Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

A delayed diffusive predator-prey system with nonmonotonic functional response subject to Neumann boundary conditions is introduced in this paper. First, we analyze the associated characteristic equation to research the conditions for local stability of the positive equilibrium point and the occurrence of Turing instability induced by diffusion in the absence of delay. Second, we provide conditions for the existence of Hopf bifurcation driven by time delay. By utilizing the normal theory and center manifold theorem, we derive explicit formulas for Hopf bifurcation properties such as direction and stability from the positive equilibrium. Third, a hybrid controller is added to the system. By judiciously adjusting the control parameters, we effectively enhance the stability domain of the system, resulting in a modification of the position of the Hopf bifurcation periodic solutions. Numerical simulations demonstrate the presence of rich dynamical phenomena within the system. Moreover, sensitivity analysis was conducted using Latin hypercube sampling (LHS)/partial rank correlation coefficient (PRCC) to explore the impact of parameter variations on the output of prey and predator populations.