一阶内向反应网络
First order endotactic reaction networks
AI总结：
本文研究了一阶内向反应网络的特性，证明了其在稳定性方面的结论，并扩展到更高阶反应网络的质量作用系统。
摘要：
反应网络是一种广泛用于建模不同科学学科中多样现象的通用框架。赋予质量作用动力学的反应网络的动态过程是一个由有向图定义的微分方程系统，这种图被称为“反应图”。内向性是一种用于研究质量作用系统的持久性和永久性的图属性。在本文中，我们对一阶内向反应图进行了详细表征。此外，我们还提供了一个关于非一阶反应网络内向性的充分条件。对一阶内向反应图的表征导致了反应图邻接矩阵的谱性质。作为结果，我们证明了每一个一阶内向的质量作用系统作为线性微分方程系统都有一个弱可逆的零缺陷实现，并且在每个（正）化学计量类中有一个唯一的平衡点，该平衡点在指数意义上全局渐近稳定（并且是正的）。利用渐近自治微分方程的稳定性结果，构造了例子以说明全局稳定性结果可以扩展到由非线性微分方程表示的更高阶反应网络的质量作用系统，这些系统不一定是内向的。与经典的证明全局渐近稳定性的方法不同，本证明不依赖于构造Lyapunov函数。本文可能成为表征更高阶内向反应图和研究一般质量作用系统全局稳定性的起点。
英文摘要：
Reaction networks are a general framework widely used in modeling diverse phenomena in different science disciplines. The dynamical process of a reaction network endowed with mass-action kinetics is a mass-action system as an ODE defined by a directed graph, the so-called ``reaction graph''. Endotacticity is a graph property used to study persistence and permanence of mass-action systems. In this paper, we provide a detailed characterization of first order endotactic reaction graphs. Besides, we provide a sufficient condition for endotacticity of reaction networks which are not necessarily of first order. Such a characterization of a first order endotactic reaction graph yields the spectral property of the adjacency matrix of the reaction graph. As a consequence, we prove that every first order endotactic mass-action system as a linear ODE has a weakly reversible deficiency zero realization, and has a unique equilibrium which is exponentially globally asymptotically stable (and is positive) in each (positive) stoichiometric compatibility class. Using a stability result for asymptotically autonomous differential equations, examples are constructed to illustrate that the global stability results can be extended to mass-action systems of higher order reaction networks modeled by nonlinear ODEs, which are not necessarily endotactic. Different from the classical approaches for proving global asymptotic stability, the proof does not rely on the construction of a Lyapunov function. This paper may serve as a starting point of characterizing endotactic reaction graphs of higher orders and studying global stability of mass-action systems in general.
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