Stochastic difference equations with the Allee effect

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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