Classification for dynamics of Markov chains on non-negative integers with arbitrary transition rates and its application

TL;DR

The paper develops a Lyapunov-based framework to classify the long-term behaviors of one-dimensional continuous-time Markov chains on the non-negative integers with arbitrary bounded jump sizes, using drift and variance $m(x)$ and $v(x)$. It introduces Laurent-type asymptotics for transition rates and shows that the dynamical features—explosivity, recurrence types, and exponential ergodicity—are completely determined by the quartet $(\alpha, \gamma, \vartheta, R)$. The authors provide general criteria in terms of $H_p(x)$ and $J(x)$, and apply them to key kinetic models, including mass-action, Michaelis–Menten, and Haldane kinetics, yielding complete (if-and-only-if) classifications in many cases. They also demonstrate how high-dimensional stochastic reaction networks can be reduced to one-dimensional rational-rate models, preserving dynamical properties and enabling practical analysis through the Laurent-type framework. The results offer a unifying, computable approach for understanding the qualitative behavior of stochastic biochemical systems and their reductions.

Abstract

Continuous-time Markov chains on non-negative integers can be used for modeling biological systems, population dynamics, and queueing models. Qualitative behaviors of birth-and-death models, typical examples of such one-dimensional continuous-time Markov chains, have been substantially studied. For one-dimensional Markov chains with polynomial transition rates, recent studies provided criteria for their long-term behavior. In this paper, we provide a classification of Markov chains on non-negative integers when the transition rates are arbitrary functions. The criteria are written with asymptotics of the transition rates. This classification implies their dynamical properties, including explosivity, recurrence, positive recurrence, and exponential ergodicity. As an application, we derive a complete classification (if and only if conditions) for those dynamical features when the transition rates have certain expansion forms, which include all rational functions, so that our classifications cover mass-action kinetics, Michaelis-Menten kinetics, and Haldane equations. Our classification solely relies on easily computable quantities: the maximal degree of the expansion of the transition rates, the mean and the variance of the transition rates. We demonstrate the utility of this classification framework using the approximation of high-dimensional mass-action systems by a one-dimensional reaction system with general rational kinetics.

Figures (2)

• Figure 1: a, b: $\log(||P(X(t)\in \cdot) - \pi(\cdot)||_{tv})$ for the cases $(V_1,K_1,V_2,K_2)=(1,1,1,1)$ and $(n_1,c_1,n_2,c_2)=(3,2,2,1)$ (a), and $(V_1,K_1,V_2,K_2)=(3,1,2,3)$ and $(n_1,c_1,n_2,c_2)=(4,2,1,1)$ (b) of \ref{['eq : birth-death-mm']} with different initial values $A(0)$. While $\log(||P(X(t)\in \cdot) - \pi(\cdot)||_{tv})$ decays with almost identical slopes in the left plot across initial values, the decay rates differ in the right plot, indicating exponential and non-exponential ergodicity, respectively. Each curve was obtained from 50,000 simulated trajectories using the Gillespie algorithm Gillespie77. The plateau in each plot emerges due to Monte-Carlo errors.
• Figure 2: a, b: The mean (a) and variance (b) of $X$ in the original model \ref{['ex:reduction row 1']}–\ref{['ex:reduction row 2']} and in the approximation \ref{['ex:reduced']}, with $U=10^2$ and $V=10^3$ based on $10^4$ trajectories. c: Sample trajectories of $1/(X_A(t)(X_A(t)-1)+1)$ and $X_B(t)/V$ in \ref{['ex:reduction row 1']}–\ref{['ex:reduction row 2']}, showing that $X_B(t)/V$ rapidly converges to $1/(X_A(X_A-1)+1)$ after each jump of $X_A$.

Theorems & Definitions (45)

• Theorem 2.1: Theorem 1, karlin1957classification
• Remark 2.1
• Remark 3.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Proposition 3.1
• Corollary 3.2
• Theorem 4.1