Energy efficiency and allometry of movement of swimming and flying animals

Rahul Bale, Max Hao, Amneet Pal Singh Bhalla, Neelesh A. Patankar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

71 Scopus citations


Which animals use their energy better during movement? One metric to answer this question is the energy cost per unit distance per unit weight. Prior data show that this metric decreases with mass, which is considered to imply that massive animals are more efficient. Although useful, this metric also implies that two dynamically equivalent animals of different sizes will not be considered equally efficient. We resolve this longstanding issue by first determining the scaling of energy cost per unit distance traveled. The scale is found to be M2/3 or M1/2, where M is the animal mass. Second, we introduce an energy-consumption coefficient (CE) defined as energy per unit distance traveled divided by this scale. CE is a measure of efficiency of swimming and flying, analogous to how drag coefficient quantifies aerodynamic drag on vehicles. Derivation of the energy-cost scale reveals that the assumption that undulatory swimmers spend energy to overcome drag in the direction of swimming is inappropriate. We derive allometric scalings that capture trends in data of swimming and flying animals over 10-20 orders of magnitude by mass. The energy-consumption coefficient reveals that swimmers beyond a critical mass, and most fliers are almost equally efficient as if they are dynamically equivalent; increasingly massive animals are not more efficient according to the proposed metric. Distinct allometric scal-ings are discovered for large and small swimmers. Flying animals are found to require relatively more energy compared with swimmers.

Original languageEnglish (US)
Pages (from-to)7517-7521
Number of pages5
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number21
StatePublished - May 27 2014


  • Cost of transport
  • Froude efficiency

ASJC Scopus subject areas

  • General


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