In this paper, we propose a class of discontinuous Galerkin methods under the scalar auxiliary variable framework (SAV-DG) to solve a biological patterning model in the form of parabolic-elliptic partial differential equation system. In particular, mixed-type discontinuous Galerkin approximations are used for the spatial discretization, aiming to achieve a balance between the high resolution and computational cost. Second and third order backward differentiation formulas are considered under SAV framework for discrete energy stability. Numerical experiments are provided to show the effectiveness of the fully discrete schemes and the governing factors of patterning formation.