Before revision:  Moreover, the largest open conjecture related to deterministic models, the so-called Global Attractor Conjecture, focuses on models with a deficiency of zero. In the stochastic reaction network setting it has been shown that a deficiency of zero together with a certain connectivity property of the associated reaction network implies that the stationary distribution of the model is a product of Poissons.
After revision:  Moreover, the largest open conjecture related to deterministic models, the species. In this paper, we consider these networks in an Erd\H os-R\'enyi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero together with a certain connectivity property of the associated reaction network implies that the stationary distribution of the model is a product of Poissons.