人口统计随机性与环境随机性下的种群动力学
Population dynamics under demographic and environmental stochasticity
AI总结：
本文研究同时受人口统计与环境随机性影响的单物种种群扩散，证明外部 Lyapunov 指数决定准平稳分布极限、灭绝时间尺度及多尺度灭绝行为。
摘要：
本文致力于研究模拟同时经历人口统计随机性和环境随机性的单一物种种群的扩散过程的长期动力学。在我们的设定中，不存在人口统计随机性时扩散过程的长期动力学由外部 Lyapunov 指数 $Λ_0$ 的符号决定，具体如下：$Λ_0<0$ 意味着（渐近）灭绝，而 $Λ_0>0$ 意味着收敛到唯一的正平稳分布 $μ_0$。如果对于很小的 $ε>0$（人口统计随机性的强度），系统规模为 $\frac{1}{ε^{2}}$，则人口统计效应将使灭绝时间几乎必然有限。这表明，为了理解其动力学，应当分析该系统的准平稳分布（QSD）$μ_ε$。在温和假设下，QSD 的存在性与唯一性是众所周知的。我们考察当种群规模趋于无穷大，即 $ε\to 0$ 时会发生什么。我们证明外部 Lyapunov 指数仍起关键作用：1）若 $Λ_0<0$，则 $μ_ε\to δ_0$，平均灭绝时间为 $|\ln ε|$ 量级，且与 QSD $μ_ε$ 相关的灭绝率具有 $\frac{1}{|\lnε|}$ 量级的下界；2）若 $Λ_0>0$，则 $μ_ε\to μ_0$，平均灭绝时间是 $\frac{1}{ε^{2}}$ 的多项式，灭绝率是 $ε^{2}$ 的多项式。此外，当 $Λ_0>0$ 时，我们能够证明系统表现出多尺度动力学：过程首先快速趋近 QSD $μ_ε$，随后在那里停留多项式长度的时间，再弛豫到灭绝状态。我们给出了在 $μ_ε$ 附近所停留时间关于 $ε$ 的尖锐渐近。
英文摘要：
The present paper is devoted to the study of the long term dynamics of diffusion processes modelling a single species that experiences both demographic and environmental stochasticity. In our setting, the long term dynamics of the diffusion process in the absence of demographic stochasticity is determined by the sign of $Λ_0$, the external Lyapunov exponent, as follows: $Λ_0<0$ implies (asymptotic) extinction and $Λ_0>0$ implies convergence to a unique positive stationary distribution $μ_0$. If the system is of size $\frac{1}{ε^{2}}$ for small $ε>0$ (the intensity of demographic stochasticity), demographic effects will make the extinction time finite almost surely. This suggests that to understand the dynamics one should analyze the quasi-stationary distribution (QSD) $μ_ε$ of the system. The existence and uniqueness of the QSD is well-known under mild assumptions. We look at what happens when the population size is sent to infinity, i.e., when $ε\to 0$. We show that the external Lyapunov exponent still plays a key role: 1) If $Λ_0<0$, then $μ_ε\to δ_0$, the mean extinction time is of order $|\ln ε|$ and the extinction rate associated with the QSD $μ_ε$ has a lower bound of order $\frac{1}{|\lnε|}$; 2) If $Λ_0>0$, then $μ_ε\to μ_0$, the mean extinction time is polynomial in $\frac{1}{ε^{2}}$ and the extinction rate is polynomial in $ε^{2}$. Furthermore, when $Λ_0>0$ we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD $μ_ε$ and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in $ε$ for the time spent close to $μ_ε$.
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