Realizations through Weakly Reversible Networks and the Globally Attracting Locus

We investigate the possibility that for any given reaction rate vector $k$ associated with a network $G$, there exists another network $G'$ with a corresponding reaction rate vector that reproduces the mass-action dynamics generated by $(G,k)$. Our focus is on a particular class of networks for $G$, where the corresponding network $G'$ is weakly reversible. In particular, we show that strongly endotactic two-dimensional networks with a two dimensional stoichiometric subspace, as well as certain endotactic networks under additional conditions, exhibit this property. Additionally, we establish a strong connection between this family of networks and the locus in the space of rate constants of which the corresponding dynamics admits globally stable steady states.