RESEARCH IN PATTERN FORMATION
IN
BIOLOGICAL AND CHEMICAL SYSTEMS

Peter Moore (Mathematics - SMU)
Werner Horsthemke (Chemistry - SMU)

In 1952 Alan Turing proposed a simple reaction-diffusion model for pattern formation in biological systems.  In 1990 De Kepper et al (V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Phys. Rev. Lett. 64, (1990), 2953) and Swimmey et al (Q. Ouyand, H.L. Swinney, Nature 352, (1991), 610) observed Turing patterns in the chlorite-iodide-malonic acid (CIMA) reaction-diffusion system.  We are investigating the Lengyel-Rabai-Epstein model of the CDIMA reaction.  The CDIMA system is a simpler model for which the system of reaction-diffusion equations is given by (I. Lengyel, G. Rabai, I.R. Epstein, J. Amer. Chem. Soc. 112, (1990), 4606, 9104, S. Setayeshgar, M.C. Cross,  Phys. Rev. E 58, (1998), 4485):

Lengyel-Rabai-Epstein Model
The cofficients and boundary conditions are given in Setayeshgar and Cross.  The Starch Triiodide concentration obtained by solving this system in one dimension using our adaptive finite element code is (compare with Figure 3, Setayeshgar, Cross, Phys. Rev. E 59, (1999), 4258) where 0 < x < 0.3,
one dimensional starch triiodide
Solving in three dimensions on [0,0.3] x [0,0.133] on a 32x32x1 uniform grid with 4th order elements and initial conditions a small perturbation in y of the one-dimensional steady state we obtain the Starch Triiodide concentration (compare with Figure 1b, Setayeshgar, Cross, 1999)
two-dimensional results

The results of our three-dimensional calculations for Starch Triiodide on a 32x32x32 uniform grid with 4th order elements and initial conditions a small perturbation in y and z of the one-dimensional steady state are (slices through the pattern)
3d-1
3d2
3d3
3d4
3d5