RESUMEN
In this paper, we study the dynamic behaviors of a fractional order predator-prey system, in which the prey population has three effects: Allee effect, fear effect, and shelter effect. First, we prove in detail the positivity, existence, uniqueness, and boundedness of the solutions of the model from the perspective of mathematical analysis. Second, the stability of the system is considered by analyzing the stability of all equilibria and possible bifurcations of the system. It is proved that the system undergoes Hopf bifurcation with respect to four important parameters at the positive equilibrium point. Third, through stability analysis of the system, we find that: (i) as long as the initial density of the prey population is small enough, it will enter the attraction region of an extinction equilibrium point, making the system population at risk of extinction; (ii) we can eliminate the limit-cycle to make the system achieve stable coexistence by appropriately increasing the fear level or refuge rate, or reducing the prey natality or the order of fractional order systems; (iii) fractional order system is more stable than integer order systems, when the system has periodic solution, the two species can coexist stably by increasing the fear level or refuge rate appropriately. The threshold of fear level and refuge rate in fractional order systems is smaller than that in integer order systems. Finally, the rationality of the research results is verified by numerical simulation.
RESUMEN
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.