Abstract
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.
Highlights
The predator–prey model is one of the basic models in the theoretical studies of ecology, and it has been studied extensively
Xiao et al [18] considered the global stability of a stage structure predator–prey system with prey refuge, and pointed out that the model undergoes a Hopf bifurcation when the delay crosses some critical values
We show the influence of the Allee effect, fear effect and prey refuge on the dynamics behavior of the system (5)
Summary
The predator–prey model is one of the basic models in the theoretical studies of ecology, and it has been studied extensively (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and the references cited therein). Chen et al [15] investigated the stable property of a predator–prey system with a constant number of prey refuges. The authors [16] showed that prey refuge has no influence on the stability of the system. Khajanchi and Banerjee [17] studied the uniform persistence and global asymptotic stability of a stage structured predator–prey model with prey refuge. Xiao et al [18] considered the global stability of a stage structure predator–prey system with prey refuge, and pointed out that the model undergoes a Hopf bifurcation when the delay crosses some critical values. Xie et al [19] studied the persistence and stability of a modified Leslie–Gower predator–prey model with prey refuge, and showed that the prey refuge has a positive effect on the persistence property. For more details in this direction, see [20,21,22,23]
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