Unstable subsystems cause Turing instability

Abstract

We study Turing instabilities in 3-component reaction-diffusion systems. The existence of a complementary pair of stable-unstable subsystems always gives rise to Turing instability for suitable diagonal diffusion matrices. There are two types of Turing instability, one called steady instability and the other wave instability. To determine which of the two types of instability actually occurs, easily verifiable conditions on unstable subsystems are given. A complementary pair of unstable-unstable subsystems in a stable full system also leads to steady instability. Our results give a perspective to the rich variety and complexity of pattern dynamics in 3-component systems of reaction-diffusion equations at the onset.