Using Differential Equations to Model Pigment Patterns
Abstract
Math in biology is typically thought of as statistics, relegated to the
study of population distributions and genetics; however, the scope of
mathematical biology is far more complex. In the area of developmental
biology, differential equations play a key role in understanding the
formation of pigment patterns. At the core of pattern formation are
a series of complex interactions between three types of molecules:
activators, inhibitors, and substrates. The rates at which these
substances are produced and diffuse, as well as the area over which
these molecules can act, all factor in the final development of a
particular pigment pattern. Hans Meinhardt has proposed a series
of differential equations modeling the formation of varied seashell
patterns (2003). Additionally, others have explored in depth
diffusion reaction mechanisms that model pattern formation in
larger vertebrates, such as giraffes (Murray, 1989). The work here
seeks to explore the development and derivation of these models, as
well as illustrate these models through computer software.
Table of Contents
Complete List of References
- [BP1]
Blanchard, P., R.L. Devaney and G.R. Hall.
Differential Equations 2nd ed.
Pacific Grove: Brooks/Cole, 2002.
- [MH1]
Meinhardt, H.
The Algorithmic Beauty of Sea Shells.
New York: Springer, 2003.
- [MH2]
Meinhardt, H.
Models for pattern formation and the position-specific activation of genes
in "On Growth, Form, and Computers." San Diego: Elsevier Academic Press, 2003, pp. 135—155.
- [MJ1]
Murray, J.D.
Mathematical Biology.
New York: Springer-Verlag, 1989.
- [NR1]
Newman, R. and M. Boles.
The Golden Relationship: Art, Math & Nature. Universal Patterns 2nd rev. ed.
Bradford: Pythagorean Press, 1992.
- [SI1]
Stewart, I.
Broken symmetries and biological patterns
in "On Growth, Form, and Computers." San Diego: Elsevier Academic Press, 2003, pp. 181—202.
