Multi-parameter bifurcations in a discrete Ricker-type predator-prey model with prey immigration

This study examines a discrete-time prey-predator model featuring a Ricker-type growth function and immigration effects to uncover the dynamics shaping ecosystem stability. Through detailed bifurcation analysis, we identify codimension-one bifurcations, including transcritical, period-doubling, and Neimark-Sacker bifurcations, as well as codimension-two bifurcations involving 1:2, 1:3, and 1:4 resonances. Our results reveal that low immigration rates stabilize the system, ensuring predictable population dynamics, while exceeding critical thresholds induces complex behaviors, such as periodic oscillations and chaos. We numerically analyze the dynamics associated with 1:2, 1:3, and 1:4 resonances, utilizing two-parameter bifurcation diagrams and basins of attraction to illustrate the transitions and stability boundaries. These findings highlight the dual role of immigration in stabilizing and destabilizing ecosystems, offering valuable insights for ecological modeling, management, and conservation strategies.