An analysis of the stability and bifurcation of a discrete-time predator–prey model with the slow–fast effect on the predator

In many environments, predators have significantly longer lives and meet several generations of prey, or the prey population reproduces rapidly. The slow-fast effect can best describe such predator-prey interactions. The slow-fast effect ε can be considered as the ratio between the predator's linear death rate and the prey's linear growth rate. This paper examines a slow-fast, discrete predator-prey interaction with prey refuge and herd behavior to reveal its complex dynamics. Our methodology employs the eigenvalues of the Jacobian matrix to examine the existence and local stability of fixed points in the model. Through the utilization of bifurcation theory and center manifold theory, it is demonstrated that the system undergoes period-doubling bifurcation and Neimark-Sacker bifurcation at the positive fixed point. The hybrid control method is utilized as a means of controlling the chaotic behavior that arises from these bifurcations. Moreover, numerical simulations are performed to demonstrate that they are consistent with analytical conclusions and to display the complexity of the model. At the interior fixed point, it is shown that the model undergoes a Neimark-Sacker bifurcation for larger values of the slow-fast effect parameter by using the slow-fast effect parameter ε as the bifurcation parameter. This is reasonable since a large ε implies an approximate equality in the predator's death rate and the prey's growth rate, automatically leading to the instability of the positive fixed point due to the slow-fast impact on the predator and the presence of prey refuge.