Realizations of quasi-polynomial systems and application for stability analysis

Quasi-polynomial systems are strongly related to mass action systems. In this paper, we extend mass action systems for the first. Then, we prove that any quasi-polynomial system with nonnegative exponents is realizable, i.e., there exists an extended mass action system can realize its dynamics. An algorithm is designed to get the realization. The input of the algorithm is the matrices characterizing the mathematical structure of the quasi-polynomial system, while the output is the realization. Since extended complex balanced mass action systems are locally asymptotically stable, quasi-polynomial systems with complex balanced realizations are also locally asymptotically stable. Based on the initial realization produced by the algorithm, we propose a systematic method for computing complex balanced realization of quasi-polynomial system. Lastly, we illustrate the efficiency of the method with two numerical examples.