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Logistic Growth Described by Birth-Death and Diffusion Processes

Author

Listed:
  • Antonio Di Crescenzo

    (Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano, Italy)

  • Paola Paraggio

    (Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano, Italy)

Abstract

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.

Suggested Citation

  • Antonio Di Crescenzo & Paola Paraggio, 2019. "Logistic Growth Described by Birth-Death and Diffusion Processes," Mathematics, MDPI, vol. 7(6), pages 1-28, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:489-:d:235123
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    References listed on IDEAS

    as
    1. Tan, W. Y., 1986. "A stochastic Gompertz birth-death process," Statistics & Probability Letters, Elsevier, vol. 4(1), pages 25-28, January.
    2. Weam Alharbi & Sergei Petrovskii, 2018. "Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics," Mathematics, MDPI, vol. 6(4), pages 1-15, April.
    3. De Lauro, E. & De Martino, S. & De Siena, S. & Giorno, V., 2014. "Stochastic roots of growth phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 207-213.
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    Cited by:

    1. Antonio Di Crescenzo & Paola Paraggio & Serena Spina, 2023. "Stochastic Growth Models for the Spreading of Fake News," Mathematics, MDPI, vol. 11(16), pages 1-23, August.
    2. Ahmed Nafidi & Ghizlane Moutabir & Ramón Gutiérrez-Sánchez, 2019. "Stochastic Brennan–Schwartz Diffusion Process: Statistical Computation and Application," Mathematics, MDPI, vol. 7(11), pages 1-16, November.
    3. Virginia Giorno & Amelia G. Nobile, 2019. "Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions," Mathematics, MDPI, vol. 7(6), pages 1-19, June.
    4. Antonio Di Crescenzo & Paola Paraggio & Patricia Román-Román & Francisco Torres-Ruiz, 2023. "Statistical analysis and first-passage-time applications of a lognormal diffusion process with multi-sigmoidal logistic mean," Statistical Papers, Springer, vol. 64(5), pages 1391-1438, October.

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