Dynamics in a Fractional-Order Four-Species Food Web System with Top Predator and Delays

The predator–prey model is a fundamental mathematical tool in ecology used to understand the dynamic relationship between predator and prey populations. This study develops a fractional-order delayed dynamical model for a four-species food web, which includes an intermediate predator feeding on two prey species and a top predator preying on all three species. The boundedness of the system’s solutions is first rigorously established using the Laplace transform method. Next, a nonlinear dynamical analysis is performed to determine the existence conditions and local stability of both the trivial and positive equilibrium points. In particular, by treating the time delay as a bifurcation control parameter, explicit criteria for the onset of Hopf bifurcation are derived. Theoretically, when the delay magnitude exceeds a critical threshold, the system loses stability and exhibits sustained oscillatory behavior. Finally, systematic numerical simulations are performed under specific parameter settings. The effects of varying fractional orders and delay magnitudes on the system’s dynamics are quantitatively explored, and the results show strong agreement with the theoretical predictions.