Stability and bifurcation analysis in a discrete plant–herbivore system under proportional harvesting

This paper investigates a discrete-time plant–herbivore system incorporating proportional plant harvesting, which reflects realistic resource management where extraction depends on population size. Unlike existing studies that mainly consider unharvested systems or treat harvesting qualitatively, harvesting is introduced as a key dynamical parameter and analyze its role in shaping system behavior. The existence and local stability of biologically feasible equilibria are established, and it is shown that harvesting can significantly alter system dynamics. Using bifurcation theory, it is demonstrated that harvesting intensity can induce transcritical, period-doubling, and Neimark–Sacker bifurcations, leading to periodic, quasi-periodic, and chaotic dynamics. In addition, the model exhibits resonance structures and Arnold tongues, highlighting complex interactions between harvesting and intrinsic population dynamics. A comparative analysis between harvested and non-harvested systems reveals that harvesting can either stabilize or destabilize population dynamics depending on parameter values. These findings provide new insights into the role of harvesting as a control mechanism in ecological systems and offer useful implications for sustainable resource management.