具有Allee效应的随机差分方程
Stochastic difference equations with the Allee effect
摘要：
对于截断随机扰动方程$x_{n+1}=\max\{ f(x_n)+lχ_{n+1}, 0 \}$与$f(x)<x$对$(0,m)$，这对应于Allee效应，我们观察到，对于非常小的扰动幅度$l$，最终的行为是类似于一个非扰动的情况下：有灭绝小的初始值在$(0,m-\varepsilon)$和持久性为$x_0 \in (m+δ, H]$的一些$H$满足$H>f(H)>m$。随着幅度的增长，初始值的区间$(m-\varepsilon, m+δ)$出现并扩展，使得以一定的概率，$x_n$维持在$[m, H]$中，并且可能最终以正概率进入区间$(0,m-\varepsilon)$。这些概率的估计值较低。如果$H$足够大，随着扰动幅度的增加，Allee效应消失：对于任何正的初始值，解都存在。
英文摘要：
For a truncated stochastically perturbed equation $x_{n+1}=\max\{ f(x_n)+lχ_{n+1}, 0 \}$ with $f(x)<x$ on $(0,m)$, which corresponds to the Allee effect, we observe that for very small perturbation amplitude $l$, the eventual behavior is similar to a non-perturbed case: there is extinction for small initial values in $(0,m-\varepsilon)$ and persistence for $x_0 \in (m+δ, H]$ for some $H$ satisfying $H>f(H)>m$. As the amplitude grows, an interval $(m-\varepsilon, m+δ)$ of initial values arises and expands, such that with a certain probability, $x_n$ sustains in $[m, H]$, and possibly eventually gets into the interval $(0,m-\varepsilon)$, with a positive probability. Lower estimates for these probabilities are presented. If $H$ is large enough, as the amplitude of perturbations grows, the Allee effect disappears: a solution persists for any positive initial value.
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