# K-variable kinetics

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".

## References

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