广义质量作用系与实指数和符号指数多项式方程的正解
Generalized Mass-Action Systems and Positive Solutions of Polynomial Equations with Real and Symbolic Exponents
摘要：
化学反应网络理论研究的对象是具有质量作用动力学的化学反应网络所产生的动力系统。特别是，该理论提供了关于所有速率常数和初始条件下正稳态的唯一性、存在性和稳定性的陈述。对于相应的多项式方程，所得结果保证了对所有正参数，正解的存在唯一性。我们解决了最近的扩展CRNT，称为广义质量作用系统，反应速率被允许在浓度的幂律。特别地，（真实）动力学级数可以不同于（整数）化学计量系数。与质量作用动力学一样，复杂的平衡平衡是由底层网络的图形拉普拉斯决定的，可以用二项方程表征，并用单项式参数化。在代数方面，我们专注于一个建设性的表征多项式方程的正解的实指数和符号。所有速率常数和初始条件的唯一性和存在性还取决于化学计量和动力学序子空间的符号向量。这导致了伯奇定理的推广，该定理对于指数中的某些扰动是鲁棒的。在这种情况下，我们讨论了多重复杂的平衡均衡的发生。我们通过一个运行的例子来说明我们的结果，并提供了一个MAPLE工作表与所有算法方法的实现。
英文摘要：
Dynamical systems arising from chemical reaction networks with mass action kinetics are the subject of chemical reaction network theory (CRNT). In particular, this theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions. In terms of the corresponding polynomial equations, the results guarantee uniqueness and existence of positive solutions for all positive parameters. We address a recent extension of CRNT, called generalized mass-action systems, where reaction rates are allowed to be power-laws in the concentrations. In particular, the (real) kinetic orders can differ from the (integer) stoichiometric coefficients. As with mass-action kinetics, complex balancing equilibria are determined by the graph Laplacian of the underlying network and can be characterized by binomial equations and parametrized by monomials. In algebraic terms, we focus on a constructive characterization of positive solutions of polynomial equations with real and symbolic exponents. Uniqueness and existence for all rate constants and initial conditions additionally depend on sign vectors of the stoichiometric and kinetic-order subspaces. This leads to a generalization of Birch's theorem, which is robust with respect to certain perturbations in the exponents. In this context, we discuss the occurrence of multiple complex balancing equilibria. We illustrate our results by a running example and provide a MAPLE worksheet with implementations of all algorithmic methods.
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