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.

In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

In this paper, we consider a fear effect predator–prey model with mutual interference or group defense. For the model with mutual interference, we show the interior equilibrium is globally stable, and the mutual interference can stabilize the predator–prey system. For the model with group defense, we discuss the singular dynamics around the origin and the occurrence of Hopf bifurcation, and find that there is a separatrix curve near the origin such that the orbits above which tend to the origin and the orbits below which tend to limit cycle or the interior equilibrium.