Impact of Allee and fear effects in a fractional order prey–predator system incorporating prey refuge

Abstract

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.