A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems
A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems
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DOI：
10.3934/mbe.2019417
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发表时间：
2019-01-01
影响因子：
2.6
通讯作者：
Yu, Polly Y.
中科院分区：
文献类型：
--
作者：
Craciun, Gheorghe;Mueller, Stefan;Yu, Polly Y.
质量作用动力学及其推广出现在（生物）化学反应网络、种群动力学和流行病学的数学模型中。由有向图产生的动力系统通常是非线性的，难以分析。研究它们的一种方法是在网络上找到暗示或排除某些动力学性质的条件。例如，广义质量作用系统的顶点平衡稳态是通过图的每个顶点的净通量为零的状态。特别是，这样的稳定状态承认一个单项参数化。顶点平衡定态的存在唯一性问题可以用两种不同的方式重新表述，一种是与统计学中的伯奇定理有关，另一种是与广义多项式映射的双射性有关，类似于几何建模中出现的映射。我们给出了一个推广的Birch定理，通过提供一个充分条件的存在性和唯一性的顶点平衡的平衡状态。
Mass-action kinetics and its generalizations appear in mathematical models of (bio)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.