Stability and bifurcation in a single species logistic model with additive Allee effect and feedback control

Yangyang Lv,Lijuan Chen,Fengde Chen

doi:10.1186/s13662-020-02586-0

Abstract

In this paper, we propose a single species logistic model with feedback control and additive Allee effect in the growth of species. The basic aim of the paper is to discuss how the additive Allee effect and feedback control influence the above model’s dynamical behaviors. Firstly, the existence and stability of equilibria are discussed under three different cases, i.e., weak Allee effect, strong Allee effect, and the critical case. Secondly, we prove the occurrence of saddle-node bifurcation and transcritical bifurcation with the help of Sotomayor’s theorem. The above dynamical behaviors are richer and more complex than those in the traditional logistic model with feedback control. We find that both Allee effect and feedback control can increase the species’ extinction property. We also reveal some new bifurcation phenomena which do not exist in the single-species model with feedback control (Fan and Wang in Nonlinear Anal., Real World Appl. 11(4):2686–2697, 2010 and Lin in Adv. Differ. Equ. 2018:190, 2018).

Highlights

IntroductionEspecially the logistic model, the dynamical behaviors, such as the stability, persistence, permanence, extinction, and existence of positive periodic solutions, have been extensively studied during the last decades
For single species model, especially the logistic model, the dynamical behaviors, such as the stability, persistence, permanence, extinction, and existence of positive periodic solutions, have been extensively studied during the last decades
2.2 Stability of the equilibria E0, E1, E2, and E3 Firstly, we investigate the stability of the trivial equilibrium E0

Summary

Introduction

Especially the logistic model, the dynamical behaviors, such as the stability, persistence, permanence, extinction, and existence of positive periodic solutions, have been extensively studied during the last decades. Lin [2] proposed a single species logistic model with Allee effect and feedback control dx x. One can naturally propose the following interesting and meaningful question: How can the additive Allee effect influence the logistic system with feedback control? Does the logistic system with additive Allee effect and feedback control still admit a unique positive equilibrium? In this paper, we shall investigate the following single species logistic model with additive Allee effect and feedback control: dx m. To the best of the authors’ knowledge, this paper is the first work which proposes and investigates the logistic model with additive Allee effect and feedback control.

The existence of positive equilibria

Saddle-node bifurcation

Findings

Conclusion