Before revision: It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have bounded, persistent trajectories (i.e., for each trajectory x(t)\in ^d_{\ge 0, the expression i=1^d |\ln(x_i(t))| is uniformly bounded in time) \mbox{%DIFAUXCMD AndBounded_ONE2011,AndGAC_ONE2011}0pt%DIFAUXCMD }  . It is conjectured that the stochastically modeled analogs of these systems are positive recurrent.
After revision: It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have bounded, persistent trajectories (i.e., for each trajectory x(t)\in ^d_{\ge 0, the expression i=1^d |\ln(x_i(t))| is uniformly bounded in time) \mbox{%DIFAUXCMD AndBounded_ONE2011,AndGAC_ONE2011}0pt%DIFAUXCMD } trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components) . It is conjectured that the stochastically modeled analogs of these systems are positive recurrent.