Asymmetric autocatalytic reactions and their stationary distribution

We consider a general class of autocatalytic reactions, that has been shown to display stochastically switching behaviour (Discreteness Induced Transitions) in some parameter regimes. This behaviour was shown to occur when either the overall species count is low, or when the rate of inflow and outflow of species is relatively much smaller than the rate of autocatalytic reactions. The long-term behaviour of this class was analyzed in Bibbona, Kim, Wiuf (J.Roy.Soc.Interface, 2020) with analytic formula for the stationary distribution in the symmetric case. We focus on the case of asymmetric autocatalytic reactions and provide a formula for an approximate stationary distribution of the model. We show this distribution has different properties corresponding to the distinct behaviour of the process in the three parameter regimes, in the Discreteness Induces Transitions regime, the formula provides the fraction of time spent at each of the stable points., The data consists of simulations based on the Gillespie stochastic algorithm for Markov chain models of chemical reactions, the description of which is available on Wikipedia: https://en.wikipedia.org/wiki/Gillespie_algorithm
The deposited file \"GillespieSSa.nb\" is a structured document of basic code for such an algorithm created by Wolfram Research Mathematica, an application used for mathematical and scientific computation.
Instructions for opening such files are available in the documentation of the Mathematica application: https://reference.wolfram.com/workbench/index.jsp?topic=/com.wolfram.eclipse.help/html/tasks/notebook/edit.html
\"To open a notebook file in the Mathematica source editor one should select the notebook and right-click, choosing Open With > Mathematica Source Editor. When the notebook has opened it should appear in the editor pane. At this stage one can work with the textual form of the notebook.
 , , # Asymmetric autocatalytic reactions and their stationary distribution

[https://doi.org/10.5061/dryad.kh18932g1](https://doi.org/10.5061/dryad.kh18932g1)

# Description of the data and file structure

The paper uses simulated data generated by a stochastic simulation algorithm. We used Mathematica code to implement a standard Gillespie algorithm for the chemical reaction network modeled by a continuous Markov chain.

## Code/Software

The file includes the complete Mathematica code for the simulation algorithm we used to generate data in our paper. The deposited file GillespieSSa.nb is a structured document created by Wolfram Research Mathematica, an application used for mathematical and scientific computation.

Instructions for opening such files are available in the documentation of the Mathematica application: https://reference.wolfram.com/workbench/index.jsp?topic=/com.wolfram.eclipse.help/html/tasks/notebook/edit.html

\"To open a notebook file in the Mathematica source editor one ...