In recent decades, there have been many studies on Hopf bifurcation and population stability with time delay. However, the stability and Hopf bifurcation of fractional-order population systems with time delay are lower. In this paper, we discuss the dynamic behavior of a fractional-order three-population model with pregnancy delay using Laplace transform of fractional differential equations, stability and bifurcation theory, and MATLAB software. The specific conditions of local asymptotic stability and Hopf bifurcation for fractional-order time-delay systems are determined. A fractional-order proportional–integral–derivative (PID) controller is applied to the three-population food chain system for the first time. The convergent speed and vibration amplitude of the system can be changed by PID control. For example, after fixing the values of the integral control gain ki and the differential control gain kd, the amplitude of the system decreases and the convergence speed changes as the proportional control gain kp decreases. The effectiveness of the PID control strategy in complex ecosystem is proved. The numerical simulation results are in good agreement with the theoretical analysis. The research in this paper has potential application values concerning the management of complex population systems. The bifurcation theory of fractional-order time-delay systems is also enriched.

Complex Dynamics and PID Control Strategies for a Fractional Three-Population Model