论文标题
人口和环境随机性下的人口动态
Population dynamics under demographic and environmental stochasticity
论文作者
论文摘要
本文致力于研究扩散过程的长期动力学，该过程模拟了一种经历人口统计学和环境随机性的单一物种。在我们的环境中，在没有人口统计学随机性的情况下，扩散过程的长期动态由$λ_0$（外部lyapunov指数）的迹象确定，如下：$λ_0<0 $ inmant <0 $ inmandiment（渐近）灭绝和$λ_0> 0 $ 0 $暗示对唯一正面的正式的常规代理$ $ $ $ $ $μ_0$。如果系统的尺寸为$ \ frac {1} {ε^{2}} $对于小$ $ε> 0 $（人口统计学的强度），人口统计学效果将使灭绝时间几乎可以肯定。这表明，要了解系统应该分析系统的准平台分布（QSD）$μ_ε$。 QSD的存在和独特性在轻度假设下是众所周知的。 我们查看当人口大小发送到无穷大时，即$ε\至0 $时会发生什么。我们表明，外部Lyapunov指数仍然起着关键作用：1）如果$λ_0<0 $，则$μ_ε\ \toΔ_0$，平均灭绝时间是$ | \lnε| $ uss $ | \lnε| $，并且与QSD $μ_ε$相关的灭绝率是$ \μ_ε$的下限$ \ frac $ \ frac $ \ frac $} | | | | | | | | | | | | 2）如果$λ_0> 0 $，则$μ_ε\至μ_0$，平均灭绝时间为$ \ frac {1} {1} {ε^{2}} $，而灭绝率是$ε^{2} $的多项式。此外，当$λ_0> 0 $ $时，我们能够证明该系统表现出多尺度动力学：首先，该过程很快接近QSD $μ_ε$，然后在那里度过多个多条件的时间后，它会放松灭绝状态。我们在接近$μ_ε$上花费的时间以$ε$的价格提供尖锐的渐近技术。
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 $μ_ε$.
