Fear Effect on a Predator-Prey Model with Non-Differential Fractional Functional Response

In this paper, we study the factor of the fear effect in a predator-prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different from the dynamics of a classical predator-prey system. The persistence, the stability and the existence of the steady states are investigated. We examine the Hopf bifurcation at the unique positive equilibrium. Direct Hopf bifurcation is studied via the central manifold theorem. When the value of the fear factor decreases and is less than a threshold ?H, the limit cycle appears, and it disappears through a loop of heteroclinic orbits when the value of the fear factor is equal to a value ?(het).