Bifurcation and control of a delayed Leslie-Gower fractional order predator-prey model with fear effect and prey refuge

In this paper, a delayed Leslie-Gower fractional order predator-prey model with prey refuge, fear effect, and Holling-II functional response is established. First, the nonnegativity, boundedness, and uniqueness of solutions are studied. Then, the stability of equilibriums of such a model without delay are discussed. Next, the existence and stability of all equilibriums of such a model with delay are given, some conditions to ensure the occurrence of Hopf bifurcation are obtained by choosing time delay as a bifurcation parameter, and the normal form of Hopf bifurcation is calculated to get the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. Furthermore, two different kinds of feedback controllers are considered to control Hopf bifurcation. Finally, numerical simulations are showed to validate the theoretical results.