  This paper studies the relations among system parameters, uniqueness, and
stability of equilibria, for kinetic systems given in the form of polynomial
ODEs. Such models are commonly used to describe the dynamics of nonnegative
systems, with a wide range of application fields such as chemistry, systems
biology, process modeling or even transportation systems. Using a flux-based
description of kinetic models, a canonical representation of the set of all
possible feasible equilibria is developed. The characterization is made in
terms of strictly stable compartmental matrices to define the so-called family
of solutions. Feasibility is imposed by a set of constraints, which are linear
on a log-transformed space of complexes, and relate to the kernel of a matrix,
the columns of which span the stoichiometric subspace. One particularly
interesting representation of these constraints can be expressed in terms of a
class of monotonous decreasing functions. This allows connections to be
established with classical results in CRNT that relate to the existence and
uniqueness of equilibria along positive stoichiometric compatibility classes.
In particular, monotonicity can be employed to identify regions in the set of
possible reaction rate coefficients leading to complex balancing, and to
conclude uniqueness of equilibria for a class of positive deficiency networks.
The latter result might support constructing an alternative proof of the
well-known deficiency one theorem. The developed notions and results are
illustrated through examples.
