We study real steady state varieties of the dynamics of chemical reaction networks. The dynamics are derived using mass action kinetics with parametric reaction rates. The models studied are not inherently parametric in nature. Rather, our interest in parameters is motivated by parameter uncertainty, as reaction rates are typically either measures with limited precision or estimated. We aim at detecting toricity and shifted toricity, using a framework that has been recently introduced and studied for the non-parametric case over both the reals and the complex numbers. While toricity requires that the variety specifies a subgroup of the direct power of the multiplicative group of the underlying field, shifted toricity requires only a coset. In the presence of parameters, we are not faced with decision problems anymore. Instead, we derive necessary and sufficient conditions in the parameters for toricity or shifted toricity to hold. Technically, we use real quantifier elimination methods. Our computations on biological networks here once more confirm shifted toricity as a relevant concept, while toricity holds only for degenerate parameter choices.

中文翻译：

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反应网络稳态变量的参数复曲面

我们研究化学反应网络动力学的真实稳态变化。使用具有参数反应速率的质量动力学来推导动力学。研究的模型本质上并不是固有的参数化。相反，我们对参数的兴趣是由参数不确定性引起的，因为反应速率通常是精度有限的措施或估计的措施。我们旨在使用最近引入并针对实数和复数的非参数情况引入并研究的框架来检测复曲面和移位复曲面。虽然复曲面要求该变种指定基础字段的乘法组的直接幂的一个子组，但转换复曲面仅需要一个陪集。在存在参数的情况下，我们不再面临决策问题。反而，我们在参数中得出了必要的充分条件，以保持复律性或转移的复律性。从技术上讲，我们使用实际的量词消除方法。我们在此处对生物网络的计算再次证实了复曲性是一个相关概念，而复曲性仅适用于简并参数选择。