How to generate species with positive concentrations for all positive times?

Abstract

Given a reaction (network) we are looking for minimal sets of species starting from which all the species will have positive concentrations for all positive times in the domain of existence of the solution of the induced kinetic differential equation. We present three algorithms to solve the problem. The first one essentially checks all the possible subsets of the sets of species. This can obviously work for only a few dozen species because of combinatorial explosion. The second one is based on an integer programming reformulation of the problem. The third one walks around the state space of the problem in a systematic way and produces all the minimal sets of the advantageous initial species, and works also for large systems. All the algorithms rely heavily on the concept of Volpert indices, used earlier for the decomposition of overall reactions [Kovács \textit{et al., Physical Chemistry Chemical Physics} 2004, 6, 1236]. Relations to the permanence hypothesis, possible economic or medical uses of the solution of the problem are analyzed, and open problems are formulated at the end.