Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

Pages: 1605 – 1642



Andrea Agazzi (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Jonathan C. Mattingly (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)


We present three examples of chemical reaction networks whose ordinary differential equation scaling limits are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences and characterize the invariant measures by Lyapunov function techniques. In particular, we design a natural set of such functions which scale homogeneously to infinity, taking advantage of the same scaling behavior of the reaction rates.


jump Markov processes, chemical reaction networks, recurrence

2010 Mathematics Subject Classification

60G52, 60J75, 92C37

Received 19 December 2018

Accepted 20 March 2020

Published 4 November 2020