Stochastic difference equations with the Allee effect

Abstract

For a truncated stochastically perturbed equation $x_{n+1}=\max \{f(x_n)+l{\chi }_{n+1},0\}$ 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 $(0,m-\epsilon )$ and persistence for $x_0\in (m+\delta ,H]$ for some $H$ satisfying $H>f(H)>m$. As the amplitude grows, an interval $(m-\epsilon ,m+\delta )$ 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 $(0,m-\epsilon )$, 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.

17 pages, 15 figures, to appear in Dynamics of Continuous and Discrete Systems - Series A