  A chemical reaction network (CRN) is composed of reactions that can be seen
as interactions among entities called species, which exist within the system.
Endowed with kinetics, CRN has a corresponding set of ordinary differential
equations (ODEs). In Chemical Reaction Network Theory, we are interested with
connections between the structure of the CRN and qualitative properties of the
corresponding ODEs. One of the results in Decomposition Theory of CRNs is that
the intersection of the sets of positive steady states of the subsystems is
equal to the set of positive steady states of the whole system, if the
decomposition is independent. Hence, computational approach using independent
decompositions can be used as an efficient tool in studying large systems. In
this work, we provide a necessary and sufficient condition for the existence of
a nontrivial independent decomposition of a CRN, which leads to a novel
step-by-step method to obtain such decomposition, if it exists. We also
illustrate these results using real-life examples. In particular, we show that
a CRN of a popular model of anaerobic yeast fermentation pathway has a
nontrivial independent decomposition, while a particular biological system,
which is a metabolic network with one positive feedforward and a negative
feedback has none. Finally, we analyze properties of steady states of reaction
networks of specific influenza virus models.
