Hopf bifurcation and controller design for a predator–prey model with double delays

In recent years, establishing time-delay dynamic models to investigate the intrinsic laws and dynamic behavior among different species has become a popular topic. In this study, we establish a new predator–prey model that includes two time delays. The existence, uniqueness, non-negativity, and boundedness of solutions of the constructed time-delay predator–prey system are evidenced through the fixed-point theorem, Lipschitz condition, and inequality techniques. By selecting the time delay as a bifurcation parameter and analyzing the associated characteristic equation of the model, its dynamic properties (including stability and Hopf bifurcation) are investigated. On account of the stability theory and bifurcation theory of time delay differential equations, we obtain the sufficient conditions on the stability and onset of Hopf bifurcation of the 3D predator–prey model with two time delays. The stability domain and time of Hopf bifurcation onset of the time delay predator–prey model are effectively regulated by utilizing the hybrid controller and the time-delay feedback controller. The effect of distinct delays on stabilizing the system and controlling Hopf bifurcation is demonstrated. The rationality of the critical outcomes obtained in this paper is validated by means of numerical simulations.