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

Qifa Lin

doi:10.1186/s13662-018-1647-2

Abstract

A single species logistic model with Allee effect and feedback control \t\t\tdxdt=rx(1−x)xβ+x−axu,dudt=−bu+cx,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\frac{dx}{dt} = rx(1-x)\\frac{x}{\\beta+x}-axu, \\\\& \\frac{du}{dt} = -bu+cx, \\end{aligned}$$ \\end{document} 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.

Highlights

1 Introduction The aim of this paper is to investigate the dynamic behaviors of the following single species logistic model with Allee effect and feedback control: dx x
Since x∗ = br < 1 = x∗∗∗, ac + br Theorems 3.1 and 3.2 show that, for the traditional logistic equation, the feedback control variable only changes the position of the positive equilibrium, and has no influence on the stability property of the positive equilibrium, i.e., the species will coexist in another stable state
At first sight, the Allee effect has no influence on the dynamic behaviors of the single species logistic system

Summary

Introduction

The aim of this paper is to investigate the dynamic behaviors of the following single species logistic model with Allee effect and feedback control: dx x. = –bu + cx, dt where β, r, a, b, and c are all positive constants. Gopalsamy and Weng [1] for the first time proposed the following single species feedback control ecosystem: dn(t) = rn(t) 1 –. Du(t) = –au(t) + bn(t – τ ), dt where a1, a2, c, a, b, and τ are all positive constants. In [1], under the assumption a1 = 0, the authors investigated the stability property of the positive equilibrium. Li and He [3] investigated the Hopf bifurcation of the following single-species food-limited system with feedback control: r(K – u(t))

Objectives

Conclusion