On the problem of evaluating quasistationary distributions for open reaction schemes

Abstract

A number of recent papers have been concerned with the stochastic modeling of autocatalytic reactions. In some instances the birth and death model has been criticized for its apparent inadequacy in being able to describe the long-term behavior of the catalyst, in particular the fluctuations in the concentration of the catalyst about its macroscopically stable state. This criticism has been answered, to some extent, with the introduction of the notion of a quasistationary distribution; a number of authors have established the existence of limiting conditional distributions that can adequately describe these fluctuations. However, much of the work appears only to be appropriate for dealing with closed systems, for attention is usually restricted to finite-state birth and death processes. For open systems it is more appropriate to consider infinite-state processes and, from the point of view of establishing conditions for the existence of quasistationary distributions, extending the results for closed systems is far from straightforward. Here, simple conditions are given for the existence of quasistationary distributions for Markov processes with a denumerable infinity of states. These can be applied to any open autocatalytic system. The results also extend to explosive processes and to processes that terminate with probability less than 1.