Modeling the Allee effect and fear effect in predator–prey system incorporating a prey refuge

In this paper, we consider a predator–prey model with Allee effect, fear effect and prey refuge. By considering the prey refuge as a parameter, we give the threshold condition for the stability of the system, and prove that the system undergoes a supercritical Hopf bifurcation. We show that increasing the prey refuge or Allee effect can make the dynamical behavior of the system more complicated; the fear effect or Allee effect has no influence on the prey density, but can lead to a decrease of the predator population at positive equilibrium.