Fractals and Allometry

 

Advisor:  Jack Heidel

 

Description:  Quantitative (or mathematical) biology made a big advance with the publication of D’Arcy Thompson’s On Growth and Form in 1917.  He provided many examples of regularity and symmetry in biology although few formulas.  After this it gradually became noticed that many processes of life can be described by allometric scaling laws of the form

 

                                                              Y=YoXb

 

which hold true over many orders of magnitude.  Perhaps the best known of all these scaling relationships is

 

                                                              B=BoM3/4

 

where B is the basal metabolic rate and M is the mass of a particular species (Bo is an experimentally determined constant), and the formula is presumed to hold over a huge range of species from the very large to the very small. 

Power law relationships such as these have a simple geometrical interpretation in terms of fractals.  The purpose of this project is to investigate the several known theories for the origin of allometric relationships and explain them in terms of fractal geometry. 

 

References: 

      1.  Jayanth Banavar et al, Supply-demand balance and metabolic scaling,

            P roc Nat Acad Sci 99(2002)10506.

  1. Adrian Bejan,  Shape and Structure from Engineering to Nature, Cambridge 2000.
  2. James Brown and Geoffrey West,  Scaling in Biology, Oxford 2000.
  3. Olaf Dreyer,  Allometric scaling and central source systems,  Phys Rev Lett 87(2001)038101.
  4. D’Arcy Thompson, On Growth and Form, Cambridge 1917.
  5. Geoffrey West and James Brown, Life’s universal scaling laws, Physics Today, Sept 2004, 36.
  6. Tamas Vicsek,  Fluctuations and Scaling in Biology, Oxford 2001.

 

Prerequisites:  Good academic record in all math courses completed including Calc I,II and III and Differential Equations.  Take Math 3500, Introduction to Computer Assisted Research, in Spring 2005.