  In biochemical networks, complex dynamical features such as superlinear
growth and oscillations are classically considered a consequence of
autocatalysis. For the large class of parameter-rich kinetic models, which
includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we
show that certain submatrices of the stoichiometric matrix, so-called unstable
cores, are sufficient for a reaction network to admit instability and
potentially give rise to such complex dynamical behavior. The determinant of
the submatrix distinguishes unstable-positive feedbacks, with a single
real-positive eigenvalue, and unstable-negative feedbacks without real-positive
eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive
feedbacks that are Metzler matrices. Thus there are sources of dynamical
instability in chemical networks that are unrelated to autocatalysis. We use
such intuition to design non-autocatalytic biochemical networks with
superlinear growth and oscillations.
