Impacts of Predation-Driven Allee Effect and Refuge on the Dynamics of a Predator–Prey Model in a Parameter Plane

In population biology, prey species try to maintain a delicate balance between reproduction and safety. The prey species subject to the mate-finding Allee effect may face a tough choice: risking life for more certain mating opportunities or decreasing mating activity to evade predators in the hope that a longer lifespan will compensate the reduced mating opportunities. In this paper, we consider a discrete-time predator–prey model where the logistically growing prey species experiences the mate-finding Allee effect. We assume that as predator density increases, prey individuals become more vigilant, leading to a decrease in their mate searching activity. We also consider that a constant proportion of prey individuals seek refuge from their predators due to the fear of predation risk. Our numerical simulations reveal that refuge has a stabilizing effect on the population dynamics, while for a certain amount of prey refuge, a decrease in the searching efficiency of mating partners due to the fear of predation risk has a destabilizing effect. We vary the parameters related to the predation-driven Allee effect and refuge simultaneously to explore system dynamics in the parameter plane and observe various intricate and striking properties of the system dynamics. The most significant discovery of our study is the novel organization of Arnold tongues in quasiperiodic and chaotic regimes. The organization of Arnold tongues looks like a “wing of a flapping bird”. Furthermore, we observe bistability between coexisting periodic attractors and notice intriguing patterns in the basins of attraction of these attractors. Lastly, we analyze the effect of the parameters related to the predation-driven Allee effect and refuge on the equilibrium densities of both species and also explore the dynamics of the analogous continuous-time system in the parameter plane.