Optimal migration theory is the established paradigm in the study of avian migration (Alerstam and Lindström 1990, Weber and Houston 1997, Hedenström 2008), and most recent studies of migration strategies and tactics are performed within this framework (e.g., Eichhorn et al. 2009, Henningsson et al. 2009). The central assumption of this theory is that the flight-range equation is a diminishing return function of added fuel mass:Alerstam and Lindström 1990, Hedenström 2008). Weber and Houston (1997) generalized Equation 1 and showed that different flight-cost estimates available to them could be summarized as
Another relationship of profound importance for optimal migration theory is the one between flight speed and the power of flight. This relationship is widely accepted as being described by the following formula:Pennycuick 1989; Hedenström 2002, 2008). This function is U-shaped (Fig. 1), which means that there is a single minimum power speed value, above and below which energy expenditure increases.
Both of these crucial relationships are based on mechanical flight theory (Pennycuick 1975,1989; Lindhe Norberg 2004; Hedenström 2008). However, neither of them is fully supported by recent experimental data.
Four recent studies measured energy costs of several hours of flight in wind tunnels in relation to intraspecific variation in body mass (Kvist et al. 2001; Engel et al. 2006; Schmidt-Wellenburg et al. 2007, 2008). Flight costs in Rosy Starlings (Sturnus roseus) were shown to increase with m0.55(95% confidence interval [CI] of the scaling exponent: 0.36–0.75; Engel et al. 2006). In another study of the same species, the scaling exponent was 0.57 (95% CI: 0.40–0.74) if the data for birds carrying harnesses and those not carrying were pooled for analysis, and 0.47 (95% CI: 0.18–0.76) if only the birds with harnesses (loaded and unloaded) were included (Schmidt-Wellenburg et al. 2008). In the Barn Swallow (Hirundo rustica), the scaling exponent was 0.58 (95% CI: 0.19–0.97; Schmidt-Wellenburg et al. 2007), and in the Red Knot (Calidris canutus) it was 0.35 (95% CI: 0.08–0.67; Kvist et al. 2001). All these values are clearly below 1; that is, flight costs increase much less steeply than assumed by the accepted aerodynamic theory (scaling exponent: 1.1–1.6; Norberg 1990, 1996; Pennycuick 1975; Rayner 1990). The flight-range equation can be obtained by integrating the flight-power equation (Weber and Houston 1997: equation 9):Equation 2). Clearly, the calculations of flight and stopover parameters are affected by these results, because this relationship deviates less from the linear function than Equation 1 (Fig. 2).
Most reviews have claimed that the U-shaped relationship between flight speed and flight power is well supported by the experimental data (Schmidt-Nielsen 1997, Blem 2000, Harrison and Roberts 2000). Only Ellington (1991) emphasized the scarcity of data to support this claim. Here, I update Ellington's (1991) review, including more recent measurements, and discuss implications for optimal migration theory. Until recently, the U-shaped relationship predicted by mechanical flight theory (Fig. 1) has been supported by a single study involving a single species, the Budgerigar (Melopsittacus undulatus; Tucker 1968), whereas other studies showed either a weak positive relationship, as in the Laughing Gull (Leucophaeus atricilla; Tucker 1972), Fish Crow (Corvus ossifragus; Bernstein et al. 1973), and European Starling (Sturnus vulgaris; Ward et al. 2004), or similar flight power across a wide range of flight speeds, as in European Starling (Torre-Bueno and Larochelle 1978), Barnacle Goose (Branta leucopsis), and Bar-headed Goose (Anser indicus; Ward et al. 2002). A recent wind-tunnel study showed independence of flight costs from flight speed in Rosy Starlings despite a 55% increase in flight speed (Engel et al. 2006). A clearly U-shaped relationship between flight power and speed was shown by in vivo measurements of muscle force in Cockatiels (Nymphicus hollandicus), whereas the relationship was weakly U-shaped in Ringed Turtle-Doves (Streptopelia risoHa; Tobalske et al. 2003) and flat in Black-billed Magpies (Pica pica; Dial et al. 1997). Recently, a U-shaped relationship was reported for the Budgerigar and Cockatiel (Bundle et al. 2007). It is noteworthy that both cases of the unequivocally U-shaped relationship involved species that do not migrate long distances. Berger (1985) found J-shaped relationships in the Sparkling Violetear (Colibri coruscans) and Green Violetear (C. thalassinus): their flight metabolism did not vary significantly between hovering speeds and up to 7 m·s-1 (which is a quite significant speed for a bird of this size). At even higher flight speeds, metabolic rate increased.
Thus, both main theoretical assumptions on which optimal migration theory is based are not supported by most empirical data. When fuel stores are low, the relationship between potential flight range and relative fuel stores deviates from the linear proportionality less than assumed by Equation 1 (Fig. 2). Under such conditions, flight cost is low, and it becomes higher and in better agreement with the predictions of the current aerodynamic flight theory with increasing fuel stores. This is supported by the data on escape flights of Blackcaps (Sylvia atricapilla; Kullberg et al. 1996) and Sedge Warblers (Acrocephalus schoenobaenus; Kullberg et al. 2000). The authors claimed that flight speed and acceleration decreased with increasing fuel load, but this effect was apparent only when fuel load exceeded 30% of lean body mass (Kullberg et al. 1996: fig. 3, Kullberg et al. 2000: fig. 1). Current calculations of optimal migration theory may approach reality for birds that cross large ecological barriers (e.g., the Sahara or the Gulf of Mexico) and carry large fat stores, but this is an interesting special case of avian long-distance migration. Most passerines that migrate over suitable habitats with continuous stopover possibility usually carry moderate fuel stores <30% of their lean body mass (Bairlein 1991, 1997).
U-shaped flight-power curves appear to be just a special case probably confined to some birds of low aerodynamic capacity, or at least to those whose annual cycle does not include longdistance migration. Most birds that spend a significant proportion of their time flying seem to be able to fly equally efficiently across a wide range of speeds. The data on parrots and hummingbirds emphasize that this ability is certainly not unlimited, but migrants seem to fly quite efficiently across the range of speeds they routinely employ. Aerodynamic calculations that predict mass exponents ranging between 1.1 and 1.6 (Pennycuick 1975, 1978; Rayner 1990; Norberg 1996) and a U-shaped flight-power curve are based on fixed-wing theory and are apparently not relevant for modeling avian flight (Dolnik 1995, Videler 2005). This means that the concepts of distinct minimum-power speed, maximum-range speed, and speed of time-minimizing migration (e.g., Hedenström 2008) also need to be reevaluated and probably revised.
I am grateful to M. Bass and V. Kosarev for their advice during the preparation of the manuscript. Critical comments by B. Tobalske and M. Murphy helped me improve an earlier draft.
- T. Alerstam , and Å. Lindström . 1990. Optimal bird migration: The relative importance of time, energy, and safety. Pages 331–351 in Bird Migration ( E. Gwinner , Ed.). Springer, Berlin. Google Scholar
- F. Bairlein 1991. Body mass of Garden Warblers (Sylvia borin) on migration: A review of field data. Vogelwarte 36:48–61. Google Scholar
- F. Bairlein 1997. Spatio-temporal course, ecology and energetics of Western Palaearctic-African songbird migration. Summary Report on the Scientific Network, 1994–1996. Institut für Vogelforschung, Wilhelmshaven, Germany. Google Scholar
- M. Berger 1985. Sauerstoffverbrauch von Kolibris (Colibri coruscans und C. thalassinus) beim Horizontalflug. Pages 307–314 in BIONA Report 3 ( W. Nachtigall , Ed.). G. Fischer, Stuttgart, Germany. Google Scholar
- M. H. Bernstein , S. P. Thomas , and K. Schmidt-Nielsen . 1973. Power input during flight of the Fish Crow, Corvus ossifragus. Journal of Experimental Biology 58:401–410. Google Scholar
- C. R. Blem 2000. Energy balance. Pages 327–341 in Sturkie's Avian Physiology ( G. C. Whittow , Ed.). Academic Press, San Diego, California. Google Scholar
- M. W. Bundle , K. S. Hansen , and K. P. Dial . 2007. Does the metabolic rate-flight speed relationship vary among geometrically similar birds of different mass? Journal of Experimental Biology 210:1075–1083. Google Scholar
- K. P. Dial , A. A. Biewener , B. W. Tobalske , and D. R. Warrick . 1997. Mechanical power output of bird flight. Nature 390:67–70. Google Scholar
- V. R. Dolnik 1995. Energy and Time Resources in Free-Living Birds. [In Russian.] Nauka Press, St. Petersburg. Google Scholar
- G. Eichhorn , R. H. Drent , J. Stahl , A. Leito , and T. Alerstam . 2009. Skipping the Baltic: The emergence of a dichotomy of alternative spring migration strategies in Russian Barnacle Geese. Journal of Animal Ecology 78:63–72. Google Scholar
- C. P. Ellington 1991. Limitations on animal flight performance. Journal of Experimental Biology 160:71–91. Google Scholar
- S. Engel , H. Biebach , and G. H. Visser . 2006. Metabolic costs of avian flight in relation to flight velocity: A study in Rose-coloured Starlings (Sturnus roseus, Linnaeus). Journal of Comparative Physiology B 176:415–427. Google Scholar
- J. F. Harrison , and S. P. Roberts . 2000. Flight respiration and energetics. Annual Review of Physiology 62:179–205. Google Scholar
- A. Hedenström 2002. Aerodynamics, evolution and ecology of avian flight. Trends in Ecology and Evolution 17:415–422. Google Scholar
- A. Hedenström 2008. Adaptations to migration in birds: Behavioural strategies, morphology and scaling effects. Philosophical Transactions of the Royal Society of London, Series B 363:287–299. Google Scholar
- P. Henningsson , H. Karlsson , J. Bäckman , T. Alerstam , and A. Hedenström . 2009. Flight speeds of swifts (Apus apus): Seasonal differences smaller than expected. Proceedings of the Royal Society of London, Series B 276:2395–2401. Google Scholar
- C. Kullberg , T. Fransson , and S. Jakobsson . 1996. Impaired predator evasion in fat Blackcaps (Sylvia atricapilla). Proceedings of the Royal Society of London, Series B 263:1671–1675. Google Scholar
- C. Kullberg , S. Jakobsson , and T. Fransson . 2000. High migratory fuel load impair predator evasion in Sedge Warblers. Auk 117:1034–1038. Google Scholar
- A. Kvist , Å. Lindström , M. Green , T. Piersma , and G. H. Visser . 2001. Carrying large fuel loads during sustained bird flight is cheaper than expected. Nature 413:730–732. Google Scholar
- U. M. Lindhe Norberg 2004. Bird flight. Acta Zoologica Sinica 50:921–935. Google Scholar
- U. M. Norberg 1990. Vertebrate Flight: Mechanics, Physiology, Morphology, Ecology and Evolution. Springer, Berlin. Google Scholar
- U. M. Norberg 1996. Energetics of flight. Pages 199–249 in Avian Energetics and Nutritional Ecology ( C. Carey , Ed.). Chapman & Hall, New York. Google Scholar
- C. J. Pennycuick 1975. Mechanics of flight. Pages 1–75 in Avian Biology, vol. 5 ( D. S. Farner , J. R. King , and K. C. Parkes , Eds.). Academic Press, New York. Google Scholar
- C. J. Pennycuick 1978. Fifteen testable predictions about bird flight. Oikos 30:165–176. Google Scholar
- C. J. Pennycuick 1989. Bird Flight Performance: A Practical Calculation Manual. Oxford University Press, New York. Google Scholar
- J. M. V. Rayner 1990. The mechanics of flight and bird migration performance. Pages 283–299 in Bird Migration ( E. Gwinner , Ed.). Springer, Berlin. Google Scholar
- K. Schmidt-Nielsen 1997. Animal Physiology: Adaptation and Environment, 5th ed. Cambridge University Press, New York. Google Scholar
- C. A. Schmidt-Wellenburg , H. Biebach , S. Daan , and G. H. Visser . 2007. Energy expenditure and wing beat frequency in relation to body mass in free flying Barn Swallows (Hirundo rustica). Journal of Comparative Physiology B 177:327–337. Google Scholar
- C. A. Schmidt-Wellenburg , S. Engel , and G. H. Visser . 2008. Energy expenditure during flight in relation to body mass: Effects of natural increases in mass and articial load in Rose-coloured Starlings. Journal of Comparative Physiology B 178:767–777. Google Scholar
- B. W Tobalske , T. L. Hedrick , K. P. Dial , and A. A. Biewener . 2003. Comparative power curves in bird flight. Nature 421: 363–366. Google Scholar
- J. R. Torre-Bueno , and J. Larochelle . 1978. The metabolic cost of flight in unrestrained birds. Journal of Experimental Biology 75:223–229. Google Scholar
- V. A. Tucker 1968. Respiratory exchange and evaporative water loss in the flying Budgerigar. Journal of Experimental Biology 48: 67–87. Google Scholar
- V. A. Tucker 1972. Metabolism during flight in the Laughing Gull, Larus atricilla. American Journal of Physiology 222:237–245. Google Scholar
- J. J. Videler 2005. Avian Flight. Oxford University Press, Oxford, United Kingdom. Google Scholar
- S. Ward , C. M. Bishop , A. J. Woakes , and P. J. Butler . 2002. Heart rate and the rate of oxygen consumption of flying and walking Barnacle Geese (Branta leucopsis) and Bar-headed Geese (Anser indicus). Journal of Experimental Biology 205:3347–3356. Google Scholar
- S. Ward , U. Möller , J. M. V. Rayner , D. M. Jackson , W. Nachtigall , and J. R. Speakman . 2004. Metabolic power of European Starlings Sturnus vulgaris during flight in a wind tunnel, estimated from heat transfer modelling, doubly labelled water and mask respirometry. Journal of Experimental Biology 207:4291–4298. Google Scholar
- T. P. Weber , and A. I. Houston . 1997. Flight costs, flight range and the stopover ecology of migrating birds. Journal of Animal Ecology 66:297–306. Google Scholar