A simple method of parameter space determination for diffusion-driven instability with three species

A very simple, practical, necessary and sufficient condition for diffusion- driven linear instability and parameter space determination in nonlinear re action systems with three species is presented. With respect to the stabili ty matrix A from linearization near a stable steady-state of a reaction sys tem, two necessary conditions are given: (i) A is stable but not negative definite; and (ii) either the largest diagonal elements of A is positive or the smallest diagonal cofactors of A is negative. Condition (i) can be generalized to any number of species but (ii) is a str onger condition which, in fact, is shown to be sufficient for diffusion-dri ven instability. As an example, the result is applied to the three-species Oregonator, the model for the Belousov-Zhabotinsky reaction. (C) 2001 Elsev ier Science Ltd. All rights reserved.