The present article formulates and dissects a single-species model with intraspecific cooperation and stochastic perturbations. Sufficient criteria for extinction and permanence are provided. In addition, the existence of a unique ergodic stationary distribution (UESD) is dissected, and the explicit form of the density function of the UESD is obtained under some conditions. Finally, the theoretical outcomes are used to study the growth of gypsy moth (Lymantria dispar) in North America, and some critical impacts of stochastic perturbations are uncovered and numerically illustrated.

中文翻译：

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种内合作的随机单物种模型分析

本文制定并剖析了具有种内合作和随机扰动的单物种模型。提供了足够的灭绝和永久性标准。此外，剖析了唯一遍历平稳分布（USED）的存在，并在一定条件下得到了UED密度函数的显式形式。最后，将理论成果用于研究北美舞毒蛾 ( Lymantria dispar ) 的生长，揭示随机扰动的一些关键影响并进行数值说明。