Very often, models in biology, chemistry, physics and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or powerlaw dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems. redirect the reaction z → y * . Instead of the reaction z → y * with flux J z→y * , we have M reactions z → y j with fluxes J ′ z→y j = J y * →y j . Let (G ′ , J ′ ) denote this newest flux system. Recall that flux equivalence means Equation (11) holds at each vertex of G and G ′ . Here we only need to look at the vertex z to show that (G ′ , J ′ ) ∼ (G, J ). Note that y j − z = w j − w 0 . From P (G,J) (y * ) = 0, we also have M j=1 J y * →y j = J z→y * . Thus, the weighted sum of vectors coming out of z isFinally, we prove that the potentials are unchanged. Trivially P (G,J ) (y * ) = P (G ′ ,J ′ ) (y * ) = 0. Also P (G,J) (y j ) = J y * →y j = J ′ z→y j = P (G ′ ,J ′ ) (y j ) for j = 1, 2, . . . , M . Last but not least,J y * →y j = J z→y * = −P (G,J) (z).We have shown that the resulting flux system (G ′ , J ′ ) is flux equivalent to the original flux system (G, J ), and the potential at each vertex is preserved.Remark. In Lemma 4.3, the source vertex z may not be distinct from y j .
