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.