On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Crescenzo, Antonio Di; Martinucci, Barbara

doi:10.3390/sym1020201

Cited by 13 publications

(4 citation statements)

References 32 publications

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“…It should be mentioned that closed-form results on bilateral birth-death processes have been obtained in the past only in few solvable cases, such as those in the above mentioned papers, and those given in Di Crescenzo [5], Di Crescenzo and Martinucci [9], Pollett [16].…”

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confidence: 99%

On a bilateral birth-death process with alternating rates

2011

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We consider a bilateral birth-death process characterized by a constant transition rate $\lambda$ \ud from even states and a possibly different transition rate $\mu$ from odd states. We determine \ud the probability generating functions of the even and odd states, the transition probabilities, \ud mean and variance of the process for arbitrary initial state. \ud Some features of the birth-death process confined to the non-negative integers by a reflecting \ud boundary in the zero-state are also analyzed. In particular, making use of a Laplace \ud transform approach we obtain a series form of the transition probability from state 1 to the \ud zero-state

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confidence: 99%

On a bilateral birth-death process with alternating rates

2011

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“…Among the families of continuous-time stochastic processes typically adopted to describe population dynamics a special role is played by birth-death processes. Indeed, despite their apparently simple structure, birth-death processes are suitable to model even complex behaviors, for instance described by bimodal transition probabilities (see [9]). We recall also the recent contribution by Crawford and Suchard [10], where an efficient algorithm for computing their transition probabilities is proposed, with relevant applications in ecology, evolution, and genetics.…”

Section: Analysis Of a Special Time-inhomogeneous Linear Birth-death mentioning

confidence: 99%

Logistic Growth Described by Birth-Death and Diffusion Processes

Crescenzo

Paraggio

2019

Mathematics

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We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

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“…Generally, the transition probabilities of this process exhibits a bimodality. See Hongler and Parthasarathy [23] and Di Crescenzo and Martinucci [13] for the symmetry property and other results.…”

Section: Bilateral Processesmentioning

confidence: 99%

A review on symmetry properties of birth-death processes

Di Crescenzo,

Martinucci

2015

Preprint

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In this paper we review some results on time-homogeneous birth-death processes. Specifically, for truncated birth-death processes with two absorbing or two reflecting endpoints, we recall the necessary and sufficient conditions on the transition rates such that the transition probabilities satisfy a spatial symmetry relation. The latter leads to simple expressions for first-passage-time densities and avoiding transition probabilities. This approach is thus thoroughly extended to the case of bilateral birth-death processes, even in the presence of catastrophes, and to the case of a two-dimensional birth-death process with constant rates.

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On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Cited by 13 publications

References 32 publications

On a bilateral birth-death process with alternating rates

On a bilateral birth-death process with alternating rates

Logistic Growth Described by Birth-Death and Diffusion Processes

A review on symmetry properties of birth-death processes

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