  We consider a class of matrices with a specific structure that arises, among
other examples, in dynamic models for biological regulation of enzyme synthesis
(Tyson and Othmer, 1978). We first show that a stability condition given in
(Tyson and Othmer, 1978) is in fact a necessary and sufficient condition for
diagonal stability of this class of matrices. We then revisit a recent
generalization of (Tyson and Othmer, 1978) to nonlinear systems given in
(Sontag, 2005), and recover the same stability condition using our diagonal
stability result. Unlike the input-output based arguments employed in (Sontag,
2005), our proof gives a procedure to construct a Lyapunov function. Finally we
study static nonlinearities that appear in the feedback path, and give a
stability condition that mimics the Popov criterion.
