An Investigation of the Discrete-Time Model Subject to Immigration, Harvesting, and Allee Effect

This paper deals with the dynamic behaviors of a discrete-time fractional-order predator–prey model in the presence of both the Allee effect and immigration on the prey population and in the presence of harvesting on the predator population. The existence and uniqueness and parametric conditions for local asymptotic stability of fixed points of the discrete-time fractional-order model are studied. Moreover, using the center manifold theorem and bifurcation theory, it is shown that the considered model undergoes flip and Neimark–Sacker bifurcations in a small neighborhood of the interior fixed point. Then, the direction of bifurcation is calculated. A feedback controller is implemented in the proposed model to control chaos thanks to the emergence of the Neimark–Sacker bifurcation. Furthermore, numerical analysis confirms the theoretical analysis with the help of Matlab software.