Stability analysis of a single species logistic model with Allee effect and feedback control

Abstract

A single species logistic model with Allee effect and feedback control $$\begin{aligned}& \frac{dx}{dt} = rx(1-x)\frac{x}{\beta+x}-axu, \\& \frac{du}{dt} = -bu+cx, \end{aligned}$$ where β, r, a, b, and c are all positive constants, is for the first time proposed and studied in this paper. We show that, for the system without Allee effect, the system admits a unique positive equilibrium which is globally attractive. However, for the system with Allee effect, if the Allee effect is limited ( $\beta<\frac{b^{2}r^{2}}{ac(ac+br)}$ ), then the system could admit a unique positive equilibrium which is locally asymptotically stable; if the Allee effect is too large ( $\beta>\frac{br}{ac}$ ), the system has no positive equilibrium, which means the extinction of the species. The Allee effect reduces the population density of the species, which increases the extinction property of the species. The Allee effect makes the system “unstable” in the sense that the system could collapse under large perturbation. Numeric simulations are carried out to show the feasibility of the main results.