Dissipative structures in reaction–diffusion systems: Numerical determination of bifurcations in the sphere

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Abstract
The steady state spatial patterns arising spontaneously in open nonlinear reaction–diffusion systems beyond an instability point of the thermodynamic branch are studied numerically for a simple kinetic scheme. The set of nonlinear partial differential equations, describing the system, is converted to a (large) set of ordinary differential equations. It is stressed that the resulting system is stiff, and must be solved accordingly. An efficient algorithm is outlined, based on Stiff predictor–corrector formulas and sparse matrix techniques, which yield a gain of a factor 470 in computing time over nonstiff methods. The developed algorithm is used to determine quantitatively the primary and first few secondary bifurcations in the sphere, thus simulating a biological cell or early blastula. Spontaneous gradient formation and ’Chemical hysteresis’, connected to the occurrence of multiple steady states, is encountered. The succession of stable patterns found for increasing size of the sphere is suggested to act as an ideal mechanism underlying the process of mitosis.

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