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 ...我们考虑一类通用的自催化反应体系，该体系在部分参数区间内已被证实存在随机切换行为，即离散诱导转变（Discreteness Induced Transitions）。该行为出现的条件为以下二者之一：一是体系内总物种数较低；二是物种的流入、流出速率远小于自催化反应的速率。针对该类体系的长期行为，Bibbona、Kim与Wiuf于2020年发表于《英国皇家学会界面期刊（J. Roy. Soc. Interface）》的研究中，给出了对称情形下稳态分布的解析表达式。本研究聚焦于非对称自催化反应场景，并给出了该模型近似稳态分布的表达式。我们证实，该近似稳态分布会呈现出不同的性质，对应三类参数区间内体系过程的不同行为模式；在离散诱导转变区间内，该表达式可给出体系在每个稳定点上的停留时间占比。 本数据集基于针对化学反应马尔可夫链模型的吉莱斯皮（Gillespie）随机算法生成的模拟数据，该算法的详细说明可参见维基百科页面：https://en.wikipedia.org/wiki/Gillespie_algorithm 本次提交的存档文件“GillespieSSa.nb”是由沃尔夫勒姆研究公司（Wolfram Research）开发的Mathematica软件编写的该算法基础代码结构化文档，Mathematica是一款用于数学与科学计算的专业应用软件。 关于此类文件的打开方法，可查阅Mathematica软件的官方文档：https://reference.wolfram.com/workbench/index.jsp?topic=/com.wolfram.eclipse.help/html/tasks/notebook/edit.html “若需在Mathematica源代码编辑器中打开笔记本文件，请选中该文件并右键单击，选择「打开方式 > Mathematica源代码编辑器」。文件打开后将显示在编辑器面板中，此时即可对笔记本的文本形式进行操作。” # 非对称自催化反应及其稳态分布 https://doi.org/10.5061/dryad.kh18932g1 # 数据与文件结构说明 本研究使用随机模拟算法生成的模拟数据，我们通过Mathematica代码实现了针对连续马尔可夫链建模的化学反应网络的标准吉莱斯皮（Gillespie）算法。 ## 代码与软件 本文件包含了本研究中用于生成数据的模拟算法的完整Mathematica代码。本次提交的存档文件“GillespieSSa.nb”是由沃尔夫勒姆研究公司（Wolfram Research）开发的Mathematica软件生成的结构化文档，该软件专为数学与科学计算设计。 关于此类文件的打开方法，可查阅Mathematica软件的官方文档：https://reference.wolfram.com/workbench/index.jsp?topic=/com.wolfram.eclipse.help/html/tasks/notebook/edit.html “若需在Mathematica源代码编辑器中打开笔记本文件，请……”