We consider the problem of analyzing chemical reaction networks that may allow multiple positive steady states. We use tools from “classical” computer algebra (Gröbner bases over a parametrized domain, computation of a discriminant variety, graphical and mathematical analysis of solution sets, cylindrical decomposition) to help determine regions of stoichiometric compatibility classes that have multiple steady states. Hybrid symbolic-numeric tools are employed to determine stability of equilibria and also to determine or rule out the possibility of boundary equilibria. Symbolic and numeric methods of perturbation analysis are also developed in order to assess robustness of equilibria and equilibria counts to changes in parameters. A brief consideration of parameter identification for the kinetic rate constants is also presented. Computational effectiveness is demonstrated on a nontrivial benchmark example from biochemistry.

我们考虑分析可能允许多个正稳态的化学反应网络的问题。我们使用来自“经典”计算机代数的工具（Gröbner基于参数化域，判别变量的计算，解集的图形和数学分析，圆柱分解）来帮助确定具有多个稳态的化学计量兼容性类别的区域。使用混合符号数字工具确定平衡的稳定性，并确定或排除边界平衡的可能性。还开发了符号和数值扰动分析方法，以评估均衡的稳健性和针对参数变化的均衡计数。还简要介绍了动力学速率常数的参数识别方法。