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Open AccessArticle

Logistic Growth Described by Birth-Death and Diffusion Processes

Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano, Italy
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Mathematics 2019, 7(6), 489; https://doi.org/10.3390/math7060489
Received: 3 May 2019 / Revised: 21 May 2019 / Accepted: 22 May 2019 / Published: 28 May 2019
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. View Full-Text
Keywords: logistic model; birth-death process; first-passage-time problem; transition probabilities; Fano factor; coefficient of variation; diffusion processes; Itô equation; Stratonovich equation; diffusion in a potential logistic model; birth-death process; first-passage-time problem; transition probabilities; Fano factor; coefficient of variation; diffusion processes; Itô equation; Stratonovich equation; diffusion in a potential
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Di Crescenzo, A.; Paraggio, P. Logistic Growth Described by Birth-Death and Diffusion Processes. Mathematics 2019, 7, 489.

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