Integrating flight mechanics, energetics and migration ecology in vertebrates

To obtain necessary resources for survival and reproduction, many animals depend on swift and efficient movements between different locations within the home range as well as seasonal migrations between continents. Long-distance migration occurs in animal swimmers, flyers and runners, where the mode of locomotion, body size and shape determine the energy cost of transport and therefore the propensity to evolve a migratory lifestyle (Schmidt-Nielsen, 1972; Alerstam et al., 2003). The physical principles of all three modes of locomotion can be approached by biomechanics (Alexander, 2003; Biewener, 2003), from which relationships between power required to move in relation to speed can be obtained. Locomotion imposes elevated metabolic rates and hence energy consumption as a result of the cyclic muscle contractions needed to swing legs or flap flukes, fins or wings. In the case of vertebrate flyers, instantaneous metabolic rates reach the highest levels recorded in the animal kingdom, which in large birds can surpass 15 times the basal metabolic rate (BMR) (Ward et al., 2004; Hedenström, 2008a). However, thanks to the relatively high speeds of flight, the energy cost of transport (energy cost per unit distance) is relatively low for flyers, allowing the evolution of impressive migrations (Alerstam et al., 2003; Hedenström, 2010). The biomechanics of animal flight has been an active research focus for more than 50 years, where in particular two groundbreaking papers by Colin J. Pennycuick and Vance A. Tucker published in this journal ignited a line of research that continues to this day (Pennycuick, 1968; Tucker, 1968; see Hedenström and Lindström, 2017). The flight mechanical theory developed by Pennycuick (1968) describes the power required to fly in relation to airspeed as the sum of three components: the induced power (P_ind) due to the cost of generating lift, the profile power (P_pro) required overcome the drag of the wings, and parasite power (P_par) due to the drag of the body. The inertial power that may arise to accelerate the wings in flapping flight is usually small and therefore ignored when considering forward flight (Pennycuick, 2008). The sum of the three main power components typically yields a U-shaped relationship when plotted against airspeed – the so-called power curve. The Pennycuick flight model is based on an actuator disc, which is an imaginary circle swept out by the extended wings, where the flapping wings induce a downwards deflected momentum jet into the wake. This drastic simplification of the otherwise complicated aerodynamics of flapping wings has proven remarkably useful, although assumptions and measurements of parameters and coefficients of the model have undergone several revisions (Tucker, 1973; Pennycuick, 1989; Pennycuick et al., 1996, 2013).

The flight mechanical theory has served as an inroad for studies of flight performance in birds and bats (Norberg, 1990), and in combination with ecological objectives it has provided input to optimal migration theory (Pennycuick, 1969; Alerstam and Lindström, 1990; Alerstam and Hedenström, 1998; Hedenström, 2008b). The validity of predictions and in some cases even qualitative predictions derived from optimal migration theory depends on two fundamental biomechanical relationships. For migration and other movements, it is the power(speed) [P(U)] relationship and the flight range equation (potential flight range in relation to stored fuel load) that constitute the basic principles from which optimal behaviours are derived. In this Review, I will revisit these relationships and provide an update by considering current knowledge. The aerodynamics and flight range for birds using gliding/soaring flight are described by different functions and a treatment of those would inflate the paper beyond reason, so therefore gliding/soaring flight is omitted. The recent developments in biologging have allowed new approaches to measure and follow flight activity throughout the annual cycle even for relatively small birds and bats. Biologging has not only provided new insights about how birds (and to a lesser extent bats) migrate but also new ways of testing which of alternative migration strategies are used. As with all research, new experiments often result in new questions that require further investigation, and flight ecology and migration are no exceptions. I will therefore end the paper by mentioning topics that need further experimental efforts.

Further developments of flow visualization, notably particle image velocimetry (PIV), have provided even more realistic wake topologies, which typically involves some changes in relation to flight speed as well as shedding of cross-stream vorticity and hence differing circulation between downstroke and upstroke (Spedding et al., 2003; Henningsson et al., 2008). A recent model incorporates a more realistic wake topology to model the power required for vertebrate flight (Klein Heerenbrink et al., 2015), also available as an R-script (https://github.com/MarcoKlH/afpt-r/). While the flight mechanical theory was widely applied to various ecological problems, especially following the publication of an associated computer program (Pennycuick, 1989), it received empirical support mainly through measurements of flight metabolic rate (see below). Direct measurements of muscle work and aerodynamic power output were lagging measurements of metabolic power (P_met) as a result of methodological obstacles.

The P_mech(U) curve is the mechanical power required to fly steadily (Fig. 1A), which is directly related to the rate of work done by the flight muscles to flap the wings. The main pectoralis muscle pulls the humerus downwards during the downstroke, while the function of the smaller supracoracoideus is partly to rotate the wing for the upstroke (Poore et al., 1997). In slow flight, the upstroke is aerodynamically largely inactive (feathered), but with increasing speed the wing also generates aerodynamic force during the upstroke (Spedding et al., 2003). By measuring the strain rate and stress of the flight muscle, Biewener et al. (1992) devised a work-loop approach to determine the mechanical power of bird flight. When applied to birds flying at different speeds in a wind tunnel, this approach yielded predicted U-shaped P_mech curves (Table 1; Fig. S2; Dial et al., 1997; Tobalske et al., 2003). By using in vivo measurement of strain rate by sonomicrometry and in vitro force measurements, Askew and Ellerby (2007) obtained characteristic U-shaped P_mech curves for zebra finches, Taeniopygia guttata, and budgerigars, Melopsittacus undulatus (Table 1; Fig. S2). An alternative non-intrusive approach was used by Pennycuick et al. (2000), where the force applied to the humerus by the flight muscles was estimated from kinematics and the vertical accelerations by the body throughout a wing stroke, which applied to a barn swallow, Hirundo rustica, yielded an increase of P_mech from a minimum in the speed range 6–11 m s⁻¹. This can be interpreted as the right-hand side of the curve at speeds greater than the optimal speed of minimum power (>U_mp).

A different approach was taken by von Busse et al. (2014) by using flow visualization to estimate the rate of kinetic energy added into the wake by flapping wings. This method requires 3D-flow visualization of high quality to capture the signal amidst background noise. However, two efforts involving vertebrate flight have so far yielded a relatively flat P_mech(U) relationship in small bats Carollia perspicillata (Table 2; Fig. S3; von Busse et al., 2014) and a shallow L-shape in the pied flycatcher, Ficedula hypoleuca (Table 1; Fig. S2; Johansson et al., 2018).

On balance, when combined, the various approaches designed to measure power output in flying vertebrates (mainly birds) are in support of the theoretical U-shaped relationship between power and airspeed (see Figs S2 and S3).

Engel et al. (2010) reviewed the literature published until 2009 and found general agreement for a U-shaped P_met(U) relationship for birds. As there had been some controversy whether the relationship should be U-, L- or J-shaped or even flat (Alexander, 1997), one observation was that in cases where deviation from a U-shaped relationship was claimed, those studies usually covered narrow speed ranges and therefore probably missed the low and high end of the speed range where power should be high (Engel et al., 2010). Two additional studies since Engel et al.’s (2010) review confirmed a U-shaped P_met curve for the cockatiel, Nymphicus hollandicus (Morris et al., 2010), and the blackcap, Sylvia atricapilla (Hedh et al., 2020).

The empirical support for a U-shaped P_met curve in bats is less clear than for birds. Measurements using mask respirometry on relatively large bats show either flat or weakly U-shaped P_met curves (Table 2; Fig. S3; Carpenter, 1985, 1986; Thomas, 1975), while measurements on smaller species using the C¹³-labelled sodium bicarbonate method (Hambly et al., 2002) resulted in U-shaped curves (Table 2; Fig. S3; von Busse et al., 2013; Troxell et al., 2019). It should be noted that speed ranges of bat studies are relatively short (mean 4.3 m s⁻¹; Table 2), which probably influences the interpretation of the shape if the low and high speeds are cut off (see Engel et al., 2010).

A comparative study including birds and bats and based on P_met measurements and calculated P_mech suggested that η increases with body mass among species and with flight speed within species (Guigueno et al., 2019). That η increases with speed was confirmed for the migratory Nathusius' pipistrelle Pipistrellus nathusii (Table 3; Fig. S4F; Currie et al., 2023), while another study of the short-tailed fruit bat Carollia perspicillata indicated a slight decrease in η with increasing flight speed (Table 3; Fig. S4E; von Busse et al., 2014). Tucker's (1972) data for the budgerigar indicated a slight convex U-shape of the relationship η versus speed (Table 3; Fig. S4A). Interestingly, combining data for P_mech (Askew and Ellerby, 2007) with P_met from another study (Bundle et al., 2007) also yielded a convex curve with a minimum at intermediate speeds (Table 3; Fig. S4B). Given the diverging patterns of how η may vary with speed (Table 3; Fig. S4), I calculated the consequence of different η(U) relationships on P_met curves, also including a fourth hypothetical case where η(U) shows a concave parabolic relationship (Fig. S5), i.e. with a maximum at an intermediate speed (i.e. optimal speed at maximum range, U_mr), a speed often used by migratory birds. The resulting P_met(U) curves for the four cases are shown in Fig. 2. From these curves it appears that case 1 [η(U) increasing] and case 3 [η(U) convex parabola] are the most unlikely because they result in P_met curves with shapes rarely or never observed. The two remaining cases are difficult to separate as they both yield U-shaped P_met curves similar to those measured in many studies. Even if case 4 [η(U) is a concave parabola] has never been observed, it is perhaps the most likely relationship if natural selection has generated maximum efficiency at speeds most relevant to the ecology of birds during transport and migration. However, even if the current empirical data available show diverging patterns for η(U), one comes to think of Rayner's (1979) conclusion when pondering the complex interplay between muscle physiology, energy transport within the bird, wing morphology, kinematics and aerodynamics, i.e. that ‘efficiency varies with speed in an effectively unpredictable way’. However, I think that the confusing situation is rather the result of the patchy and uncoordinated efforts to determine η(U) both within and between species. As this is such a fundamental property of bird flight models used to predict behaviours, it should be a future research priority to clarify the relationship between P_mech and P_met. If allowed to make a bold prediction about how efficiency varies with flight speed, I would bet my money on case 4 even if not yet observed.

For isometrically scaled ideal birds and bats, any characteristic flight speed is expected to scale proportional to m^1/6 (Pennycuick, 1975), and although measurements of flight speeds of birds on migration show an increase with increasing body size, the scaling exponent is less than 1/6 in real birds (Alerstam et al., 2007; Pennycuick et al., 2013). The reason appears to be a combination of phylogeny and allometric scaling of wing shape, i.e. aspect ratio (b²/S), which increases with body mass (Alerstam et al., 2007). However, in an analysis of flight speed in five tern species, with similar wing shape but a 10-fold difference in body mass between the smallest and the largest species, flight speed scaled as the predicted m^1/6 (Hedenström and Åkesson, 2016).

The power curve immediately suggests two ‘optimal’ flight speeds: U_mp for minimum power and U_mr for maximum range (Fig. 1A). By combining the P(U) curve and energy intake at stopovers with the objective of minimizing the overall speed of migration, which includes accounting for time for refuelling at stopovers, an alternative optimal flight speed emerges associated with minimum overall time of migration U_mt (>U_mr), which depends on the rate of fuel deposition (Alerstam and Lindström, 1990; Hedenström and Alerstam, 1995). In analogy, if the objective is to deliver food (energy) to young in a nest, the optimal flight speed is also >U_mr, depending on the rate of energy gained when foraging at a site away from the nest (Norberg, 1981). The analysis can be extended to flight transport between food patches by a net maximizing forager (Fig. 3; Hedenström and Alerstam, 1995). In this model, the objective is to maximize the difference between energy intake (E) and the transport cost (C) between patches divided by the combined patch residence time (t_p) and flight time between patches (t_t). The transport cost is DP/U, where D is the distance between patches. By constructing a mutual tangent to the foraging gain curve and the transport cost curve, we obtain both the optimal patch residence time (t_p^*) and the optimal flight speed U* (Fig. 3). When there is a positive energy balance, i.e. (E−C)>0, the optimal flight speed U*>U_mr, but with increasing flight distance between patches, the energy balance will approach zero and U*→U_mr (Fig. S6A). If the average rate of food gain increases, it will be optimal to also increase the flight speed (Fig. S6B). For further alternative flight speeds in different ecological situations, see Hedenström and Alerstam (1995). It should be noted that in cases where the animal operates at a metabolic ceiling, the best policy when foraging is to maximize the efficiency, defined as the gain/cost ratio, and the associated flight speed is U_mr (Hedenström and Alerstam, 1995).

The relevant question now is do birds and bats obey theoretical recommendations provided by flight mechanics? I have compiled a few examples of mainly birds flying in ecological contexts when one or other of the optimal flight speeds is expected (Table 4). In general, it seems as though birds do select contextually relevant flight speeds, even if discrepancies occur if observed speeds are directly compared with calculated optimal speeds from models. Therefore, strong inference is obtained by comparing the same species (or preferably individuals) when flying in different situations, such as redshanks, Tringa totanus, during migration, display and flight between food patches (Table 4; Hedenström, 2024).

Flight speed is not only dependent on the input parameters to flight mechanical models and ecological context but also contingent on winds and social context. To maintain a characteristic speed (e.g. U_mr, U_mt) and a constant track over ground, the bird (or bat) should simultaneously adjust airspeed with respect to head/tail winds and the side wind (Liechti et al., 1994). Birds generally show appropriate adjustments of airspeed in relation to wind (e.g. Hedenström et al., 2002), and if following a coastline, speed is also adjusted to the side wind component (Hedenström and Åkesson, 2016, 2017b; Hedenström, 2024).

If birds save energy by flying in a flock formation (e.g. Lissaman and Schollenberger, 1970; Weimerskirch et al., 2001; Portugal et al., 2014; Friman et al., 2024), the power curve would be lowered and characteristic speed should be slower than that of a single individual, although quantitative effects are little known. However, there is mounting evidence that, contrary to expectation, speed instead increases with increasing flock size (Noer, 1979; Hedenström and Åkesson, 2016, 2017b). Therefore, we must find another explanation for why flight speed increases with flock size rather than a purely aerodynamic one. When analysing speed in relation to multiple factors, it appears that a trait (airspeed) that superficially should be a rather simple behavioural decision has a very complex background, where birds simultaneously factor in multiple intrinsic and extrinsic factors. It remains to be clarified what cues birds sample to determine and maintain their flight speed. Optic flow of ground features can be used at low altitudes (Serres et al., 2019), but how does a bird determine whether a wind change that requires adjustment of airspeed when flying at high altitude occurred? This and other unsolved puzzles about flight speed will also keep scientists busy in the near future.

Bar-tailed godwits Limosa lapponica baueri migrate by a single non-stop flight between Alaska and New Zealand in autumn, a flight of up to 11,500 km completed in about 8 days (Gill et al., 2009). Before departing, the godwits accumulate large fuel stores (Piersma and Gill, 1998), which, if adopting flight range according to Eqn 10, equates to an estimated x=0.42% h⁻¹ (i.e. 0.42% of body mass is consumed as fuel for every hour of flight; Hedenström, 2010). The Arctic tern, Sterna paradisaea, has a longer migration than the bar-tailed godwit, but the terns can potentially feed along the route even if they probably also cover long distances on stored fuel reserves (Egevang et al., 2010). The common swift, Apus apus, shows an even more extreme flight endurance by being airborne for up to 10 months during its entire non-breeding period (Hedenström et al., 2016), but swifts are adapted to an airborne lifestyle and of course maintain a balanced energy budget by feeding on the wing. The flight range (Eqns 5, 7, 8) refers to flight distance on stored fuel, but, nevertheless, not even the bar-tailed godwit may have reached the maximum limit according to simulations (Pennycuick and Battley, 2003). These simulations assumed that body drag coefficient (C_Db) is as low as 0.1, but if C_Db is higher than this, as suggested by wind tunnel studies (Tucker, 1990; Klein Heerenbrink et al., 2016), flight range of the godwits may be close to the limit after all.

As an individual consumes fuel during flight, the power required to fly for an ideal bird (sensuPennycuick, 1975), should scale as m^7/4. This relatively high scaling exponent assumes that fuel is stored homogeneously around the body without affecting its length, which implies that the projected body frontal area is directly proportional to the fuel load. A few wind tunnel studies using the doubly labelled water method to estimate P_met obtained a weaker dependence on body mass than the aerodynamic prediction, with mean mass exponents falling in the range 0.35–0.58 (Kvist et al., 2001; Engel et al., 2006; Schmidt-Wellenburg et al., 2007). Using mass loss as proxy for fuel consumption, Kvist et al. (1998) obtained a scaling exponent of 2.98, i.e. even higher than that expected. If we, for the sake of argument, accept that within-subject P_met∝m^p, where the scaling exponent p is lower than expected, this means that the penalty for carrying fuel is less severe than suggested by Eqn 5. The reason for this remains unknown, whereas Kvist et al. (2001) suggested that the muscle conversion efficiency is mass related, with elevated efficiency at high fuel loads. The mechanism for such variation remains unknown, but if it is related to the work rate by the muscles, it may be analogous to the increased efficiency in relation to flight speed, as found in bats by Currie et al. (2023). A contributing factor could be how fat is stored in the body and how this affects the projected frontal area, and hence the body drag (see Eqn 2). If fat can be stored in body cavities and at the front and end of the body (Wirestam et al., 2008), the drag penalty will be less than assumed when deriving Eqn 5. Be that as it may, P_met shows a within-individual mass dependence and therefore the potential flight range will be a function of diminishing returns (Eqns 5 and 7), which is prerequisite for many predictions related to optimal migration strategies. However, it remains to be determined more precisely how flight costs are affected by the consumption of fuel (fat and protein) during long-haul migratory flights.

The shortest route between two locations on a sphere (Earth) is a great circle (orthodrome) and, especially if migrating at high latitudes along routes with east or west directions, the great circle can be significantly shorter than the rhumb line (loxodrome) of constant compass direction. Birds setting out on long-haul flights in the high Arctic in autumn depart with initial directions consistent with great circles (Alerstam et al., 2001, 2008). However, many migrants fly along other geometric routes, and often along different routes between autumn and spring migration, i.e. making so-called loop migrations. Often, such loops are attributed to predictable wind patterns or seasonally occurring food availability (Shaffer et al., 2006; Klaassen et al., 2010; Åkesson et al., 2012). An alternative explanation of route selection may be based purely on biomechanical considerations, i.e. that the crossing of a vast ecological barrier requires a significant fuel load. Because of the diminishing returns utility in flight range of fuel load (Eqns 5 and 7; Fig. 1B; Fig. S1), there is a break-even detour of the same energy cost as a direct flight across the barrier, i.e. a tolerable detour, if dividing the journey into multiple steps requiring small fuel loads (Fig. 4; Alerstam, 2001). In the limit of infinitesimally small fuel loads, the range is given by Y_max=cf/2 (see Eqn 5; Fig. 4). Alerstam (2001) used this argument to analyse whether different known migrations involving detours could be explained by the cost of carrying fuel and found that many migrations are consistent with this criterion. A detour can also be accepted if the detour involves the crossing of the barrier where the barrier is reduced in distance. The argument can also be applied to time-selected migration, especially if the detour includes stopovers that provide improved fuelling conditions (Alerstam, 2001).

Norevik et al. (2020) combined detour analysis with wind conditions along the route for different populations of European nightjars, Caprimulgus europaeus, breeding at different longitudes in Europe, where western breeding populations showed longer detours during spring migration than those breeding further to the east (Norevik et al., 2020). A combination of winds and shortening of the barrier distance explained the different detours between the populations. Another interesting migration detour is that by juvenile sharp-tailed sandpipers, Calidris acuminata, which make a first flight from the breeding area in central north Siberia to Alaska, where they accumulate fuel for a trans-Pacific flight to wintering areas in Australia and New Zealand (Handel and Gill, 2010). This detour results in the inclusion of a longer barrier (The Pacific) than if they had followed the east Asian flyway as adults do, which may still be a favourable strategy for a time minimizer (see next paragraph) as fuel rates are extremely high in Alaska (Lindström et al., 2011). Gill et al. (2009) found that for bar-tailed godwits flying from Alaska to New Zealand across the Pacific in autumn, a detour along the east Asian flyway would require that migration is split into 5–8 flights to be energetically advantageous. Perhaps there are not so many suitable stopovers available along this route, or the godwits are time minimizers as juvenile sharp-tailed sandpipers seem to be? In spring, however, the godwits make a detour via the Yellow Sea (Battley et al., 2012), which is a major hub for migrating shorebirds. By this detour, they have a shorter second flight than a reverse direct flight across the Pacific, which may be important in spring if they need to arrive with some stored energy reserves to allow onset of breeding (capital breeding; Drent and Daan, 1980).

The Nathusius' bat migrates from northern Europe towards the southwest in autumn (Petersons, 2004), where populations from Latvia could either cross the Baltic Sea or detour along the southern Baltic coast. An analysis showed that a direct flight is optimal in this case (Hedenström, 2009), which is consistent with the passage of this species on Öland in the autumn. Combining analyses of flight range (Eqns 7 and 9), the geographical distribution of ecological barriers and prevailing wind patterns allows a powerful approach for understanding the evolution of migration routes in birds and bats, which is not least of importance in a world of rapid climate change that may affect the conditions for many migratory species and populations.

Depending on the overall objective of migration performance, a bird may execute the journey according to alternative strategies, which typically are the minimization of energy cost (transport or total cost of migration), the time or the predation risk (Alerstam and Lindström, 1990; Hedenström and Alerstam, 1997). Two or more currencies may be combined (Houston, 1998), but central to energy and time minimization is the flight range equation and the power curve (Fig. 1). The flight range is used to determine optimal stopover duration and departure fuel load, and hence also flight step lengths. These behavioural decisions are contingent on search/settling time and energy costs when arriving to new stopovers, as well as expected gradients of expected fuel deposition rates along the route and the distribution of suitable stopover habitats. The power curve is central for behavioural decisions while airborne, i.e. the selection of optimal flight speed in relation to winds and overall strategy.

In experimental studies of stopover behaviour where fuelling rate was manipulated by providing food, responses were generally in agreement with a time-minimization strategy (Hedenström, 2008b; Alerstam, 2011), with an exception indicative of energy minimization (Dänhardt and Lindström, 2001). By measuring flight step lengths in autumn and spring using accelerometery loggers, little ringed plovers, Charadrius dubius, showed increasing (autumn) and decreasing (spring) flight steps (Hedenström and Hedh, 2024). This is consistent with a time-minimization strategy if there are increasing and decreasing resource gradients along the route in autumn and spring, respectively.

Flight (air) speed is predicted to be higher than U_mr in time-minimizing migration (see above). Observed flight speeds are sometimes higher in spring than in autumn, which has been interpreted as indicating that birds migrate according to a time-minimization strategy in spring and an energy-minimization strategy in autumn (Nilsson et al., 2013), whereas flight step adjustments during autumn and spring migration in little ringed plover suggest that time minimization applies in both seasons, and so different seasonal flight speeds may be due to other factors.

Mortality rates in birds are generally higher during migration compared with periods of residency (Sillett and Holmes, 2002; Klaassen et al., 2014), which is probably due to incessant arrival into unfamiliar habitats where risk factors are unknown, but also because birds' (and bats') manoeuvrability is adversely affected by heavy fuel loads (Hedenström, 1992; Lind et al., 1999; Burns and Ydenberg, 2002) and the need to forage intensely (Lima and Dill, 1990). Some shorebirds appear to have adjusted the timing of autumn migration to avoid predation risk by peregrines, Falco peregrinus, which have a similar migration route (Lank et al., 2003). It is probably naive to believe that birds or bats strictly follow a simple rule of time, energy or predation minimization when migrating, but depending on the season and ecological context, elements of different strategies are probably adopted with the overall goal to maximize lifetime reproductive output. Nonetheless, the alternative migration strategies serve as benchmarks against which to evaluate observed behaviours in real animals, not least using the opportunities provided by modern biologging devices.

The execution of a migratory flight should convert stored fuel into distance, preferably taking the bird/bat as far as possible. Traditionally, the flight has been likened to that of an aeroplane, consisting of an initial climb to a cruising altitude where the bird/bat remains until the end of the flight (Hedenström and Alerstam, 1994), possibly with a slow cruise climb to maintain maximum effective lift/drag ratio (Pennycuick, 1978). Observations suggest that rate of climb, U_z, for birds is lower than the biomechanical maximum, which is consistent with the notion of an optimal climb rate that depends on the expected wind assistance at the cruising altitude (Hedenström and Alerstam, 1994; Hedenström, 2024). However, evidence from studies using radar and bio-loggers with altimeters suggests that migrating birds make frequent altitude shifts (Mateos-Rodríguez and Liechti, 2012; Bowlin et al., 2015; Norevik et al., 2021). The reason for such altitude shifts remains unknown, but a popular interpretation is that they are aimed at probing whether there are improved wind conditions at a different altitude from at the current one.

Migratory birds tend to depart from a stopover when winds are favourably aligned with the intended migration direction (Åkesson and Hedenström, 2000; Åkesson et al., 2002; Gill et al., 2009), while they remain at the stopover during periods of adverse wind conditions even if they have reached the optimal fuel load (Delingat et al., 2008). This is adaptive because winds modify the exchange rate of fuel into distance (Weber et al., 1998; Liechti and Bruderer, 1998). In situations when winds vary randomly between days, it could be favourable to allow drift with cross-winds when far from the goal if the remaining distance to the goal is shorter than if compensating (Alerstam, 1979a). Also, if wind strength increases with increasing altitude, a single flight can be divided into a first segment at high altitude allowing partial drift, followed by a second segment at low altitude with overcompensation (Alerstam, 1979b).

Radar studies suggest that birds concentrate at altitudes with favourable wind support (Mateos-Rodríguez and Liechti, 2012; Horton et al., 2016), but birds seem to be content with a local optimum within a wider altitude range. More recent studies involving altimeters on individual migrants have documented frequent altitude shifts during nocturnal flights (Bowlin et al., 2015; Norevik et al., 2021). Whether these shifts represent probatory excursions in search of improved wind conditions remains an open question, but it seems likely to serve this function. To conduct climbing flight followed by a descent to the original altitude does not cost a lot of extra energy (as is often assumed), as the potential energy gained during the climb can be used to overcome drag during the descent, which in nightjars is a powered descent (Norevik et al., 2021). Hence, it may be worth spending a little extra energy now and then if the pay-off comes as an improved wind support. By aerodynamic modelling, Sachs (2022) suggested that it may even be economically advantageous to alternate climbing flight followed by ‘powered glides’ (descending flight by flapping flight at reduced power) in relation to flying the same horizontal distance at constant altitude. This model contains some untested assumptions regarding the flight mode-related lift coefficients and a constant η, and the prediction of a continuous climb/descent flight pattern seems to be at odds with the altitude profiles of European nightjars (Norevik et al., 2021). An alternative hypothesis for flight altitude selection is to balance trade-offs between energy cost of flight (winds) and evaporative water loss (Carmi et al., 1992), although radar data of altitude selection in migrants were best explained by winds (Liechti et al., 2000).

Recent efforts involving biologging tags with altimeters have revealed diurnal altitude shifts, where migratory great reed warblers, Acrocephalus arundinaceus, and great snipes, Gallinago medea, ascend to high altitude (≥5 km) at dawn and maintain high altitude throughout the daytime, and descend to lower altitudes at dusk (Sjöberg et al., 2021; Lindström et al., 2021). A favoured explanation for such diel shifts in flight altitude is that flying in colder air at high altitudes help to control body temperature due to solar radiation (Sjöberg et al., 2023). In contrast to great reed warblers and snipes, when European nightjars cross the Mediterranean Sea and Baltic Sea in daylight, they fly very low (Norevik et al., 2023). When flying low over water, nightjars also shift flight mode from continuous flapping flight to an energy-saving flap–gliding flight, possibly also exploiting the ground effect.

There are several different flight strategies regarding when and how to execute migratory flights in birds and bats. It is certain that winds play a major role, but additional factors such as altitude, solar radiation, humidity (Gerson and Guglielmo, 2011) and surface structure (flat or rugged) contribute to the complexity and diversity in flight strategies observed in birds. How bats conduct migratory flights remains largely unknown, but flight altitudes are likely to be much lower than in most birds.

In this Review, I have attempted to illustrate how vertebrate flight can be approached using biomechanics and energetics when analysing flight performance and migration strategies in real animals. Research in this field is genuinely interdisciplinary, drawing knowledge from biomechanics, aerodynamics, physiology and ecology to generate a theoretical framework that combines fundamental principles with adaptive optimization of resources such as energy, time and survival in migratory vertebrates. However, as evident from the above, many uncertainties remain, not least regarding the two fundamental equations of Fig. 1. Obviously, building a layer of ecological theory using Eqns 3 and 5 as input will be shaky if these equations are already afflicted with uncertainty. Therefore, I urge scientists to continue investing interest and efforts in research aimed at refining our understanding about how the power curve and range equation are contingent on body size and shape, fuel load and flight speed, which eventually will provide an even better understanding about locomotion and migration performance and strategies in animals.

Wind tunnels have been an important tool for assessing properties of aerodynamics and energetics, and they will continue to play a major role in this research. However, the current revolution in miniaturization and the sophistication of multisensory bio-loggers means data can be generated on flight duration, altitudes and effort with unprecedented detail. New discoveries about impressive migratory flights over entire annual cycles shed new light on flight and migration strategies, also allowing scientists to evaluate which of alternative strategies animals use. We live in a fascinating time where research about biomechanics, energetics and ecological adaptations in migrants receives renewed injections from all perspectives. I end by suggesting a few outstanding questions that I think offer rewarding research opportunities. (1) How does energy conversion efficiency vary in relation to flight speed, body size, wing shape and fuel load? (2) How is body drag (C_Db) affected by body size, body shape and fuel load? (3) Are kinematics and wing shape (including camber) adjusted across flight speeds to optimize flight efficiency? (4) When cruising at high altitude, can a bird determine its airspeed and make adaptive adjustments to wind change? (5) If yes to (4), how?

I am grateful to the organisers of the JEB symposium ‘Integrating Biomechanics, Energetics and Ecology in Locomotion’ for inviting me to this very interesting meeting and prompting me to write this Review, and to two anonymous reviewers for encouraging and constructive comments.