Hopf Bifurcation and Control of a Fractional-Order Delay Stage Structure Prey-Predator Model with Two Fear Effects and Prey Refuge

Abstract

A generalized delay stage structure prey-predator model with fear effect and prey refuge is considered in this paper via introducing fractional-order and fear effect induced by immature predators. Hopf bifurcation and control of this system are investigated though regarding the delay as the parameter. Firstly, by using the method of linearization and Laplace transform, the roots of the characteristic equation of the linearized system of the original system are discussed, and the sufficient conditions for the system exhibits an unstable state of symmetrical periodic oscillation (Hopf bifurcation) are explored. Secondly, a linear delay feedback controller is added to the system to increase the stability domain successfully. Thirdly, numerical simulations are performed to validate the theoretical analysis, and the various impacts on the dynamical behavior of the system occurring by fear effects, prey refuge, and each fractional-order are illustrated, respectively. Furthermore, the influence of feedback gain on the bifurcation critical point is analyzed. Finally, an analysis based on the results and in-depth research about this system under the biological background is stated in the conclusion.