Bifurcations of a delayed predator-prey system with fear, refuge for prey and additional food for predator

Taking into account the impacts of the fear by predator, anti-predation response, refuge for prey, additional food supplement for predator and the delayed fear induced by the predator, we establish a delayed predator-prey model in this paper. We analyze the persistence and extinction of species and the existence and uniqueness of a coexistence fixed point. Particularly, we investigate the local asymptotic stability of the equilibrium by use of the characteristic equation theory of a variational matrix. Applying the Hopf bifurcation theorem, we investigate and obtain the bifurcation thresholds of the parameters of fear, refuge coefficient, the quality and quantity of additional food and the anti -predation delayed response produced by prey. Finally we give some examples to verify our theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal that these parameters play an important role in the long-term behaviors of species and should be applied correctly to preserve the continuous development of species.

Automated summary

In this paper, a delayed predator-prey model is established, taking into account the impacts of predator fear, anti-predation response, refuge for prey, additional food supplement for predator, and the delayed fear induced by the predator. The persistence and extinction of species, as well as the existence and uniqueness of a coexistence fixed point, are analyzed. The local asymptotic stability of the equilibrium is investigated using the characteristic equation theory of a variational matrix. By applying the Hopf bifurcation theorem, the bifurcation thresholds of various parameters are obtained, including fear, refuge coefficient, quality and quantity of additional food, and anti-predation delayed response produced by prey. Examples are given to verify the theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal the important role of these parameters in the long-term behaviors of species and emphasize the need for their correct application to preserve the continuous development of species.