Characterize the parameter locus for WR realizations of a given network

Determine, for a fixed reaction network N, the full set of positive rate constant vectors κ such that the mass-action system (N, κ) admits a realization by a weakly reversible deficiency-zero network (i.e., the associated mass-action ODE system has a WR realization).

Background

The paper develops an algorithm (Algorithm 2) deciding whether a network is WR-realizable for all choices of rate constants and proving that, when WR-realizable, the realizing WR network is unique. However, for networks that are not WR-realizable for all parameter values, understanding precisely which rate constant sets yield a WR realization remains unresolved.

Identifying this parameter region likely involves symbolic convex cone computations and may require tools from real algebraic geometry. The authors indicate plans to investigate special classes where such regions can be characterized, as illustrated by simple examples in the paper.

References

Question 3. Given a reaction network $N$, what is the set of rate constants $\kappa$ such that the mass-action system $(N,\kappa)$ has a $WR$ realization? This seems to be a hard question, which involves symbolic convex cone computation and may require tools from real algebraic geometry.

— Weakly reversible deficiency zero realizations of reaction networks (2502.17461 - Buxton et al., 10 Feb 2025) in Section 6, Discussion and Future Work