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

A very simple, practical, necessary and sufficient condition for diffusion-driven linear instability and parameter space determination in nonlinear reaction systems with three species is presented. With respect to the stability matrix A from linearization near a stable steady-state of a reaction system, two necessary conditions are given: 1. A is stable but not negative definite; and 2. 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 stronger conditions which, in fact, is shown to be sufficient for diffusion-driven instability. As an example, the result is applied to the three-species Oregonator, the model for the Belousov-Zhabotinsky reaction.