Qualitative Properties and Optimal Control Strategy on a Novel Fractional Three-Species Food Chain Model

In this study, the dynamics of a novel three-species food chain model featuring the Sokol–Howell functional response are explored. The fear of predators is incorporated into prey reproduction, and refuge is integrated into the middle predators within the framework of the Caputo fractional derivative. Theoretical aspects such as the existence and uniqueness of equilibria, their boundedness, and stability analysis are encompassed in the investigation. To examine the existence of chaos, Lyapunov exponents are computed. The optimal control measure concerning the growth of the prey population was considered, and the conditions that must be met for the optimal response to exist in the optimal control issue were determined using Pontryagin’s Maximum Principle. The theoretical outcomes were validated by using numerical simulation powered by the Adams–Bashforth–Moulton type predictor-corrector technique. Numerical justifications are provided for the influences of fear and refuge factors. When fear is absent, a numerical analysis is conducted on the global stability of the system for fractional order derivative.