Arnold tongues, shrimp structures, multistability, and ecological paradoxes in a discrete-time predator-prey system

This paper explores a discrete-time system derived from the well-known continuous-time Rosenzweig-MacArthur model using the piecewise constant argument. Examining the impact of increasing carrying capacity and harvesting efforts, we uncover intricate phenomena, such as periodicity, quasiperiodicity, period-doubling, period-bubbling, and chaos. Our analysis reveals that increasing the carrying capacity of prey species can lead to both system stabilization and destabilization. We delve into normal forms associated with different bifurcation types, accompanied by numerical examples, observing multistabilities with intricate basin structures. Bistable, tristable, and quadruple attractors characterize the model's multistable states. Additionally, we find that enriching prey species negatively affects predator abundance, and increasing carrying capacity can lead to a sudden jump in predator population to the brink of extinction. Examining the two-parameter space of predator and prey harvesting efforts, we identify organized periodic structures: Arnold tongues and shrimp-like structures within quasiperiodic and chaotic regions. Arnold tongues exhibit a sequence of periodic adding. The shrimp structures indicate the existence of period-doubling and period-bubbling phenomena. Discussions on ecological interpretations of predator harvesting, including the paradoxical hydra effect, are provided.