mass in bats and limits to maximum bat size

Flying birds span four orders of magnitude in size, from about 1.5 g to around 16 kg (Pennycuick, 2008), whereas bats span only three, from 1.9 g to 1.6 kg (Roberts, 1977). The extinct giant vulture Argentavis had a wingspan of 6.5–7.5 m and weighed about 70 kg (Chatterjee et al., 2007). Pterosaurs ranged from about 4 g to possibly as much as 100 kg (Wellnhofer, 1991). The largest extant flying birds thus weigh about 10 times more than the largest bats and the largest pterosaurs were almost 70 times heavier. By contrast, the lower size limit is strikingly similar for all three groups of flying vertebrates.

Because active flight is functionally complex and energetically very expensive, flying animals require highly advanced morphological adaptations in the structure and function of the skeleton, muscles and cardiorespiratory system. The much smaller maximum size of bats than of birds and pterosaurs might depend on morphological and metabolic constraints, inherent to their mammalian structure.

But the oxygen capacity in flight is about the same in bats as in birds (Maina, 2000). Both have a large heart with huge cardiac output, high erythrocyte volume fraction, high haemoglobin concentration and high blood oxygen-carrying capacity, as well as efficient blood supply to the flight muscles. Bats, however, have a tidally, dead-end ventilated lung, whereas birds have a more efficient one-way, through-flow respiratory system. Maina suggested that this might explain why the largest bats weigh only a tenth the mass of the largest birds (Maina, 2000). But although birds and bats have respiratory organs of fundamentally different structure, they have converged functionally so that bats achieve equally high oxygen uptake rates as birds owing to a remarkably large lung with a large pulmonary capillary blood volume and a remarkably extensive alveolar surface area (Maina, 2000).

Here, we present new data on wingbeat frequencies that we have collected from 27 bat species, 25 of which were in free flight in the field, and also data taken from the literature for 38 additional species, giving a total of 65 species. We fitted power functions to the whole data set as well as to data from bats in free flight in the field only, and to subsets of different bat taxa. A published function for birds is used for comparison. Bats use lower wingbeat frequencies than birds of similar masses. Bats also have lower flight muscle mass than equally large birds. Therefore, we explored the possibility that the lower flight muscle mass in bats limits their wingbeat frequency, which in turn limits the power available for flight. Based on the theory of animal flight, a comparison between geometrically similar animals of different sizes shows that the power required to fly increases faster with increasing body mass than does the power available (Pennycuick, 1975; Pennycuick, 2008). Therefore, the maximum attainable wingbeat frequency, which determines the maximum power available, may limit bat maximum size to only one-tenth of that for birds.

We will now compare the size ranges of extant birds and bats, sorted by diet, to see in which diet categories birds and bats attain similar maximum body sizes and where they do not.

The majority of bats are insectivorous and hunt flying prey. They are all rather small. Capture of flying insects requires high agility and manoeuvrability, which depend on the ability to make swift rolls and tight turns (Andersson and Norberg, 1981; Norberg, 1986; Thollesson and Norberg, 1993). Insect catching also requires fairly slow flight to reduce the distance travelled during the reaction time, from the point of short-range prey detection by echolocation to the initiation of a capture manoeuvre. The shorter this reaction-time flight distance is, and the shorter the minimum turning radius, the farther out from the line of flight the bat can reach insect prey.

The ability to make tight manoeuvres and to fly slowly is favoured by low wing loading (weight divided by wing area, M_bg/S). Among geometrically similar animals the radius r of a balanced turn increases in direct proportion to wing loading, r∝M_bg/S (Norberg and Norberg, 1971; Norberg, 1990), whereas any characteristic flight speed V increases in proportion to the square root of wing loading, V∝(M_bg/S)^1/2. Among geometrically similar animals, wing loading itself increases as M_b^1/3, so V∝(M_bg/S)^1/2∝(M_b/M_b^2/3)^1/2∝M_b^1/6. Therefore, small bats can make tight turns and also fly slowly, whereas large bats with their higher wing loadings are at a disadvantage, which imposes an upper size limit on aerial insect hunters.

It has been suggested that coupling of echolocation calls with the wingbeat cycle might constrain the maximum size of aerial insectivorous bats (Jones, 1994). Because wingbeat frequency decreases with increasing body size, large bats would be unable to echolocate at sufficiently high rates for insect capture. But several insectivorous bats produce more than one echolocation pulse per wingbeat, particularly during the close-in phase at insect capture (Kalko, 1994), so the dependence on echolocation would not seem to limit the maximum body size of aerial insect-feeding bats.

The majority of insectivorous bats weigh less than 50 g, but the largest species of the genus Cheiromeles weigh up to 135 g. By comparison, the majority of birds that forage for insects during continuous flight (as opposed to sallying flights from perches) also weigh less than 50 g, whereas some nightjars and swifts weigh up to 100 g. A few nightjars (Caprimulgidae; genus Eurostopodus and Podager, nighthawk) may weigh up to 180 g and a few swifts (Apodidae; genus Hirundapus, needletail) up to 200 g. These masses, similar for birds and bats, probably indicate an ecologically viable upper size for aerial insectivores.

Nectar-feeding bats use hovering flight extensively. It is extremely power consuming and can be practised only by small flying animals. The power required for hovering, like the power for level flight, increases faster with increasing body mass than the power the muscles can generate. But because hovering requires so much power, sustained hovering (as opposed to brief anaerobic bursts) becomes impossible at a much lower body mass than the maximum mass compatible with cruising flight. Nectar-feeding bats thus need to be small to be able to hover in front of flowers while foraging and also to facilitate manoeuvring flight among vegetation. In addition, in view of the small food reward per visited flower and the large energy cost for getting it, the nectar diet cannot support large flying animals. Nectar-feeding bats weigh between 6.5 and 45 g, whereas their counterparts among birds – the hummingbirds – weigh between 1.5 and 50 g.

The heaviest carnivorous bat weighs about 160 g (Vampyrum spectrum) and fish eaters weigh up to 60 g. They use echolocation to detect ripples on the water surface made by fish, which they catch with hooked claws by skimming the water with their feet as they fly down close to the surface. The largest bird predators of mammals weigh up to 9 kg (the harpy eagle, Harpia harpyja), and the largest fish-eating bird up to 15 kg (the Dalmatian pelican, Pelecanus crispus). The largest raptorial birds are thus 56 times larger than any carnivorous bat, and the largest fish-eating bird is as much as 250 times heavier than the largest fish-eating bat.

The largest bats belong to suborder Megachiroptera, the flying foxes, most of which are mainly frugivores. Only four or five species weigh more than 1 kg. Pteropus giganteus is reported to be the heaviest (1.3–1.6 kg), whereas Acerodon jubatus, Pteropus vampyrus and Pteropus mahagans weigh about 1.2 kg (Nowak, 1991). No megachiropteran bat echolocates in order to find food, so they are not size limited by a phase-lock between wingbeat frequency and echolocation rate. One of the largest flying fruit-eating birds, the Toco toucan (Ramphastos toco), weighs up to 860 g. So the largest fruit-eating bats are a little larger than their avian counterparts.

The largest bat species thus match the body sizes of the largest birds in the diet categories insect eaters, nectar feeders and frugivores. Among carnivores and fish eaters, however, the largest birds weigh 50–250 times more than the largest bats. And there are no bats at all corresponding to albatrosses, bustards, turkeys, cranes, geese, swans, and scavengers like vultures and condors. So bats have undergone much less evolutionary diversification than birds and fill fewer ecological niches.

The basic morphology of bats may restrict their potential to adapt to terrestrial locomotion and swimming. Adaptation to these locomotion modes might be hindered by the wing membrane, which in bats attaches along the entire length of the hindleg, down to the foot. It continues across the tail (however short it is in Megachiroptera), so the hindlegs and the tail are engaged for support of the flight membrane.

Ground-dwelling life and running are indeed rare among bats, but occur in the New Zealand omnivorous and echolocating bat Mystacina tuberculata (12–35 g) (Riskin et al., 2006). The blood-eating Desmodus rotundus (30–40 g) also moves adeptly on the ground. Apart from normal walking in Mystacina and Desmodus, with a forefoot and a hindfoot on opposite sides of the body swinging in unison, D. rotundus also uses a strange jumping gait in which the two arms produce the power stroke in unison, whereas the hindlimbs merely take up the impact force on landing, hindlimbs first, just before the forelimbs touch-down to immediately become engaged in the power stroke (Riskin and Hermanson, 2005). The forelimbs produce the most powerful stroke because they are so much stronger than the hindlimbs, a condition that seems awkward for terrestrial locomotion. But this dilemma must have been shared by pterosaurs – and it did not prevent them from evolving into big sizes.

Pterosaurs also had a flight membrane that obviously attached to the hindlegs. This is particularly obvious in Pterodactylis cochi, whose flight membrane attached almost down to the ankle (Wellnhofer, 1991). But pterosaurs nonetheless evolved very large body sizes and radiated into a wide variety of niches. Pteranodon ingens had an estimated wingspan of 8 m and a mass of 17 kg (Bramwell, 1971), and the largest pterosaur, Quetzalcoatlus northropi, is estimated to have had a wingspan of 11–12 m and a weight of 100 kg (Wellnhofer, 1991).

The largest extinct birds and pterosaurs thus had much larger body masses than the maximum of 16 kg observed among extant birds. It is not known what flight modes these large extinct flyers could manage. Powered flight at such large body masses would require a large proportion of the body mass to be made up of flight muscles. But low wing loading would permit slow flight with concomitantly reduced power requirements (see Eqn 11). Pterosaurs did actually have larger wing areas than birds and bats of comparable sizes [see fig. 10.2 in Norberg (Norberg, 1990)], and pterosaurs with body masses over 0.1 kg had lower wing loadings than any bird or bat group [pterosaur wing area estimated from the fossil record on the assumption that the wing membrane attached along the hindleg down to the ankle; fig. 10.3 in Norberg (Norberg, 1990)].

Our scaling of wingbeat frequency with body mass in bats is based on data from 65 morphologically diverse bat species, representing 11 families and ranging in size from the smallest known bat, the 2.0 g Craseonycteris thonglongyai (native to Burma and Thailand), to an 870 g Pteropus alecto, a 435-fold difference in body mass (Tables 1, 2, 3). We present original wingbeat frequency data for 27 of the 65 species in the sample. Information for the 38 remaining species was taken from the literature.

The new wingbeat frequency data come from analysis of high-speed motion picture films that we have taken of 23 species in the field and two species in the laboratory. We have also examined two Samoan species, Pteropus samoensis and Pteropus tonganus, from video recordings made in the field by Anne Brooke. Our new data thus come from two species flying in the lab and from 25 species of wild bats in free flight in the field. Data from our 25 outdoor-flying species are combined with literature data on 26 additional bat species in free flight in the field (51 species listed in Tables 1 and 2). Laboratory and wind tunnel data are listed in Table 1 (two species) and Table 3.

The bats that we filmed in flight in the field had been captured and weighed and were released for filming outdoors in daylight. The laboratory films were made of bats flying in a tunnel made of fine net. We filmed with a Kodak 16 mm high-speed film camera (run at 200–1000 frames s^–1 and with timing light-marks on the film), a Photosonics 1VN 16 mm high-speed film camera (run at 100 or 200 frames s^–1 with 2% inaccuracy of the set film rate and with LED timing light-marks on the film), and a spring-driven Pathé 16 mm film camera (run at 87–88 frames s^–1 as established by filming a stopwatch). The two Samoan Pteropus species were video recorded by Anne Brooke with a standard video camera at 30 frames s^–1, and analysed by us. Table 1 lists the camera used for filming each of the 27 species, for which we determined wingbeat frequency.

We measured wingbeat frequency from film sequences showing horizontal un-accelerated flight with no manoeuvres. Films were run on a NAC Model DF-16C stop-motion analysis projector, enabling single-frame film advancement. The video sequences were also analysed frame-by-frame. The laboratory films of Glossophaga soricina (Table 3) and Plecotus auritus (Table 1) cover a range of flight speeds. A background string grid in the flight tunnel was used as a scale for determining flight speed, after correcting for the different camera distances to bat and grid. The wingbeat frequencies for these two species were taken from regressions of frequency vs flight speed and applied to the minimum power speed for the respective species as estimated from Pennycuick’s program 1A (Pennycuick, 1989).

Wingbeat frequency and body mass were averaged across individuals for each species. We log-transformed the raw data for wingbeat frequency vs body mass and subjected them to linear least squares regression, but present the results as power functions.

The largest aerodynamic reaction forces that can ever be elicited by the wings of an animal are limited by the maximal muscle forces that can be exerted. So, except for accidental impact forces, the largest stresses that skeletal elements can ever be exposed to during flight are controlled and limited by maximal muscle forces during maximum performance events, such as extreme manoeuvres (Norberg and Wetterholm Aldrin, 2010). When different sized animals are geometrically similar, the bending and twisting stresses set up in skeletal elements by maximal muscle forces are the same regardless of animal size (Norberg and Wetterholm Aldrin, 2010).

The safety factor against breakage of skeletal elements depends on bone dimensions and structure and on bone material mechanical strength. The wing skeleton of bats (the humerus) has lower safety factors for bending and twisting than those of birds (Kirkpatrick, 1994). But the mechanical strength of bone is similar in birds and bats and does not vary with body size (Alexander, 1981; Biewener, 1982; Kirkpatrick, 1994). Therefore, the lower safety factors of bat humeri must be due to differences in wing skeleton structure between bats and birds.

If the cross-sectional linear dimensions of wing muscles and wing bones scale isometrically to one another and to wing bone longitudinal length, i.e. if muscles and bones maintain geometric similarity to one another, but not necessarily with respect to the body mass, then the safety factor against breakage under loads due to maximal muscle forces will be the same regardless of body mass (Norberg and Wetterholm Aldrin, 2010). There seems to be no information on how wing bone and flight muscle dimensions scale directly to one another in bats. We will therefore explore their scaling relationships to one another indirectly by comparing how they scale with respect to body mass.

Wingspan scales as M_b^0.35 in Megachiroptera, as M_b^0.33 in Microchiroptera and as M_b^0.32 in the two suborders combined (Norberg and Rayner, 1987), which suggests that wing bone lengths are also isometric with respect to body mass in different sized bats, i.e. scaling as M_b^0.33. We used dimensional data on bat wing skeleton from Mar Mar (Mar Mar, 2003) and calculated the following regression functions for humerus and radius lengths vs body mass: l_humerus=0.128M_b^0.345 and l_radius=0.197M_b^0.342 (N=23, among which several species belong to the same genus). The information available thus shows perfect isometry between wing bone longitudinal lengths and body mass.

As detailed below, the mass of the downstroke wing muscles of bats scales isometrically with respect to body mass (Bullen and McKenzie, 2004). Therefore, if the muscles are of similar shapes in different sized bats, as seems likely, their cross-sectional diameter and area should also scale isometrically with body mass. As the data reviewed here indicate that wing bone as well as flight muscle dimensions scale isometrically with respect to the body mass they must obviously also scale isometrically to one another, thus fulfilling the above criterion for stress similarity. Therefore, maximum muscle-controlled stress in the wing skeleton should be independent of body size in bats. For that reason, maximum body size among bats does not seem to be constrained by wing skeleton strength per se, but may instead be related to flight muscle capacity.

Below we focus on muscle and flight performance and show that the lower wingbeat frequency and the lower flight muscle mass in bats, as compared with birds of similar sizes, may explain why the upper size limit is lower among bats than in birds.

The power required for flight, and the power available from flight muscles, which depends on flight muscle mass and wingbeat frequency, can be estimated from theory. This shows that there is an upper limit to the body mass at which the muscles can just produce the power required for flapping flight (Pennycuick, 1975; Pennycuick, 2008). Here, we first review the theory for the scaling of the power required for flight and the power available vs body mass under the assumption that different sized animals are geometrically similar. Based on that, we use empirically obtained functions for wing area and wingbeat frequency vs body mass, the exponents of which depart from those expected for geometrically similar animals, and estimate the power available to real animals of various sizes in order to find what the maximum body mass might be for bats that are capable of horizontal powered flight.

Body drag acts backwards along the flight path. But the drag on the wing acts in the direction of the wing’s local relative airflow, which is the vector sum of the relative velocity, due to flapping of the wing, and the freestream velocity, due to the flight speed. And the velocity of the wing and the steepness of its path vary throughout the wingbeat cycle and along the wingspan. The ‘effective drag’ represents all these aerodynamic forces, which the muscles work against, and is a hypothetical, average, horizontal backward force, which, after being multiplied by the flight speed, would result in the mechanical power actually developed in flapping flight.

The power available (P_a) from the flight muscles may be estimated as the product of the muscle mass M_m and the muscle mass-specific power, which is the power available from a unit mass of muscle. And this power is the mass-specific work Q_m^*, defined as the work done per unit mass of muscle in one contraction, multiplied by the flapping frequency f_w. Among geometrically similar animals the muscle mass is proportional to the body mass. And when geometrically similar animals beat their wings through the same stroke angle, they exert identical muscle strain, i.e. the length over which the muscle contracts is the same proportion of the muscle’s initial length, so the strain in the myofibrils is independent of animal size. For muscles of the same type, whether large or small, exerting a given relative force, whether it is the maximum one possible or some given fraction thereof, the stress in the myofibrils is the same regardless of animal size [see p. 176 of Pennycuick (Pennycuick, 2008)]. So the mass-specific work Q_m^*, which is stress multiplied by strain in a unit mass of muscle, is independent of body mass.

From aerodynamic considerations (Pennycuick, 1975) and dimensional analysis (Pennycuick, 1990; Pennycuick, 1996) [see also p. 183 of Pennycuick (Pennycuick, 2008)], it has been shown that the wingbeat frequency in cruising flight should scale with body mass as M_b^–1/6 in geometrically similar animals. Pennycuick argued that muscle and wing characteristics determine a ‘natural’ wingbeat frequency, which is used in cruising flight and is also near the frequency for maximum power [see pp. 181, 184–185 of Pennycuick (Pennycuick, 2008)].

Because the power required (Eqn 11) increases faster than the power available (Eqn 12) as body mass increases, the two lines eventually intersect, thereby defining a mass above which aerobic powered flight would be impossible (Pennycuick, 1975; Pennycuick, 2008). The wingbeat frequency varies with body mass, but Q^*_m,max does not. As the wingbeat frequency determines the power available, it also determines the maximum potential body mass of flying vertebrates.

We use Eqn 12 to estimate the power available from the downstroke flight muscles as a function of body mass of different sized but geometrically similar animals, but instead of using f_w∝M_b^–1/6, which is the theoretical scaling of wingbeat frequency under geometric similarity, we insert the scaling exponents for empirically observed wingbeat frequencies vs body mass from Eqns 1 and 9. This gives P_a∝M_b×M_b⁰×M_b^–0.27∝M_b^0.73 for birds and P_a∝M_b×M_b⁰×M_b^–0.26∝M_b^0.74 for bats when the regression for all bats is used (Eqn 1). So the power available, as estimated for birds and bats from empirically obtained scaling exponents for wingbeat frequency vs body mass, increases even more slowly with increasing body mass than according to the theoretically determined M_b^5/6=M_b^0.83, which applies when f_w is taken to be proportional to M_b^–1/6 (Eqn 12).

In bats, a larger number of muscles are involved during flight than in birds, but bats nevertheless have less flight muscle mass relative to body mass than birds. Here, we consider only the muscles in birds and bats that power the wings in downstroke. This is because animals with body masses near the potential maximum, which we emphasize here, would be constrained to fly near their minimum power speed, at which the upstroke would most likely be rather passive, requiring little work from upstroke muscles.

The downstroke flight muscle masses in bats have been investigated previously (Hartmann, 1963; Bullen and McKenzie, 2004) and are shown in Table 4. The masses vary among species with different flight behaviour, as they do in birds. The ventral thoracic flight muscles in bats make up on average 9.1% of the body mass. The mass of the pectoralis major muscle in birds makes up 8–25% of body mass (Greenewalt, 1962), with a mean of 15.5%. (It should be noted that the higher percentages sometimes cited for muscles involved in flight often include upstroke muscles as well.) Downstroke flight muscles thus make up about 6.4% more of the body mass in birds than in bats. So the downstroke flight muscle mass of bats is on average only 59% of that in birds of comparable sizes.

In a sample of 30 Australian bat species the proportion M_m/M_b of the body mass that is made up from the total mass of all downstroke muscles scales with body mass as M_b^0.0027 in males and M_b^0.0023 in females [see fig. 6 in Bullen and McKenzie (Bullen and McKenzie, 2004)]. The total mass of downstroke muscles thus scales with body mass as M_b^1.0027 in males and M_b^1.0023 in females. Therefore, we here take downstroke flight muscle mass in bats to be directly proportional to the body mass, as it is in birds, among which M_m∝M_b^0.99 (Rayner, 1988).

The power output increases with increasing wingbeat frequency. But the wingbeat frequency is probably tuned to some natural, resonant, frequency (Greenwalt, 1960; Pennycuick, 2008), with the wings acting as damped oscillators, ruled mechanically by the dimensions, masses and stiffness of the elements involved. The cruising flight wingbeat frequency is thought to be near the frequency for maximum power generation (Pennycuick, 2008).

Indications that the wingbeat frequency varies only within a limited range for a given body size come from comparison of the Strouhal number (St) in different sized animals. St is a dimensionless number in fluid dynamics and describes oscillating flows, such as vortex shedding from a stationary object immersed in a flow. It may be defined as St=fl/V_flow, where f is frequency of vortex shedding, l is a characteristic length (like the diameter of a cylinder or wire), and V_flow is flow velocity (Katz and Plotkin, 1991). When applied to a beating wing of a flying animal, St may instead be defined as the product of flapping frequency and stroke amplitude divided by the forward speed: St=f_wh/V, where f_w is wingbeat frequency, V is flight speed, and h≈2l_wsin(θ/2) is the vertically projected height of the path of the wingtip, from its top to its bottom position, l_w is wing length and θ is dorsoventral stroke angle between the wing’s top and bottom positions (Taylor et al., 2003; Nudds et al., 2004).

The lift-based propulsive efficiency of a root-hinged wing in coupled flapping and pitching oscillations reaches its maximum within a narrow range of Strouhal numbers between 0.2 and 0.4, just as for a thin plate in coupled heaving and pitching oscillations (Taylor et al., 2003; Nudds et al., 2004). Birds and bats of different sizes fly at very similar Strouhal numbers, within or near the 0.2–0.4 range (Taylor et al., 2003; Nudds et al., 2004; Lindhe Norberg and Winter, 2006). This indicates that the wingbeat frequency, although decreasing with increasing body mass, is selected for propulsive efficiency to fall within a narrow range, characteristic for the respective body size.

Because wingbeat frequency is likely to be selected for propulsive efficiency – as indicated by the similarity in Strouhal number – the frequency is also not likely to vary much with flight speed. This is borne out by observations on a bat, Glossophaga soricina, whose wingbeat frequency was constant at flight speeds between 1.75 and 3.99 m s^–1; f_w=12.7V^0.007 (Lindhe Norberg and Winter, 2006). Likewise, Bullen and McKenzie (Bullen and McKenzie, 2002) noted: ‘Wingbeat frequency for each [of 23 bat] species was found to vary only slightly with flight speed over the lower half of the speed range. At high speeds, frequency is almost independent of velocity.’ As an extreme example from birds, the wingbeat frequency of the Archilochus colubris hummingbird was remarkably constant at 53±3 Hz during hovering, manoeuvres and top-speed flights (Greenwalt, 1960).

Change of wingbeat frequency is an obvious way of altering flight speed. But the above review shows that this option is not widely used. Other conceivable means of varying flight speed are to vary the wingbeat amplitude angle and/or shift the wingbeat plane inclination, both measures of which alter thrust but need not change the vertical, weight-supporting, aerodynamic force component. Lift, and hence thrust, can also be changed by varying the wings’ angles of attack, but this must be combined with other measures, like either of the other two mentioned above, in order to modulate the vertical force.

The convergence of different sized animals onto a narrow range of Strouhal numbers, and the near-constant wingbeat frequency in individual animals at different flight speeds, support the notion that cruising flight wingbeat frequencies are optimized for propulsive efficiency. It therefore seems justified to use the observed cruising flight wingbeat frequencies for estimating the maximum power available.

The observed maximum body mass of extant birds is about 16 kg. This is regarded to be an upper limit to the body mass at which a bird can generate just enough power for aerobic powered flight (Pennycuick, 2008). We therefore assume that the lines for power required and power available in birds intersect at 16 kg. In order to find the value of k_bird we equate Eqns 12 and 14 and solve for k_bird in terms of the muscle mass-specific work Q_m^*. The constant k_bird [=g(D′/L′)k_V] represents variables that are independent of body mass (see Eqn 11).

The maximal muscle mass-specific work Q^*_m,max is the same for vertebrate muscles of the same type [see p. 176 of Pennycuick (Pennycuick, 2008)], and should thus be the same for bird and bat flight muscles. Therefore, the power-limited, maximum size in bats, corresponding to that in birds, can be estimated by using the above value of k_bird.

We thus repeat the above procedure with bats and equate Eqns 16 and 12. But for estimation of the power required with the use of Eqn 16 we take the value 0.198Q^*_m,max for k_bird obtained above for a 16 kg bird (Eqn 17). For analytical manageability we thus assume that the constant is identical for bats and birds, k_bat=k_bird=k=g(D′/L′)k_V, and equal to 0.198Q^*_m, which means that the effective drag-to-lift ratio D′/L′ and the constant k_V (relating flight speed to body mass) would be the same for birds and bats (see Eqn 11) and D′/L′ would be independent of body mass.

Therefore, if 16 kg is the maximum potential body mass for a bird capable of sustained powered flight, the corresponding power-limited maximum body mass for bats would be about 1.99 kg. If we instead use the wingbeat frequency regression for all bats (Eqn 1), or that from our own data for field flights only (Eqn 2), the maximum body mass for bats is estimated to be 2.17 and 1.44 kg, respectively. Both values are low, only 14% and 9% of the maximum size of birds capable of sustained powered flight.

However, 16 kg birds have almost no power margin (the difference between available power and required power), and might have to use gliding and soaring as their main locomotion modes. They have difficulties manoeuvring, and preferably take off by gliding down from a height, or use the legs and skitter on water to help gain take-off speed. There are only a few bat species (megachiropterans) that occasionally use soaring flight during commuting (Lindhe Norberg et al., 2000). But they manoeuvre among branches in tree canopies to forage and roost, so probably have wider power margins relative to the power required than a 16 kg bird. A potential bat weighing 2.17 kg would not be able to manoeuvre in the way that the largest living bats do.

It has previously been suggested that 12 kg is a size limit for aerobic flight in birds (Pennycuick, 1975). If a 16 kg bird can just manage powered flight, a 12 kg bird would have some extra power available over that absolutely required and a flight capacity more similar to that of the largest living bats. Because we test the predicted maximum body mass against existing bats, comparison with a 12 kg bird may be more appropriate than comparison with a 16 kg bird. Inserting 12 kg for the body mass in Eqn 17 gives k=0.223Q_m^* for birds, and using this expression for bats together with the frequency regressions in Eqns 1, 2, 3 gives a maximum bat size of 1.12–1.67 kg. The largest megachiropteran bat weighs 1.6 kg, which is within the predicted range.

Fig. 6 shows the estimated lines for power required and power available vs body mass for bats when compared with a 12 kg and a 16 kg bird, and when using regression Eqns 1, 2, 3 for wingbeat frequency. The lines converge at body masses between 1.1 and 2.17 kg. The variation depending on our choice of reference bird mass is small (shaded zone).

From theory, the minimum required wingbeat frequency for geometrically similar animals scales as M_b^–1/6 and the maximum attainable frequency scales as M_b^–1/3 (Hill, 1950; Pennycuick, 1975; Pennycuick, 2008). The minimum frequency is that which is required to generate adequate aerodynamic propulsive force. And the maximum attainable frequency is constrained mechanically and depends on flight muscle force, wing mass and wing length. Because the maximum attainable frequency decreases faster with increasing body mass than the minimum frequency required, they coincide at a critical body mass, above which enough power for sustained flight cannot be generated.

In Fig. 7 the minimum required and maximum attainable wingbeat frequency lines are drawn with their theoretically derived slopes but adjusted in elevation so that they intersect at 2 kg, which is some average of the power-limited body masses obtained with the use of different reference maximum bird masses and wingbeat scaling functions. Under this hypothesis, bats weighing more would not be able to flap their wings fast enough to generate the force and power required for level, aerobic, flight, whereas smaller bats would use lower-than-maximum flapping frequencies in cruising flight and therefore would have the capacity to generate power in excess of that required for level flight (Pennycuick, 1975; Pennycuick, 2008). Small bats will also normally use lower-than-maximum muscle mass-specific work during cruising flight.

It is noteworthy that 62 of the 65 empirical data points for wingbeat frequency vs body mass lie in between the theoretically derived boundary frequency lines in Fig. 7. And the line fitted to the empirical data points for bats is very nearly intermediate in slope (M_b^–0.264vs M_b^–1/3=M_b^–0.33 and M_b^–1/6=M_b^–0.17), as is the slope for birds except hummingbirds [M_b^–0.27 (Rayner, 1988)].

The relationship between wingbeat frequency and body mass that we have described herein is based on all bat species (N=65) for which there are data on wingbeat frequency in cruising flight. Wingbeat frequency and body mass were averaged across individuals of each species. We used the raw, untransformed, species values and take the result to be representative for the bat order. However, for reasons explained below we here repeat the regression, using family-level data.

Related species have similar genotypes and therefore resemble each other in phenotypic traits, variously termed ‘phylogenetic signal’ or ‘phylogenetic inertia’. In across-species comparisons this non-independence of data from related species causes statistical problems in that species-rich lineages become ‘over-represented’ and gain ‘disproportionately’ large (considered undue) statistical weight in relation to species-poor lineages. To give equal statistical weight to closely related species as to others is considered statistically similar to introducing repeat values – akin to counting one species several times.

The problem is that a particular trait or, in this case, a correlated change of wingbeat frequency and body mass, is counted once for each species, even though much of that correlation evolved in the common ancestral lineage of closely related species. It leads to overestimation of the number of times that a particular trait evolved. Therefore, the confidence limits of a fitted regression line would be spuriously narrow. To remove the effect of non-independence of data due to shared phylogeny, methods have been developed that take account of phylogeny by transforming the raw data to independent contrasts (differences or comparisons) between pairs of species (and pairs of constructed internal nodes) that share a common ancestor in a bifurcating phylogeny. These contrasts can then be used for analysis (Felsenstein, 1985; Grafen, 1989; Harvey and Pagel, 1991). Insistence on the need to use phylogenetically controlled methods is motivated essentially from concerns about the reliability of tests for statistical significance (Martin et al., 2005).

For these phylogenetically controlled methods it is assumed that genetic change is random (or ‘Brownian’; like particles in suspension being driven by surrounding liquid molecules in random thermal motion) and increases in proportion to isolation time, so that character variance is proportional to branch length in the phylogeny. These assumptions about the way evolution proceeds are prerequisite for the application of the independent contrasts methods.

For several reasons, however, we do not use phylogenetically controlled statistical methods here. (1) It would introduce hidden (phylogenetic and methodological) variables, which would make the analysis less transparent. (2) The result depends on the particular reconstruction of phylogeny that is used and we cannot be certain that it would be the true one or whether it might confound the results. (3) Regarding comparisons across pairs of constructed internal nodes, the method’s explicit assumption that a trait of an ancestral species (node) equals the arithmetic mean from the pair of descendant species is untenable because species may branch off from a lineage and change while the parent species may remain unchanged. (4) Random change (Brownian motion) is not an appropriate evolutionary model for highly adaptive traits and is unlikely to apply to the radiation of flight morphology in bats. (5) Adaptive evolution need not proceed gradually but may be rapid in connection with speciation and may then slow down, so branch length in the phylogenetic tree may not be a reliable indicator of variance (or data independence). (6) Several distantly related bat lineages contain species that have converged on the same diet, causing directional selection toward similar flight morphologies. (7) Ordinary regression with no adjustment for phylogeny facilitates comparison with previous studies.

Points 4 and 6 will be further elaborated on here. Flight is extremely energy demanding and the flight mode is closely associated with the ecological niche, notably habitat choice and foraging method (Norberg and Fenton, 1988; Norberg and Rayner, 1987), so flight morphology must be under strong selection, ruled by mechanical principles. When distantly related bat species undergo convergent evolution toward the same diet and habitat choice, they also come to use similar foraging behaviour and flight modes. This selects for similar flight morphology, such as wing length, wing area and aspect ratio, which together with body mass determine wingbeat frequency for a given flight mode.

Analyses of the flight morphology of bats have shown that even distantly related species form clusters according to diet grouping (Norberg and Fenton, 1988; Norberg and Rayner, 1987). Insectivorous bats of different microchiropteran families show large similarities in wing form. Nectar-feeding bats in the genus Phyllostomidae among Microchiroptera and subfamily Macroglossinae among Megachiroptera have converged on a similar diet, foraging technique and flight morphology, even though they belong to different suborders. Similarly, carnivorous bats have evolved in different families (Nycteriidae, Megadermatidae, Vespertilionidae and Phyllostomatidae) and have converged on a similar flight morphology (Norberg and Fenton, 1988). And frugivorous bats in the suborder Megachiroptera and in the family Phyllostomidae among Microchirpotera have similar flight morphology even though they belong to different suborders [see fig. 8F in Norberg and Rayner (Norberg and Rayner, 1987)]. These examples further illustrate the prevalence of directional selection on flight morphology in bats, violating the random-change assumption.

Our concern here is not about the number of times that correlated changes in wingbeat frequency and body mass have occurred but rather how best to describe the actual relationship between wingbeat frequency and body mass among existing bats, with a view to extrapolating that relationship upwards to the largest potential bat, capable of powered flight, and predicting its likely wingbeat frequency from the regression found, given the bat characteristic scaling functions for wing length and wing area vs body mass. In a sense we are thus looking for a trend-line that specifies the likely body mass-related characteristics (in this case wingbeat frequency) of any potential new species arising.

When there are many closely related species at certain places in the phylogeny, we think this signifies evolutionary success owing to particularly good adaptation in those lineages. Clusters of closely related species may therefore represent ‘hot-spots’ that identify the dominant trend of bat adaptation better than do phylogenetically scattered species.

Even if there are many species in a cluster because of wide ecological space for the phenotypic traits of that particular lineage, those species, just because of their abundance, do indicate the actual ‘core’ of phenotypic traits among bats. So, for finding the regression equation for wingbeat frequency vs body mass that can be considered characteristic for bats, we have used raw, untransformed, values from all species for which wingbeat frequency data are available.

The family-based Eqn 19 is very similar to the species-based Eqn 1 (based on all 65 species studied), in which the constant is 3.06 and the exponent is –0.264. It is also similar to Eqn 3 for 51 species in free flight in the field, in which the constant is 2.95 and the exponent –0.267 is identical to that in Eqn 19.

When the family-based Eqn 19 for wingbeat frequency is used for estimating the largest potential bat – using the procedure outlined above and assuming that 16 kg is the maximal body mass of birds capable of powered flight – the maximum bat size compatible with powered flight comes out as 2.32 kg, to be compared with 2.17 kg obtained with the use of the species-based Eqn 1 for wingbeat frequency, but otherwise with the same assumptions. Again, using the family-based Eqn 19 for wingbeat frequency, but instead taking the reference maximum body mass for birds capable of powered flight to be 12 kg, which would leave some power margin for manoeuvrable flight, the maximum size for bats is estimated to be 1.78 kg, to be compared with 1.67 kg obtained with the use of the species-based Eqn 1, but otherwise with the same assumptions. The results thus obtained with various data and assumptions are robust.

The maximum available flight power is determined by the maximum muscle mass-specific power multiplied by flight muscle mass. And muscle mass-specific power is equal to the muscle mass-specific work multiplied by the maximum attainable wingbeat frequency. From the comparison that we have made here between bats and birds we conclude that, in general, the downstroke muscles make up a smaller proportion of the body mass in bats than in birds of the same mass. And even though the wingbeat frequency decreases at identical rates with increasing body mass among bats and birds, the frequency is generally lower in bats, which may be due to their lower flight muscle mass.

If the largest hypothetical bat, capable of powered flight, were to develop the same muscle mass-specific power as a 16 kg bird does in cruising flight, then the body mass of the largest potential bat would be 1.4–2.3 kg. This conclusion is based on the assumption that the largest birds and bats have a downstroke muscle mass like the mean value, which is 15.5% of the body mass for birds and 9.13% for bats, and also that their wingbeat frequency is as calculated from the empirically based scaling functions for wingbeat frequency vs body mass in cruising flight, given in Eqns 1, 2, 3 and 19 for bats and in Eqn 9 for birds. When 12 kg is instead chosen as the reference maximum body mass for birds, but with everything else kept equal, the maximum body mass for bats is predicted to be 1.1–2.0 kg. The body mass of the largest bat, extinct or extant, is about 1.6 kg. This is within the range expected if it is the bat characteristic flight muscle mass and wingbeat frequency that limit the maximum body mass in bats. It is only a tenth the mass of the largest flying extant bird.

 
• A_w

  wing area

 
• D

  drag

 
• f_w

  wingbeat frequency

 
• g

  acceleration due to gravity

 
• h

  height

 
• k, k_bird, k_bat

  constant [g(D′/L′)k_V] for geometrically similar animals, birds and bats

 
• k_V

  constant relating flight speed to body mass

 
• L

  lift

 
• L′/D′

  effective lift to drag ratio

 
• l_humerus

  length of humerus

 
• l_radius

  length of radius

 
• l_w

  wing length

 
• M_b

  body mass

 
• M_m

  downstroke flight muscle mass

 
• P_a

  power available

 
• P_r

  power required

 
• Q_m^*

  muscle mass-specific work; work done per unit mass of muscle in one contraction

 
• Q_m,max^*

  maximal muscle mass-specific work

 
• V

  flight speed

 
• θ

  dorsoventral stroke angle

We wish to thank all those who helped us to find and/or handle bats in the field, especially Anne Brooke, Carlos Diaz, Festo Mutere, Colin Pennycuick and Armando Rodrigues-Durán. Anne Brooke provided body mass data and video sequences of flying Pteropus samoensis and P. tonganus. Thanks also to Jens Rydell for comments on an earlier draft of this manuscript. We thank Peter Norberg for valuable discussions and two referees for thoughtful and helpful comments.