  We present three examples of chemical reaction networks whose ordinary
differential equation scaling limit are almost identical and in all cases
stable. Nevertheless, the Markov jump processes associated to these reaction
networks display the full range of behaviors: one is stable (positive
recurrent), one is unstable (transient) and one is marginally stable (null
recurrent). We study these differences and characterize the invariant measures
by Lyapunov function techniques. In particular, we design a natural set of such
functions which scale homogeneously to infinity, taking advantage of the same
scaling behavior of the reaction rates.
