K-variable kinetics

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A chemical reaction network \mathcal{N} = (\mathcal{S}, \mathcal{C}, \mathcal{R}) is said to be endowed with k-variable kinetics if the kinetic vector \vec{R}(t,\vec{c}) has entries of the form

\vec{R}_i(t,\vec{c}) = k_i(t) c_1^{y_{1i}} \cdots c_n^{y_{ni}}, \; \; \; i = 1, \ldots, r,

where \vec{y}_i = [ y_{1i}, \ldots, y_{ni} ]^T is the stoichiometric vector for the reactant complex of the i^{th} reaction and 1/\eta < k_i(t) < \eta, for i = 1, \ldots, n, for  \eta > 0.

K-variable kinetics is a generalization of mass action kinetics which allows the rate of each reaction to vary within a compact region over time. The terminology "k-variable kinetics" was introduced by Gheorghe Craciun, Casian Pantea and Fedor Nazarov[1] and has been further studied by Casian Pantea[2] and David Anderson[3], who called the kinetics "bounded mass action kinetics".


  1. Gheorghe Craciun, Casian Pantea, and Fedor Nazarov, Persistence and permanence of mass-action and power-law dynamical systems, to appear SIAM J. Appl. Math., 2012
  2. Casian Pantea, On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44(3), 2012
  3. David Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71(4):1487--1508, 2011