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

Abstract

Stochastic logistic and θ-logistic models have many applications in biological and physical contexts, and investigating their structure is of great relevance. 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.