Effect of immigration in a predator-prey system: Stability, bifurcation and chaos

In the present manuscript, a discrete-time predator-prey system with prey immigration is considered. The existence of the possible fixed points of the system and topological classification of coexistence fixed point are analyzed. Moreover, the existence and the direction for both Neimark-Sacker bifurcation and flip bifurcation are investigated by applying bifurcation theory. In order to control chaos due to the emergence of the Neimark-Sacker bifurcation, an OGY feedback control strategy is implemented. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the system, are given to support the accuracy of the analytical finding. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.