Chemical reactions network theory (CRNT) analysis and applications of poly-PL kinetic system

Evolutionary Game Theory (EGT) models the evolutionary phenomenon through a replicator system that is based on the polynomial payo functions. It is a polynomial dynamical system of ordinary differential equations that analyzes strategies that prove to be bene cial under certain conditions. In the study of Veloz et al., they proposed to analyze the dynamics of EGT models using Chemical Reaction Network Theory (CRNT) in the form of polynomial kinetics (POK). In CRNT, topological properties of the dynamical systems are studied and analyzed. From the CRNT point of view, it now becomes interesting to study a superset of POK, which we call poly-PL kinetics (PYK). This set is formed by getting nonnegative linear combinations of power law functions. Since only CRNs with mass-action (MAK) and power-law kinetics (PLK) are currently available, PYKs are transformed into PLK. In this study, we present two approaches on how these transformations can be done. In the first approach, we use a network structure-oriented transformations using the S-invariant term-wise addition of reactions, which we denoted as STAR. Whereas in the second approach, it is via Craciun\u27s Euclidean embedded graph (E- graph). Here, polynomial dynamical system can be regarded as being generated by some E-graph. Results on positive equilibria of PYK systems based on the associated E-graph are discussed. Aside from the transformation approaches, another pioneering work on PYK (to our knowledge) of this study is the qualitative analysis of certain poly-PL kinetics systems and its application to polynomial replicator systems. We are able to identify a class of poly-PL which have zero kinetic reactant efficiency, an important criteria for the existence of complex balanced equilibria