On Time-Changed Birth-Death Processes with Catastrophes

TL;DR

This work extends classical birth–death processes by introducing two time-changed models with catastrophes: one using the inverse of a stable subordinator and the other using the inverse of a tempered stable subordinator. The authors derive fractional differential equations for state probabilities, establish Laplace-transform relations to non-time-changed processes, and characterize catastrophe timings, sojourns, and first-visit events to zero, including effective catastrophes. They further specialize to time-changed linear BDPC and tempered BDPC, obtaining explicit extinction probabilities, moments, and generating functions, and provide simulation algorithms with illustrative plots. The results integrate fractional and tempered dynamics into BDPC frameworks, offering analytical and computational tools for applications in population dynamics, queues, and risk modeling where non-Markovian time changes and catastrophes are relevant.

Abstract

We study two time-changed variants of the birth-death process with catastrophe where the time-changing components are the first hitting times of the stable subordinator and the tempered stable subordinator. For both the processes, we derive the governing system of fractional differential equations for their state probabilities. The Laplace transforms of these state probabilities are obtained in terms of those of the corresponding time-changed birth-death processes without catastrophes. We obtain the distribution of catastrophe occurrence times as well as the sojourn times within non-zero states. We study distributional properties of the first visit time to state zero in a particular case. Also, the first occurrence time of an effective catastrophe is studied. Moreover, we study the time-changed linear birth-death processes with catastrophes, derive the explicit expressions for its state probabilities, expectation and variance. For a specific case, we compare the expectation plots across different parameter values and provide an algorithm for simulating sample paths with illustrative plots.

Figures (4)

• Figure 1: Plots of $\mathbb{E}(N^{\alpha,\nu}(t))$ for $\lambda=3$, $\mu=1$, $\nu=1$, and for different values of $\alpha$.
• Figure 2: Plots of $\mathbb{E}(N^{\alpha,\nu}(t))$ for $\alpha=0.5$, $\lambda=4$, $\mu=1$, and for different values of $\nu$.
• Figure 3: Sample path simulation of time-changed LBDPC. Plot (a) is for parameters $\alpha=1$, $\lambda=15$, $\mu=11$ and $\nu=2$ and Plot (b) is for parameters $\alpha=0.5$, $\lambda=15$, $\mu=11$ and $\nu=2$.
• Figure 4: Sample path simulation of time-changed LBDPC. Plot (a) is for parameters $\alpha=0.3$, $\lambda=10$, $\mu=12$ and $\nu=3$ and Plot (b) is for parameters $\alpha=0.8$, $\lambda=10$, $\mu=12$ and $\nu=3$.

Theorems & Definitions (31)

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• Proposition 3.1
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• Corollary 3.1
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