Fractional-Order Dynamics and Fear-Induced Bifurcation in Delayed Food Chain Model

This paper investigates the bifurcation problem in a fractional-order delayed food chain model that incorporates a fear effect. We observe that the fractional order significantly impacts the delayed system, influencing its stability in the presence of fear. Both the fractional order and the fear effect play crucial roles in determining the system’s stability. Furthermore, we observe stability switching induced by the fear effect while keeping the delay fixed. We identify the stability condition of the proposed model and precisely establish bifurcation points by utilizing delay as a bifurcation parameter. The system exhibits robust stability performance with smaller control parameters, and Hopf bifurcation arises as the control parameter surpasses a critical value. Additionally, through theoretical analysis and numerical simulations, we investigate the effects of fractional order, the fear effect, and time delay on the system’s stability.