Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions
Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions
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DOI：
10.1137/17m1129076
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发表时间：
2019-01-01
影响因子：
1.2
通讯作者：
Craciun, Gheorghe
中科院分区：
文献类型：
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作者：
Craciun, Gheorghe
在生物学、化学、物理学和工程学中，一些最常见的数学模型是多项式动力系统，即右边有多项式的微分方程系统。受生物化学和化学工程中反应网络分析的概念和结果的启发，我们证明了正正交上的任何多项式动力系统都可以被认为是由嵌入在R-n中的有向图生成的，称为欧几里得嵌入图。这使我们能够将关于反应网络模型的关键猜想（如全局吸引子猜想，或持久性猜想）转换成关于一些重要的多项式动力系统类的更一般的版本。然后，我们引入了环微分包体，它是一种具有显著几何结构的分段常数自治动力系统。我们证明了如果欧几里得嵌入图G具有可逆性，那么由G生成的任何多项式动力系统都可以嵌入到一个环微分包含中。我们讨论了这种嵌入如何为证明全局吸引子猜想和持久性猜想提供了一种方法。
Some of the most common mathematical models in biology, chemistry, physics, and engineering are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that have been developed for the analysis of reaction networks in biochemistry and chemical engineering, we show that any polynomial dynamical system on the positive orthant R-n > 0 can be regarded as being generated by an oriented graph embedded in R-n, called a Euclidean embedded graph. This allows us to recast key conjectures about reaction network models (such as the Global Attractor Conjecture, or the Persistence Conjecture) into more general versions about some important classes of polynomial dynamical systems. Then, we introduce toric differential inclusions, which are piecewise constant autonomous dynamical systems with a remarkable geometric structure. We show that if a Euclidean embedded graph G has some reversibility properties, then any polynomial dynamical system generated by G can be embedded into a toric differential inclusion. We discuss how this embedding suggests an approach for the proof of the Global Attractor Conjecture and Persistence Conjecture.