Before revision: As the total number of molecules in a system decreases, transitions in the qualitative behavior of autocatalytic chemical reaction dynamics may appear. By adapting a method based on  the discrete-time Markov process , we provide here a general analytic tool for the reaction dynamics of systems with a small number of molecules.
After revision: As the total number of molecules in a system decreases, transitions in the qualitative behavior of autocatalytic chemical reaction dynamics with a decrease in molecule number have attracted much attention. Here, a method based on the discrete-time Markov process , we provide here a general analytic tool for the reaction dynamics of systems with a small number of molecules.