Stochastic reaction networks (SRNs) are continuous-time Markov chain models widely used in systems biology, where transition rates are classically determined by mass-action kinetics. In many applications, however, the kinetic parameters themselves are not constant: they fluctuate in response to external conditions such as temperature, light, or the state of an upstream signalling pathway. We model this by coupling the SRN to a continuous-time Markov chain, the environment, that switches the reaction rates between finitely many regimes.

In this talk I will first introduce stochastic reaction networks and the random-environment model, and then present a set of parametric sufficient conditions for positive recurrence of the joint (network, environment) process. The conditions apply to networks that are linear in a precise sense, so that to each environmental regime one can associate a coefficient matrix (mild relaxations are given in the paper). Hurwitz stability of the averaged coefficient matrix governs the fast-switching regime, while simultaneous Hurwitz stability of each individual coefficient matrix governs the slow-switching regime; at intermediate switching rates essentially anything can happen. I will close by discussing the main limitation of this framework, namely that the environment acts on the network as a pure catalyst with no feedback, setting up Aidan Howells' subsequent talk, which allows the network to perturb its own environment.