### Mathematical Biosciences and Engineering

2019, Issue 4: 2118-2140. doi: 10.3934/mbe.2019103
Research article Special Issues

# Results on stochastic reaction networks with non-mass action kinetics

• Received: 12 December 2018 Accepted: 18 February 2019 Published: 12 March 2019
• In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons [1]. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in [1], and proved it is a well known Lyapunov function [2]. In 2016, Cappelletti and Wiuf showed the converse of the main result in [1]: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced [3]. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in [1] are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.

Citation: David F. Anderson, Tung D. Nguyen. Results on stochastic reaction networks with non-mass action kinetics[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2118-2140. doi: 10.3934/mbe.2019103

### Related Papers:

• In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons [1]. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in [1], and proved it is a well known Lyapunov function [2]. In 2016, Cappelletti and Wiuf showed the converse of the main result in [1]: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced [3]. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in [1] are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.

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