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Complex balanced mass-action systems (CBMASs) are of great importance in thefiled of biochemical reaction networks. However analyzing the persistence ofthese networks with high dimensions and time delays poses significantchallenges. To tackle this, we propose a novel approach that combines1-dimensional (1d) or 2d delayed CBMASs (DeCBMASs) and introduces inheritablecombination methods based on the relationship between semilocking sets andintersecting species. These methods account for various scenarios, includingcases where the set of intersecting species is empty, or there are no commonspecies in non-trivial semilocking sets and the intersecting species set, orwhen special forms are present. By utilizing these combination methods, wederive sufficient conditions for the persistence of high-dimensional DeCBMASs.This significantly expands the known class of delayed chemical reaction networksystems that exhibit persistence. The effectiveness of our proposed approach isalso demonstrated through several examples, highlighting its practicalapplicability in real-world scenarios. This research contributes to advancingthe understanding of high-dimensional DeCBMASs and offers insights into theirpersistent behavior.

Network Combination to Persistence of High-dimensional Delayed Complex Balanced Mass-action Systems