带乘性α-稳定Lévy噪声的随机Logistic增长模型的平均退出时间和逃逸概率
Mean Exit Time and Escape Probability for the Stochastic Logistic Growth Model with Multiplicative α-Stable Lévy Noise
摘要：
本文建立了一个同时受白噪声和非高斯噪声驱动的随机Logistic鱼类生长模型。本文重点研究了鱼类种群的平均灭绝时间、逃逸概率和Fokker-Planck方程。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。在讨论解的存在性、唯一性和稳定性之后，我们计算了这些方程解的数值逼近。对于每一个噪声模型，我们然后比较的行为的平均灭绝时间和福克-普朗克方程的增长率r，承载能力K，高斯噪声的强度$λ$，噪声强度$σ$和稳定性指数$α$的变化。如果$λ <{\sqrt2}$满足，则右边界处的区间（0，1）的MET是有限的。对于$λ > {\sqrt2}$，在该边界处来自（0，1）的MET是无限的。较大的稳定性指数$α$不太可能导致鱼类种群灭绝。
英文摘要：
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, the 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 $λ <{\sqrt2}$. For $λ > {\sqrt2}$, the MET from (0,1) at this boundary is infinite. A larger stability index $α$ is less likely to lead to the extinction of the fish population.
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