Hopf bifurcation and hybrid control in a delayed predator-prey model

In this paper, we considered a predator-prey model with self-feedback delay to investigate its Hopf bifurcation and control. First, non-negativity, boundedness, and the existence and uniqueness of solutions were discussed. Next, the conditions of the existence of the unique positive equilibrium were analyzed. Then, by using delay as the bifurcation parameter, the local stability of the positive equilibrium, and the existence and characteristics of Hopf bifurcation were investigated. Furthermore, by designing a hybrid controller integrating linear time-delayed state feedback and parametric regulation, the local asymptotic stability at the positive equilibrium was achieved within the target delay interval. Finally, numerical simulations verified the theoretical results. From the perspectives of unknown and known control gains, the stability switching mechanism and the maximum stable delay interval were explored. The results demonstrated that the proposed hybrid controller maintains system stability over a wider delay range and eliminates population extinction risks, providing a theoretical basis for stability regulation in ecosystem management.