On Persistence of Some $\mathcal {W}_{I}$-Endotactic Chemical Reaction Networks

Abstract

Motivated by the concept of the endotactic network, a kind of special geometric structure in chemical reaction networks developed for persistence analysis, we propose a new notion, named <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {W}_{I}$</tex-math></inline-formula> -endotactic network. The corresponding network set is a larger class of set than the endotactic network, let alone the weakly reversible network. Based on an energy-like function, we prove that all 1-dimensional mass-action <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {W}_{I}$</tex-math></inline-formula> -endotactic networks are persistent. Further, we prove some higher dimensional network systems also to support persistence if the system is composed of a series of 1-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {W}_{I}$</tex-math></inline-formula> -endotactic networks. We discuss two cases for the sub-systems with and without intersecting species, and present the corresponding sufficient conditions to capture persistence. Finally, we use some examples including abstract and real biochemical reaction networks to illustrate our results.