Impact of fear effect and prey refuge on a fractional order prey-predator system with Beddington-DeAngelis functional response.

In this article, a fractional-order prey-predator system with Beddington-DeAngelis functional response incorporating two significant factors, namely, dread of predators and prey shelter are proposed and studied. Because the life cycle of prey species is memory, the fractional calculus equation is considered to study the dynamic behavior of the proposed system. The sufficient conditions to ensure the existence and uniqueness of the system solution are found, and the legitimacy and well posedness in the biological sense of the system solution, such as nonnegativity and boundedness, are proved. The stability of all equilibrium points of the system is analyzed by an eigenvalue analysis method, and it is proved that the system generates Hopf bifurcation nearby the coexistence equilibrium with regard to three parameters: the fear coefficient k, the rate of prey shelters p, and the order of fractional derivative q. Compared with the integer derivative, the system dynamics in the situation of fractional derivative is more stable. We observe an interesting phenomenon through the simulation: with the increase in the level of the fear effect, the stability of the positive equilibrium point changes from stable to unstable and then to stable. At this time, there are two Hopf branches nearby the positive equilibrium point with respect to the fear coefficient k, and the system can be in a stable state at very low or high level of the fear effect. In addition, when the order of the fractional differential equation of the system decreases continuously, the stability of the system will change from unstable to stable, especially in the case of low-level fear caused by predators and low rate of prey shelters. Therefore, our findings support the view that the strong memory can promote the stable coexistence of two species in the prey-predator system, while fading memory of species will worsen the stable coexistence of two species in the proposed system.