In this work we extend the characterization of injectivity via the Jacobian criterion first developed by Craciun and Feinberg for chemical reaction networks with outflow reactions to arbitrary chemical reaction networks taken with mass action kinetics . Injective chemical reaction networks do not have the capacity to admit multiple positive steady states for any rate constants and within each stoichiometric class. It is shown that a network is injective if and only if the determinant of the Jacobian of the system of ordinary differential equations associated to the network never vanishes . The determinant  is a polynomial on the species concentrations and the rate constants , and its coefficients are fully determined. Previous works apply to chemical reaction networks whose stoichiometric space has maximal dimension. Here we present a direct route, independent of the dimension of the stoichiometric space which precludes at the same time the existence of degenerate steady states  .
We provide a Jacobian criterion that applies to chemical reaction networks taken with mass-action kinetics to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. Our work extends previous results by Craciun and Feinberg for chemical reaction networks with stoichiometric space of maximal dimension. Specifically, a network is called injective if the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws . The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states . Further, we relate injectivity of a chemical reaction network to that of the chemical reaction network obtained by adding outflow reactions for all species .
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In this work we extend the characterization of injectivity via the Jacobian criterion first developed by Craciun and Feinberg for chemical reaction networks with outflow reactions to arbitrary
We provide a Jacobian criterion that applies to
R
mass action kinetics . Injective chemical reaction networks do not have the capacity to admit
mass-action kinetics to preclude the existence of
R
for any rate constants and
within any stoichiometric class for any choice of rate constants. Our work extends previous results by Craciun and Feinberg for chemical reaction networks with stoichiometric space of maximal dimension. Specifically, a network is called injective if the species formation rate function is injective in the interior of the positive orthant
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It is shown
We show
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the system of ordinary differential equations associated to the network never vanishes
a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws
A
of the function
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on
in
R
,
(linear in the latter)
R
Previous works apply to chemical reaction networks whose stoichiometric space has maximal dimension. Here we present a direct route, independent of the dimension of the stoichiometric space which precludes at the same time the
The criterion also precludes the
A
. Further, we relate injectivity of a chemical reaction network to that of the chemical reaction network obtained by adding outflow reactions for all species