Unveiling multistability and chaos in a discretized predator–prey model incorporating harvesting and refuge: A study of Arnold tongues and shrimp-shaped periodic structures

Biological control is essential for maintaining biodiversity and reducing overpredation in ecological communities, for which several major strategies are employed. Among these, harvesting and refuge play vital roles in regulating predator–prey interactions. These measures not only help to control overexploitation of prey by the predators but also support ecological coexistence. In the present study, we apply a semi-discrete technique to construct a discrete model from a continuous predator–prey system, incorporating linear harvesting, constant ratio refuge, and Holling type-II functional response. We analyze the stability of the resulting discrete model and derive the necessary conditions for the emergence of Neimark–Sacker and flip bifurcations. Our objective is to evaluate how ecological interventions, such as harvesting and refuge, affect the intrinsic growth rate of prey, predation rate of predators, and the overall dynamics of the system. For this analysis, we use one-parameter and two-parameter bifurcations (isospike), largest Lyapunov exponent (LLE), and phase space diagrams through numerical simulations to identify and characterize the complex and diverse dynamical behaviors that arise in the system. The results show that when two parameters, such as prey growth rate and predation rate, are changed simultaneously or separately, the system exhibits stable periodic structures such as Arnold tongues and shrimp-shaped structures in the quasi-periodic and chaotic regions, respectively. These patterns are not isolated but often follow specific sequences like Farey sequence and period adding sequences. In particular, within quasi-periodic regions, Farey sequence-like patterns have been observed between adjacent Arnold tongues. These patterns can be utilized to identify and understand nonlinearly nested stable periodic windows that emerge in response to ecological interventions such as harvesting and refuge intensity. The study also reveals that the system transitions into chaotic dynamics via shrimp-shaped period-bubbling cascades, which are the hallmark of complex nonlinear behavior. These ‘shrimp-like’ structures are characterized by nested windows of underlying periodicity within chaotic regimes, which indicates the presence of complex bifurcation patterns and multistability in parameter space. The system exhibits bistability and tristability across multiple attractors, which underlines its inherent multistability. Ecologically, the results suggest that controlled levels of refuge and harvesting can maintain stable coexistence. However, beyond critical thresholds, these interventions can even destabilize the system or lead to the predator’s extinction.