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 $\theta$-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 $\theta$-logistic equations in a unified context, within which also the Gompertz model emerges from an anomalous scaling.

arXiv comment25 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

References in corpus (9)

• Exponential functionals of Brownian motion, I: Probability laws at fixed time
• Exponential functionals of Brownian motion, II: Some related diffusion processes
• Minimizing the population extinction risk by migration
• Phase diagram for a logistic system under bounded stochasticity
• Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models
• Giant fluctuations in logistic growth
• Statistical mechanics of stochastic growth phenomena
• Stochastic foundations in nonlinear density-regulation growth
• Gompertz and logistic stochastic dynamics: Advances in an ongoing quest