This work is devoted to the study of a stochastic logistic growth model with and without the Allee effect. Such a model describes the evolution of a population under environmental stochastic fluctuations and is in the form of a stochastic differential equation driven by multiplicative Gaussian noise. With the help of the associated Fokker–Planck equation, we analyze the population extinction probability and the probability of reaching a large population size before reaching a small one. We further study the impact of the harvest rate, noise intensity and the Allee effect on population evolution. The analysis and numerical experiments show that if the noise intensity and harvest rate are small, the population grows exponentially, and upon reaching the carrying capacity, the population size fluctuates around it. In the stochastic logistic-harvest model without the Allee effect, when noise intensity becomes small (or goes to zero), the stationary probability density becomes more acute and its maximum point approaches one. However, for large noise intensity and harvest rate, the population size fluctuates wildly and does not grow exponentially to the carrying capacity. So as far as biological meanings are concerned, we must catch at small values of noise intensity and harvest rate. Finally, we discuss the biological implications of our results.

中文翻译：

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乘性噪声下具有等位效应的逻辑收获模型

这项工作致力于研究具有和不具有 Allee 效应的随机逻辑增长模型。这样的模型描述了在环境随机波动下种群的演变，并且是由乘性高斯噪声驱动的随机微分方程的形式。在相关的福克-普朗克方程的帮助下，我们分析了种群灭绝的概率以及在达到小种群之前达到大种群规模的概率。我们进一步研究了收获率、噪音强度和 Allee 效应对种群进化的影响。分析和数值实验表明，如果噪声强度和收获率较小，种群呈指数增长，达到承载能力后种群规模围绕其波动。在没有 Allee 效应的随机逻辑收获模型中，当噪声强度变小（或变为零）时，平稳概率密度变得更加尖锐，其最大值接近 1。然而，对于较大的噪声强度和收获率，种群规模波动很大，并且不会与承载能力成倍增长。因此，就生物学意义而言，我们必须抓住较小的噪声强度和收获率值。最后，我们讨论了我们的结果的生物学意义。人口规模波动很大，不会随着承载能力成倍增长。因此，就生物学意义而言，我们必须抓住较小的噪声强度和收获率值。最后，我们讨论了我们的结果的生物学意义。人口规模波动很大，不会随着承载能力成倍增长。因此，就生物学意义而言，我们必须抓住较小的噪声强度和收获率值。最后，我们讨论了我们的结果的生物学意义。