Existence of a unique quasi-stationary distribution in stochastic reaction networks

Hansen, Mads Christian; Wiuf, Carsten

doi:10.1214/20-ejp445

Cited by 10 publications

(11 citation statements)

References 43 publications

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“…A scalable computational sufficient condition for ergodicity of essential SRNs has been proposed, and applied to gene regulatory networks [20]. Similar results have recently been established for ergodicity of quasi-stationary distributions (QSDs) for extinct SRNs [21]. Again, for mass-action SRNs with onedimensional stoichiometric subspace, we provide a sharp criterion for ergodicity as well as quasi-ergodicity (Theorem 4.6), also in terms of the four aforementioned parameters.…”

Section: Introductionsupporting

confidence: 57%

Classification and threshold dynamics of stochastic reaction networks

Wiuf¹,

Xu²

2020

Preprint

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Stochastic reaction networks (SRNs) provide models of many realworld networks. Examples include networks in epidemiology, pharmacology, genetics, ecology, chemistry, and social sciences. Here, we model stochastic reaction networks by continuous time Markov chains (CTMCs) and derive new results on the decomposition of the ambient space N d 0 (with d ≥ 1 the number of species) into communicating classes. In particular, we propose to study (minimal) core networks of an SRN, and show that these characterize the decomposition of the ambient space.Special attention is given to one-dimensional mass-action SRNs (1-d stoichiometric subspace). In terms of (up to) four parameters, we provide sharp checkable criteria for various dynamical properties (including explosivity, recurrence, ergodicity, and the tail asymptotics of stationary or quasi-stationary distributions) of SRNs in the sense of their underlying CTMCs. As a result, we prove that all 1-d endotactic networks are non-explosive, and positive recurrent with an ergodic stationary distribution with Conley-Maxwell-Poisson (CMP)-like tail, provided they are essential. In particular, we prove the recently proposed positive recurrence conjecture in one dimension: Weakly reversible mass-action SRNs with 1-d stoichiometric subspaces are positive recurrent. The proofs of the main results rely on our recent work on CTMCs with polynomial transition rate functions.

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Section: Introductionsupporting

confidence: 57%

Classification and threshold dynamics of stochastic reaction networks

Wiuf¹,

Xu²

2020

Preprint

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“…The classification and description of the stochastic behaviour of CRNs is complex. Many interesting results were investigated, like positive recurrence (Anderson and Kim 2018;Anderson and Nguyen 2020), non-explositivity of complex balanced CRN (Anderson and Kurtz 2018), extinction/absorption events (Johnston et al 2018;Hansen and Carsten 2020), quasi-stationary distributions (Hansen and Carsten 2020) or the classification of states of some stochastic CRNs (Xu et al 2019). However, even in situations where theorems apply, we are far from a complete characterization, see (Anderson and Kim 2018;Anderson and Kurtz 2018;Johnston et al 2018;Hansen and Carsten 2020;Xu et al 2019) for examples.…”

Section: Stochastic Modelmentioning

confidence: 99%

Stationary distributions via decomposition of stochastic reaction networks

Hoessly

2021

J. Math. Biol.

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We examine reaction networks (CRNs) through their associated continuous-time Markov processes. Studying the dynamics of such networks is in general hard, both analytically and by simulation. In particular, stationary distributions of stochastic reaction networks are only known in some cases. We analyze class properties of the underlying continuous-time Markov chain of CRNs under the operation of join and examine conditions such that the form of the stationary distributions of a CRN is derived from the parts of the decomposed CRNs. The conditions can be easily checked in examples and allow recursive application. The theory developed enables sequential decomposition of the Markov processes and calculations of stationary distributions. Since the class of processes expressible through such networks is big and only few assumptions are made, the principle also applies to other stochastic models. We give examples of interest from CRN theory to highlight the decomposition.

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“…The proofs of Proposition 5.4 and 5.5 are provided in the supplement. In the following, we provide asymptotics of the tail distributions for chains in C. To obtain the asymptotics, we first come up with an identity for stationary measures, which was first stated in an alternative form in [24]. Let ω + = max Ω + and ω − = − min Ω − .…”

Section: Stationary Distributionsmentioning

confidence: 99%

Dynamics of continuous time Markov chains with applications

Xu,

Hansen,

Wiuf

2019

Preprint

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This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular within biology. We study the classification of states for general CTMCs on the non-negative integer lattices, by characterizing the set of absorbing states (similarly, trapping, escaping, positive irreducible components and quasi-irreducible components). For CTMCs on non-negative integers with polynomial transition rates, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, positive/null recurrence, implosivity, and existence and non-existence of passage times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained. Moreover, an identity for stationary measures is established and tail asymptotics for stationary distributions is given. A similar identity as well as asymptotics is derived for quasi-stationary distributions. Finally, we apply our results to stochastic reaction networks.

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Existence of a unique quasi-stationary distribution in stochastic reaction networks

Cited by 10 publications

References 43 publications

Classification and threshold dynamics of stochastic reaction networks

Classification and threshold dynamics of stochastic reaction networks

Stationary distributions via decomposition of stochastic reaction networks

Dynamics of continuous time Markov chains with applications

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