For a class of polynomial kinetic systems, this work examines connections between system parameters, and uniqueness and stability of the resulting equilibria. Such systems are typically employed to describe nonlinear dynamics in chemical reaction networks, but over the last years they proved useful in modeling a wide range of nonlinear dynamic systems with applications in biology, process systems, economics or transportation problems. In particular , a canonical representation of the set of all possible feasible equilibrium solutions is developed. The characterization is made in terms of compartmental matrices which by construction are strictly stable and define the so-called family of solutions. Feasibility is imposed by a set of constraints, which are linear in the log-transformed space of complexes, and relate to the kernel of the stoichiometric subspace. One particularly interesting representation of these constraints can be established in terms of a class of monotonous functions which turn out to be critical to conclude uniqueness of equilibrium points in a class of deficiency one networks. One main consequence of such representation is the possibility of a simple constructive proof of the  deficiency one theorem. It also allows a precise characterization of the parameter space region of complex balance solutions we refer to as the Horn set. Future directions may involve detection or design of networks having multiple equilibria, or the use of complex balance condition to provide stabilization via feed-back control of open reaction systems .
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 .
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For a class of polynomial kinetic systems, this work examines connections between
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the resulting equilibria. Such systems are typically employed to describe nonlinear dynamics in chemical reaction networks, but over the last years they proved useful in modeling
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nonlinear dynamic systems with applications in
application fields such as chemistry, systems
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systems, economics or transportation problems. In particular
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functions which turn out to be critical to conclude uniqueness of equilibrium points in
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
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deficiency one networks. One main consequence of such representation is the possibility of a simple constructive
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It also allows a precise characterization of the parameter space region of complex balance solutions we refer to as the Horn set. Future directions may involve detection or design of networks having multiple equilibria, or the use of complex balance condition to provide stabilization via feed-back control of open reaction systems
The developed notions and results are illustrated through examples