Regularities

Last  year saw a paused expedition into the world of phyllotaxy and certain patterns in the plant world, for example in the arrangement of fir cones and pineapples. Some evidence of this is to be found at reference 1. With the pause being represented by the unread reference 2, sitting on a shelf in the study.

While today, asking Bing what was meant by the phrase 'ontogenetic growth' - in the context of looking at the evolution of early human brain capacity - I got taken to reference 3, where I was intrigued to read of a regularity in the world of animals. The top of this letter is snapped above. Probably the result of a chance encounter with notice in the NYRB.

It turns out that you can say quite a lot about the growth of an animal during its lifetime with a simple model of energy flows in cells. This is summarised in the equations snapped above, with equation 3 (bottom left) giving the rate of growth as the difference of two simple functions of size, that is to say body weight.

With the table above giving values of the parameter 'a', in that equation, for various animals.

With all this boiling down to the single, universal equation 'r = 1 - e^-τ', where the variable 'r' is to do with size and the variable 'τ' is to do with age. The goodness of fit of this equation to the animal world is illustrated in the snap above.

The story we are told is that the capacity of the network delivering energy to cells - the arterial system as it were - scales at a different rate to the number of cells. With the result that, eventually, the animal is so big that there is no energy left over for growth - the horizontal asymptote top right above.

Most animals have determinate growth which means that they quite quickly reach something close to that maximum size. While other animals, for example many fish, live out their lives on the left hand part of the curve, seemingly having no particular upper limit on their size. This is called indeterminate growth.

A nice example of relatively simple mathematics saying something important about animals.

References

Reference 1: https://psmv5.blogspot.com/search?q=douady.

Reference 2: Do plants know maths: Unwinding the story of plant spirals, from Leonardo da Vinci to now - Stéphane Douady, Jacques Dumais, Christophe Golé, Nancy Pick - 2024. Published by the Princeton University Press, printed in China.

Reference 3: A general model for ontogenetic growth – Geoffrey B. West, James H. Brown, Brian J. Enquist – 2001.