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

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).