  Complex balanced mass-action systems (CBMASs) are of great importance in the
filed of biochemical reaction networks. However analyzing the persistence of
these networks with high dimensions and time delays poses significant
challenges. To tackle this, we propose a novel approach that combines
1-dimensional (1d) or 2d delayed CBMASs (DeCBMASs) and introduces inheritable
combination methods based on the relationship between semilocking sets and
intersecting species. These methods account for various scenarios, including
cases where the set of intersecting species is empty, or there are no common
species in non-trivial semilocking sets and the intersecting species set, or
when special forms are present. By utilizing these combination methods, we
derive sufficient conditions for the persistence of high-dimensional DeCBMASs.
This significantly expands the known class of delayed chemical reaction network
systems that exhibit persistence. The effectiveness of our proposed approach is
also demonstrated through several examples, highlighting its practical
applicability in real-world scenarios. This research contributes to advancing
the understanding of high-dimensional DeCBMASs and offers insights into their
persistent behavior.
