Dynamical Analysis of a Delayed Fractional-Order Food Chain Model with the Allee Effect and Prey Refuge

In this paper, a delayed fractional-order three-trophic-level food chain model with Allee effect and prey shelter is proposed and analyzed. The model adopts the Caputo fractional derivative to describe the memory effect, and introduces two discrete time delays which represent the gestation and response delays of predators, respectively. We establish the positivity, boundedness, existence, uniqueness and continuity of solutions. By using Jacobian matrix analysis and fractional stability theory, we investigate the equilibrium points and their local stability. The corrected characteristic equation is derived, and sufficient conditions for a Hopf-type stability switch induced by the delays are obtained. In the integer-order case α=1, the critical delay and transversality condition are verified numerically, supporting a classical Hopf-bifurcation conclusion subject to the usual nondegeneracy assumptions. Numerical simulations verify the theoretical results and illustrate the combined effects of time delays, fractional memory, prey shelter and Allee effect on system dynamics. The results show that time delays may destabilize the system and produce persistent oscillatory numerical behaviour, while stronger memory effects, shelter strength and Allee effects enhance system stability and suppress oscillatory behaviors.