Stochastic difference equations with the Allee effect

TL;DR

This paper analyzes stochastic difference equations modeling the Allee effect, showing how small perturbations influence extinction and persistence, and how larger perturbations can eliminate the Allee effect altogether.

Contribution

It provides a detailed analysis of how stochastic perturbations affect the Allee effect, including probability estimates and conditions for persistence or extinction.

Findings

Small perturbations lead to extinction or persistence depending on initial values.

Increasing perturbation amplitude creates a probability interval for persistence.

Large perturbations can eliminate the Allee effect, ensuring persistence for all positive initial values.

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 $(0,m-\varepsilon)$ 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.