Computer simulations of three-dimensional Turing patterns in the Lengyel-Epstein model

We investigate numerically Turing patterns in the Lengyel-Epstein model in three dimensions. In a bulk homogeneous system under periodic boundary conditions, we obtain not only lamellar, cylindrical, and spherical structures but also several interconnected periodic structures including the Schwartz P-surface structure. In order to examine Turing patterns in the conditions accessible experimentally, we consider inhomogeneous systems where a parameter in the reaction-diffusion equations depends on the space coordinate with either Dirichlet or Neumann boundary conditions. In this situation, we find that a perforated-lamellar structure and an Fddd structure, both of which have a uniaxial symmetry, appear depending on the boundary conditions.