Dynamics and bifurcation control of a fractional-order delayed predator-prey model with an omnivore

In this study, we propose a novel fractional delayed predator-prey model that includes an omnivorous species and explore bifurcation control through a state feedback control strategy. We begin by deriving the characteristic polynomial using the Laplace transform and establish new sufficient conditions for stability analysis and Hopf bifurcation, treating the time delay \tau as a bifurcation parameter. To address the Hopf bifurcation in the uncontrolled system, we design a state feedback controller with a time delay. Our results indicate that the time delay \tau significantly affects the onset of the Hopf bifurcation. Additionally, the inclusion of a fractional order ۰<\alpha \leq ۱ enhances solution stability while adding complexity to the dynamics of the model. We find that judicious selection of the feedback gain can delay bifurcation, highlighting the critical role of control effort. To validate our theoretical findings, we present numerical simulations conducted using a modified Adams-Bashforth-Moulton predictor-corrector method. These simulations support our theoretical results and demonstrate the efficacy of our proposed control strategy in managing the dynamical behaviors of the model.

Keywords:

Fractional order , Bifurcation control , Predator–prey model , Time–delay , Hopf Bifurcation