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Matrix A is said to be additively D-stable if A-D remains Hurwitz for all nonnegative diagonal matrices D. In reaction-diffusion models, additive D-stability of the matrix describing the reaction dynamics guarantees stability of the homogeneous steady-state, thus ruling out the possibility of diffusion-driven instabilities. We present a new criterion for additive D-stability using the concept of compound matrices. We first give conditions under which the second additive compound matrix has nonnegative off-diagonal entries. We then use this Metzler property of the compound matrix to prove additive D-stability with the help of an additional determinant condition. This result is then applied to investigate stability of cyclic reaction networks in the presence of diffusion.