Systems of differential equations with polynomial right-hand sides are very common in applications. In general, these systems can give rise to very complex dynamics: multiple equilibria, oscillations, and even chaotic dynamics. Even if we restrict our attention to polynomial dynamical systems that are generated by reaction networks, all these complex dynamical behaviors are still possible. On the other hand, if a polynomial dynamical system is generated by a weakly reversible deficiency zero ( \(\text {WR}_0\) ) reaction network, then its dynamics is known to be remarkably stable: Oscillations and chaotic dynamics are ruled out, and, up to linear conservation laws, there exists a single positive steady state, which is locally asymptotically stable. Here our main focus is on reaction networks \(\mathcal {N}\) which generate dynamical systems that can also be generated by some \(\text {WR}_0\) networks. Our interest is motivated by the fact that the dynamical systems generated by such networks \(\mathcal {N}\) enjoy all the stability properties mentioned above, but without the need for \(\mathcal {N}\) to satisfy the (quite restrictive) \(\text {WR}_0\) properties. We prove that if a given reaction network \(\mathcal {N}\) has a \(\text {WR}_0\) realization for all choices of rate constants, then there exists a unique \(\text {WR}_0\) network \(\mathcal {N}'\) such that \(\mathcal {N}\) is realizable by \(\mathcal {N}'\) . Additionally, we describe an algorithm which, for any reaction network \(\mathcal {N}\) , decides if \(\mathcal {N}\) is realizable by some \(\text {WR}_0\) network \(\mathcal {N}'\) , and finds this unique network \(\mathcal {N}'\) whenever it exists.
