A Predator-Prey Model: Understanding the Role of Fear Effect

In this paper, we delve into a predator-prey model that includes fear effect, particularly focusing on the impact of fear on the predation rate of medium-sized predators. First, the existence and type of equilibria of the model are discussed. Then, by using bifurcation theory, we find that the model will experience saddle-node bifurcation, Hopf bifurcation, degenerate Hopf, double limit cycle, Bogdanov-Takens bifurcation, saddle-node bifurcation of limit cycles and saddle-node homoclinic bifurcation. The analysis results show that as the level of fear increases, the oscillation amplitude of the model gradually decreases, which actually enhances the stability of the ecosystem. However, when the level of fear is too high, the dynamic behavior of the model will undergo significant changes, and may even lead to the extinction of medium-sized predator populations, which poses a threat to the stability of the ecosystem. This suggests that fear can effectively control the density of medium predators and prevent the excessive growth of species below the food chain, thus maintaining the balance of the ecosystem.