Stability of randomly switching stochastic reaction networks with asymptotically linear transition rates

Stochastic reaction networks are mathematical models frequently used in, but not limited to, biochemistry. These models are continuous-time Markov chains whose transition rates depend on certain parameters called rate constants, which despite the name may not be constant in real-world applications. In this paper we study how random switching between different stochastic reaction networks with asymptotically linear rate functions affects the stability of the process. We give matrix conditions for both positive recurrence (indeed, exponentially ergodicity) and transience (indeed, evanescence) in both the regime with high switching rates and the regime with low switching rates. We then make use of these conditions to provide examples of processes whose stability behavior changes as the switching rate varies. We also explore what happens in the middle regime where the switching rates are neither high nor low and our theorems do not apply. Specifically, we show by examples that there can be arbitrarily many phase transitions between exponentially ergodicity and evanescence as the switching rate increases.