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arXiv · 2004.10478

Logistic and $θ$-logistic models in population dynamics: General analysis and exact results

Nicola Cufaro Petroni·Salvatore De Martino·Silvio De Siena
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Abstract

In the present paper we provide the closed form of the path-like solutions for the logistic and $θ$-logistic stochastic differential equations, along with the exact expressions of both their probability density functions and their moments. We simulate in addition a few typical sample trajectories, and we provide a few examples of numerical computation of the said closed formulas at different noise intensities: this shows in particular that an increasing randomness - while making the process more unpredictable - asymptotically tends to suppress in average the logistic growth. These main results are preceded by a discussion of the noiseless, deterministic versions of these models: a prologue which turns out to be instrumental - on the basis of a few simplified but functional hypotheses - to frame the logistic and $θ$-logistic equations in a unified context, within which also the Gompertz model emerges from an anomalous scaling.

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Nicola Cufaro Petroni, Salvatore De Martino, Silvio De Siena. 2020-05-26. Logistic and $θ$-logistic models in population dynamics: General analysis and exact results. https://doi.org/10.1088/1751-8121%2Fabb277

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