Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.

中文翻译：

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反应网络中的绝对浓度稳健性和多重平稳性：共存条件

应用中出现的许多反应网络是多稳态的，即它们具有多个稳态的能力，而一些网络表现出绝对浓度鲁棒性（ACR），这意味着某些物质浓度在所有稳态下都是相同的。多平稳性和 ACR 在生物环境中都很重要，但直到最近才将注意力集中在这些特性共存的可能性上。我们的主要结果表明，这种最多双分子网络（涵盖生物学中出现的大多数网络）的共存需要至少三个物种、五个复合物和三个反应。我们根据线性守恒定律的数量证明了一般网络反应数量的附加界限。最后，我们证明，除了少数特殊情况外，ACR 相当于小型双分子网络（更准确地说，一维或最多两个物种）的非多平稳性。我们的证明涉及稀疏多项式系统的分析，并且我们还使用化学反应网络理论的经典结果。