A numerical study of the performance of the time integration schemes: IMEX and Operator Splitting combined with the original version of the finite element discontinuous spatial discretization known as ``Local Discontinuous Galerkin'' (LDG) and its minimal dissipation version, MD-LDG is presented. The study focuses on precision and numerical stability for diffusion and nonlinear reaction systems which arise in chemical engineering problems, in particular in the modeling of Turing patterns. Numerical experiments show that discontinuous Galerkin finite element method is an excellent alternative for the simulation of this class of problems.