A sufficient condition for additive -stability and application to reaction–diffusion models

The matrix A is said to be additively D-stable if A−D remains Hurwitz for all non-negative diagonal matrices D. In reaction–diffusion models, additive D-stability of the matrix describing the reaction dynamics guarantees the 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 non-negative 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 the stability of cyclic reaction networks in the presence of diffusion. Finally, a reaction network structure that fails to achieve additive D-stability is exhibited.