Slow–Fast Fractional Commensal Model Incorporating the First Species-Dependent Carrying Capacity and Allee Effect Subject to the Second Species

In this paper, the dynamics and bifurcation of a fractional-order commensal model are studied. The existence of the fixed points of the considered model and their topological classifications are analyzed. By varying a key bifurcation parameter, it is shown that the considered model undergoes a flip bifurcation at the unique positive fixed point. The existence conditions for the flip bifurcation, as well as its direction, are established through the application of center manifold theory. The Ott–Grebogi–Yorke chaos control technique is employed to suppress chaotic dynamics and achieve stabilization of the model. Rich dynamical behaviors are illustrated through numerical simulations, including bifurcation diagrams, Lyapunov exponents, and phase portraits to support the analytical findings.