Analytic Form of the Quasi-stationary Distribution of a Simple Birth-Death Process
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Analytic Form of the Quasi-stationary Distribution of a Simple Birth-Death Process Julian Lee∗ Department of Bioinformatics and Life Science, Soongsil University, Seoul 06978, Korea (Received 5 August 2020; revised 19 August 2020; accepted 19 August 2020) I consider a simple birth-death model with an absorbing state, where the stable fixed point of the corresponding deterministic mean-field dynamics turns into a transient peak of the probability distribution due to the presence of a tiny fluctuation. The model satisfies the detailed-balance condition, enabling one not only to obtain the analytic form of a quasi-stationary distribution, but also to obtain the analytic form of the escape time under the assumption of quasi-stationarity. I argue that the quasi-steady distribution with exponentially decaying normalization is an excellent approximation of the dynamics at late times, especially for small fluctuations. The analytic expressions for the quasi-stationary distribution and the escape time are expected to be more accurate, hence more useful, for systems with larger sizes. Keywords: Stochastic process, Quasi-stationary state, Master equation, Analytic solution, Population dynamics DOI: 10.3938/jkps.77.457
I. INTRODUCTION A stochastic processes is well approximated by using deterministic mean field dynamics, when the magnitude of the stochastic fluctuation is sufficiently small [1]. In particular, the peaks and the valleys are often formed near the stable and the unstable fixed points, respectively. However, when an absorbing state is present, the stationary state is drastically modified even for tiny fluctuation [2–23]. The stable fixed point of the meanfield equation turns into a transient peak of the quasistationary distribution [2, 24–28], and the system eventually falls into the absorbing state. In this work, I consider a simple birth-death model with such a behavior, where the analytic form of the quasi-stationary distribution can be obtained under the assumption that the local equilibration between nonabsorbing states is much faster than the leakage to the absorbing state. In fact, analytic forms of the quasistationary distributions have been obtained for certain models [2] by using the generating function method. What separates the current model from most previous ones is that the current model satisfies the detailedbalance condition, which simplifies the equation for the quasi-stationary distribution to a first-order recursion relation, thereby allowing for a direct solution. The peak of the quasi-stationary distribution is, indeed, found to be formed near the stable fixed point of the deterministic equation, as expected. From the numerical computa∗ E-mail:
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pISSN:0374-4884/eISSN:1976-8524
tion, I find that the distribution of the non-absorbing states approaches the quasi-stationary distribution at early times, and that its shape agrees with the analytic expression. I also find that the shape of the distribution of the non-absorbing states is maintained throughout late times, but its overall normalization deca
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