A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes

We construct models of continuous-time Markov chain (CTMC) and Itô stochastic differential equations of population interactions based on a deterministic system of two phytoplankton and one zooplankton populations. The mechanisms of mutual interference among the predator zooplankton and the avoidance of toxin-producing phytoplankton (TPP) by zooplankton are incorporated. Sudden population extinctions occur in the stochastic models that cannot be captured in the deterministic systems. In addition, the effect of periodic toxin production by TPP is lessened when the birth and death of the populations are modeled randomly.

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Stationary distributions via decomposition of stochastic reaction networks

Journal of Mathematical Biology ◽

10.1007/s00285-021-01620-3 ◽

2021 ◽

Vol 82 (7) ◽

Author(s):

Linard Hoessly

Keyword(s):

Markov Chain ◽

Markov Processes ◽

Stochastic Models ◽

Continuous Time ◽

Reaction Networks ◽

Continuous Time Markov Chain ◽

Stationary Distributions ◽

Stochastic Reaction ◽

Stochastic Reaction Networks ◽

Sequential Decomposition

AbstractWe 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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Stochastic Models for Type Counts in a Literary Text

Journal of Applied Probability ◽

10.1017/s0021900200047744 ◽

1975 ◽

Vol 12 (S1) ◽

pp. 313-323

Author(s):

J. Gani

Keyword(s):

Markov Chain ◽

Discrete Time ◽

Stochastic Models ◽

Continuous Time ◽

Markov Chain Model ◽

Probability Generating Function ◽

Literary Text ◽

Homogeneous Markov Chain ◽

Chain Model ◽

Continuous Time Markov Chain

This paper studies a Markov chain model for type counts {Xn} in a literary text. First, a homogeneous Markov chain in discrete time is considered. This is then embedded in a continuous time Poisson process; the probability generating function for the resulting continuous time Markov chain is obtained. Expectations and variances of type counts are found for different values of the token count and various sizes M of an author's vocabulary; these results are finally tested against known data for three of Shakespeare's plays.

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