Logistic and $\theta$-logistic models in population dynamics: General analysis and exact results
Abstract
In the present paper we provide the closed form of the path-like solutions for the logistic and -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 -logistic equations in a unified context, within which also the Gompertz model emerges from an anomalous scaling.
Keywords
Cite
@article{arxiv.2004.10478,
title = {Logistic and $\theta$-logistic models in population dynamics: General analysis and exact results},
author = {Nicola Cufaro Petroni and Salvatore De Martino and Silvio De Siena},
journal= {arXiv preprint arXiv:2004.10478},
year = {2020}
}
Comments
25 pages, 8 figures. The second part of the paper (from p. 10, Section 3.1 on) is substantially improved w.r.t. the old version. As a consequence Title, Abstract, Introduction, Conclusions and References have been accordingly updated