I've seen that metabolic rate scales logarithmically as function of mass for many animals over an extremely large span of parameters. What other scaling laws exist at the individual level?

  • $\begingroup$ Obviously, one has surface-area to volume issues (e.g., it is speculated that huge dragonflies were possible in the past due to increase O2 despite lack of lungs) and cross-sectional area to volume/mass issues (e.g., bigger animals tend to require disproportionately thicker legs). $\endgroup$
    – user1858
    May 15, 2013 at 13:01

1 Answer 1


Here are some off the top of my head.

  • The height an animal can jump depends on the muscle cross-sectional area ($l^2$) and its mass ($l^3$). Mass grows faster with body size ($l$) so small animals can therefore jump higher relative to their body size ($l$) than large animals. Very similar scaling exists for strength of limbs vs. mass, ability to fly vs. mass, etc.

  • In pinhole eyes (as found in clams and nautilus) sensitivity to light depends on the size of the pupil ($l^2$), the bigger the hole the more light gets in. However the ability to focus depends on the reciprocal of the size of the hole (but I don't know the scaling, so $l^{-n}$). Therefore there is a trade off between sensitivity and ability to focus.

  • There are some recent models about foraging behaviour in a 2D terrestrial environment (with the front of the animal being a length $l$) vs. foraging in a 3D marine environment (with the front of the animal being a surface $l^2$). The model predicts different behaviours for the size of prey animals should aim for.

  • The rate at which sound attenuates scales with the frequency of the sound as $1/f^2$. The frequency of a resonator scales with the reciprocal of volume $1/l^3$. So small animals make high pitched sounds which attenuate quickly and don't travel very far.

  • Finally, the properties of fluids change with size. The Reynolds number describes viscosity and is dependent on linear scale $l$. Therefore small animals experience fluids as viscous and can "crawl" through water whereas large animals have to "swim" through water.

However, any directly physical or chemical-physical property of biology will experience scaling laws.

  • $\begingroup$ This is great! Do you have any references for some of these so I can read more, especially the first three? $\endgroup$
    – Hooked
    May 16, 2013 at 2:22
  • $\begingroup$ Are you looking for academic paper style references or something a bit softer? Here's a few papers anyway. Hope you don't get stuck with paywalls. sciencedirect.com/science/article/pii/S0003347271801343 <- Bats calls damtp.cam.ac.uk/user/gold/pdfs/purcell.pdf <- 70's paper about Reynolds jeb.biologists.org/content/208/18/3581.full <- allometry of flight $\endgroup$
    – timcdlucas
    May 16, 2013 at 2:46
  • $\begingroup$ Just for the record, jumping height depends also on the distance ($l$) along which a muscle's force delivers energy. The net result is independence ($l^0$) between jumping height and body size, but smaller animals still jump higher relative to size (by a factor of $1/l$). $\endgroup$
    – lauir
    Feb 11, 2016 at 1:57

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