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%DIFDELCMD < \R%%% ^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. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete time Markov chain.
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 %DIFDELCMD < \R%%% , 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. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete time Markov chain.
D
bounded, persistent trajectories (i.e., for each trajectory x(t)\in
D
^d_{\ge 0
A
trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components)