RESEARCH
IN PATTERN FORMATION

IN

BIOLOGICAL AND CHEMICAL SYSTEMS

Peter Moore (Mathematics - SMU)

Werner Horsthemke (Chemistry - SMU)

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):

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,

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)

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)