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

中文翻译：

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人口动力学中的逻辑模型和θ逻辑模型：一般分析和精确结果

随机逻辑模型和θ-逻辑模型在生物学和物理环境中都有许多应用，研究它们的结构具有重要的意义。在本文中，我们为logistic和θ-logistic随机微分方程提供了路径式解的封闭形式，以及它们的概率密度函数和矩的精确表示。我们还模拟了一些典型的样本轨迹，并提供了一些在不同噪声强度下上述闭合公式的数值计算示例：这特别表明，随着随机性的增加，虽然过程变得更加不可预测，但渐近趋于抑制平均物流增长。在得出这些主要结果之前，先讨论这些模型的无噪声确定性版本：