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Stochastic difference equations with the Allee effect

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  • For a 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 $(0,m-\varepsilon)$ and persistence for $x_0 \in (m + \delta, H]$ for some $H$ satisfying $H > f(H)> m$. As the amplitude grows, an interval $(m-\varepsilon, m + \delta)$ 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 $(0,m-\varepsilon)$, 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.
    Mathematics Subject Classification: Primary: 39A50, 37H10; Secondary: 93E10, 92D25.


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