Symmetric, asymmetric, and antiphase Turing patterns in a model system with two identical coupled layers.

We study Turing pattern formation in a model reaction-diffusion system with two coupled identical layers. The coupling creates a pitchfork bifurcation, which unfolds the symmetric steady state via primary Turing instability, into a pair of distinct, unstable, asymmetric steady states (a-SS). The a-SS gain stability at a reverse Turing bifurcation. The multiple stabilities created by the coupling generate a corresponding multiplicity of structures, including symmetric, asymmetric, antiphase, and localized Turing patterns. Coexistence and competition of the different types of Turing patterns are studied. A one-dimensional localized structure exhibits striking curvature effects.