Linking Network Structure and Dynamics to Describe the Set of Persistent Species in Reaction Diffusion Systems

In this work we characterize the set of persistent species in dynamical systems related to chemical reaction networks. Chemical reaction networks consist of a set of species and a set of reaction rules describing the interactions between the species. From a reaction network, by differentiation with respect to time and space, a reaction-diffusion system (RDS) can be derived describing the dynamics of the concentrations of the species. We show how double integration with respect to time and space of the solutions of the RDS conversely leads back to the reaction network and reveals all the possibly persistent subsets of species. We show that organizations as defined in chemical organization theory (COT) are strongly related to the persistent subsets of species. Organizations are subsets of species that have two properties. First, they are closed, that is, there is no reaction running on them that produces new species which are not contained in the organizations. Second, organizations are self-maintaining. By additionally allowing for the distribution of species, we generalize organizations towards distributed organizations (DOs). After introducing our concept of persistence as the first main result of this study, we prove that for a given reaction network the set of DOs is always a lattice. The second main result is that the set of persistent species of a solution of an RDS is always a DO. By linking these two results we achieve a connection between persistence concerning a single solution of an RDS and persistence with regard to all solutions of all RDSs having one and the same underlying reaction network. We show how this strongly benefits reaction network analysis. By presenting simulation results performed with MATLAB, we illustrate the discussed phenomena.