Stability scenarios and period‐doubling onset of chaos in a population model with delayed harvesting

Abstract

In this work, we investigate complete stability behaviors of the Rosenzweig–MacArthur predator–prey model with delayed harvesting of the prey. The unharvested system exhibits either a steady‐state or an oscillatory dynamics for the coexistence of the species. We explore how the delayed harvesting affects the dynamics of these two modes by analyzing the system stability in effort‐delay bi‐parameter plane. Some novel dynamical scenarios and intricate dynamics are obtained. Analytical conditions for different stability scenarios are derived by examining the associated quasi‐polynomial eigenvalue equation. For invariant harvesting effort, the time delay induces four stability scenarios: stability invariance, instability invariance, stability change, and stability switching. On the other hand, the effort instigates five stability scenarios: stability invariance, instability invariance, stability change, instability change, and instability switching, when the delay strength is fixed. Majority of literatures on harvesting reported that harvesting stabilizes predator–prey interactions. However, we will show that the delayed harvesting can destabilize the system. One of the novelties of the study is to unveil the occurrence of effort‐induced chaos via period‐doubling mechanism. Interestingly, the effort‐induced switching phenomena and chaos do not occur for non‐delayed harvesting.