Dynamics of a discrete-time predator-prey system with fear effect, prey refuge, harvesting and additional food for predators

In this paper, we study the complex dynamics of a discrete-time predator-prey system of prey, middle predator, and top predator with Crowley-Martin response function by incorporating fear effect, prey refuge, and predator’s harvesting. Moreover, a constant amount of additional food will be delivered to top predators, and middle predators will benefit from this excess food. Firstly, existence and local stability of different types of fixed points are discussed. Further, a special attention is given to the permanence result of the system. Then, we have shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation at the interior fixed point. We also calculate the Feigenbaum’s constant of the system. In addition, chaos in the sense of Marotto is proved under some appropriate parameter conditions. Particularly, we use 0-1 test to detect chaotic behaviors in the system. It is shown that fear effect and prey refuge have the ability to stabilize the system. It is also found that introducing additional food and harvesting on the top predator may help to keep the system in check. Finally, numerical simulations are performed not only to validate theoretical findings but also to show the rich dynamics of the three-species food chain system.