In this paper, a multiple delays stage-structure predator-prey model with refuge and cooperation is established. First, the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed by analyzing the characteristic equations of the system. Second, taking time delays as the bifurcation parameters, the existence of Hopf bifurcation at the positive equilibrium is given. Next, the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed based on the center manifold theorem and normal form theory. Moreover, the optimal harvesting policy of the system is showed by using Pontryagin's maximum principle. Finally, we give the global sensitivity analysis of some parameters by calculating the partial rank correlation coefficients, and some numerical simulations are performed to verify the correctness and feasibility of the theoretical results by using the MATLAB software.

In our paper, a delayed diffusive phytoplankton-zooplankton-fish model with a refuge and Crowley-Martin and Holling II functional responses is established. First, for the model without delay and diffusion, we not only analyze the existence and stability of equilibria, but also discuss the occurrence of Hopf bifurcation by choosing the refuge proportion of phytoplankton as the bifurcation parameter. Then, for the model with delay, we set some sufficient conditions to demonstrate the existence of Hopf bifurcation caused by delay; we also discuss the direction of Hopf bifurcation and the stability of the bifurcation of the periodic solution by using the center manifold and normal form theories. Next, for a reaction-diffusion model with delay, we show the existence and properties of Hopf bifurcation. Finally, we use Matlab software for numerical simulation to prove the previous theoretical results.

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.