This paper concerns the dynamical behavior of weakly reversible, deterministically modeled population processes near the facets (codimension-one faces) of their invariant manifolds and proves that the facets of such systems are "repelling." It has been conjectured that any population process whose network graph is weakly reversible (has strongly connected components) is persistent. We prove this conjecture to be true for the subclass of weakly reversible systems for which only facets of the invariant manifold are associated with semilocking sets, or siphons. An important application of this work pertains to chemical reaction systems that are complex-balancing. For these systems it is known that within the interior of each invariant manifold there is a unique equilibrium. The Global Attractor Conjecture states that each of these equilibria is globally asymptotically stable relative to the interior of the invariant manifold in which it lies. Our results pertaining to weakly reversible systems imply that this conjecture holds for all complex-balancing systems whose boundary equilibria lie in the relative interior of the boundary facets. As a corollary, we show that the Global Attractor Conjecture holds for those systems for which the associated invariant manifolds are two-dimensional.
Weakly Reversible CF-Decompositions of Chemical Kinetic Systems
本文研究分解为弱可逆复因子分解（CF）体系的化学动力学系统。在幂律动力学系统中，CF系统（表示为PL-RDK系统）是其中反应物络合物的分支反应在动力学次序矩阵中具有相同行的那些。质量作用和广义质量作用系统（GMAS）是众所周知的例子。Schmitz的全球碳循环模型是一个先前研究过的非复可因子分解（NF）幂律系统（表示为PL-NDK）。我们得到新的弱可逆CF-分解的存在条件，并提出了一个算法来验证这些条件。我们讨论了识别独立分解的方法，即，其中子网络的化学计量子空间形成直和，因为这样的分解将子网络的正平衡集与整个网络的正平衡集相关联。然后我们利用这些结果确定了PL-NDK系统的正平衡集，这些正平衡集允许一个独立的弱可逆分解为PLP型PL-RDK系统，即，正平衡是对数参数化的，这是Fortun et al.（2019）的亏零定理的广泛推广。
