  In this paper we study a class of stochastic chemical reaction networks
(CRNs) for which chemical species are created by a sequence of chain reactions.
We prove that under some convenient conditions on the initial state, some of
these networks exhibit a discrete-induced transitions (DIT) property: isolated,
random, events have a direct impact on the macroscopic state of the process. If
this phenomenon has already been noticed in several CRNs, in auto-catalytic
networks in the literature of physics in particular, there are up to now few
rigorous studies in this domain. A scaling analysis of several cases of such
CRNs with several classes of initial states is achieved. The DIT property is
investigated for the case of a CRN with four nodes. We show that on the normal
timescale and for a subset of (large) initial states and for convenient
Skorohod topologies, the scaled process converges in distribution to a Markov
process with jumps, an Additive Increase/Multiplicative Decrease (AIMD)
process. This asymptotically discontinuous limiting behavior is a consequence
of a DIT property due to random, local, blowups of jumps occurring during small
time intervals. With an explicit representation of invariant measures of AIMD
processes and time-change arguments, we show that, with a speed-up of the
timescale, the scaled process is converging in distribution to a continuous
deterministic function. The DIT analyzed in this paper is connected to a simple
chain reaction between three chemical species and is therefore likely to be a
quite generic phenomenon for a large class of CRNs.
