On The Relation Between Organizations And Limit Sets In Chemical Reaction Systems

Chemical organization theory has been suggested as a new approach to analyze complex reaction networks. Concerning the long-term behavior of the network dynamics we will study its foundations mathematically. Therefore we consider a chemical reactor containing molecules of different species reacting with each other according to a set of reaction rules. We further assume that the dynamical behavior of the concentration of each species is given by a continuous chemical ordinary differential equation. Abstracting from dedicated concentration values we consider a discrete problem: Which species can appear in the reactor after a long time? We define the limit set abstraction, which contains the subsets of species characterizing the long-term behavior. We prove that all these subsets are closed and that for all bounded limit sets, at least one of them is self-maintaining and thus is an organization. This implies for a chemical ordinary differential equation systems that any attractor that does not touch the state space boundaries (in particular, any periodic attractor) lies within one organization, that is, the set of species with positive concentrations in any state of such an attractor is an organization. This in turn explains why in a deterministic system evolving according to reaction rules one can observe species sets that are closed and self-maintaining, thus organizations.