In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker–Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker–Planck equation as growth rate r, carrying capacity K, intensity of Gaussian noise λ, noise intensity σ and stability index α vary. The MET from the interval (0, 1) at the right boundary is finite if λ < 2. For λ > 2, the MET from (0, 1) at this boundary is infinite. A larger stability index α is less likely leading to the extinction of the fish population.

中文翻译：

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具有乘法 α 稳定 Lévy 噪声的随机逻辑增长模型的平均退出时间和逃逸概率

在本文中，我们制定了一个由白噪声和非高斯噪声驱动的随机逻辑鱼生长模型。我们的研究重点是平均灭绝时间、逃逸概率以测量噪声引起的灭绝概率和鱼类种群的 Fokker-Planck 方程X(吨). 在高斯情况下，这些量满足局部偏微分方程，而在非高斯情况下，它们满足非局部偏微分方程。在讨论了存在性、唯一性和稳定性之后，我们计算了这些方程解的数值近似。然后，对于每个噪声模型，我们将平均消光时间的行为和 Fokker-Planck 方程的解作为增长率进行比较r， 承载能力ķ, 高斯噪声强度λ, 噪声强度σ稳定性指数α各不相同。来自区间的 MET(0, 1)如果在右边界处是有限的λ < 2. 为了λ > 2，MET来自(0, 1)在这个边界是无限的。更大的稳定性指数α不太可能导致鱼类种群灭绝。