Bifurcation-Driven Oscillation Mode Switching in a Time-Periodic Leslie–Gower Predator–Prey Model

Population cycling is a widespread phenomenon observed in ecosystems, closely associated with both the internal population factors and external environmental factors. This paper focuses on the oscillation modes in a time-periodic Leslie–Gower predator–prey model. We establish the existence of positive periodic solution by using the Manásevich–Mawhin continuation theorem. In addition, we investigate the dynamic behaviors of the periodic solution as the parameter continuously varies and observe the bifurcations of the periodic solutions by numerical bifurcation analysis. It reveals that the system’s internal frequency and external frequency will, respectively, govern the oscillation process, with their interplay demarcated by a Neimark–Sacker bifurcation point. This interaction gives rise to diverse oscillation patterns, such as periodic oscillation and mixed-mode oscillation, and triggers pattern switching behavior within the system.