Abstract
Autocatalysis underlies the ability of chemical and biochemical systems to replicate. Autocatalysis was recently defined stoichiometrically for reaction networks; five types of minimal autocatalytic networks, termed autocatalytic cores were identified. A necessary and sufficient stoichiometric criterion was later established for dynamical autocatalysis in diluted regimes, ensuring a positive growth rate of autocatalytic species starting from infinitesimal concentrations, given that degradation rates are sufficiently low. Here, we show that minimal autocatalytic networks in the dynamical sense, in the diluted regime, follow the same classification as autocatalytic cores in the stoichiometric sense. We further prove the uniqueness of the stationary regimes of autocatalytic cores, with and without degradation, for all types, except types II with three catalytic loops or more, for which the question remains open. These results indicate that the stationary point is robust under perturbation at low concentrations. More complex behaviours require additional non-linear couplings.