A Study on Complex Dynamics of Four-Species Fractional Food Web Model With Chaos and Bifurcation

In this research, we have examined a food web model to explore the impact of fear effects and harvesting of apex predators on dynamic behaviors. We have utilized Caputo and Caputo–Fabrizio (C–F) fractional derivatives to analyze the model's complexity and dynamics. We have investigated the existence and uniqueness of solutions through fixed point theory and evaluated the system's stability utilizing the Ulam–Hyers (U–H) criterion. We also employed numerical schemes utilizing the Caputo and C–F fractional operators to deepen our understanding of the food web model. The system analysis demonstrates chaotic dynamics, attractors, and periodic windows. Additionally, we have investigated the numerical solution using matlab. We have examined the time series graph while varying the values of the parameters and provided better, more intricate information about the solution of the food web model. We have analyzed the Hopf bifurcation by varying different parameters. Significantly, we have analyzed that the fear effect can mitigate variations in population density, aiding in stabilizing the system.