The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge

In this paper, we investigate the dynamics of an improved Leslie-Gower predator–prey model which is characterized by the reduction of the prey growth rate due to fear of the predator (i.e., antipredator behavior). The value of this study lies in two aspects: mathematically, (i) it provides the existence and the stability of the positive equilibrium; (ii) it gives the existence of the Hopf bifurcation and limit cycle; and (iii) it shows the mechanisms of the fear factor and the prey refuge on the level of the positive equilibrium. Biologically, we find that the influence of the fear factor is complex: (i) increasing the level of fear can cause the level of the population density to decrease and the prey to become extinct; (ii) the effect of the cost of fear on the stability of the positive equilibrium is rich and complex: it can either destabilize the stability and benefit the emergency of the periodic behavior or stabilize the system by excluding the existence of periodic solutions; (iii) with a fixed level of fear, the prey refuge is beneficial to the coexistence of the prey and the predator, and with the increase of the level of the prey refuge, the positive equilibrium may change from stable spiral sink to unstable spiral source to stable spiral sink. These results may enrich the dynamics of the predator–prey systems.