Dynamics in a Fractional-Order Competitive–Competitive–Cooperative System with Beddington–DeAngelis Functional Responses and Delay

This study investigates the dynamics of a delayed fractional-order competition-competition-cooperative system with Beddington–DeAngelis functional responses. First, we prove the boundedness and uniqueness of the solutions. We analyze the existence conditions and local asymptotic stability of various equilibrium points using the stability theory. Second, by taking the competition time delay τ as the bifurcation parameter, we derive explicit criteria for the stability of the system and the onset of aHopf bifurcation. Once the delay surpasses a critical threshold, the system loses its stability and displays periodic oscillatory behavior. Furthermore, the influence of the fractional order on the system dynamics is also examined. Finally, numerical simulations are performed to verify the theoretical results, providing significant insights into ecosystem complexity.