Pattern formations of a delayed diffusive predator-prey model with predator harvesting and prey social behavior

In this research, we are interested in analyzing the behavior of a diffusive predator-prey model in the presence of predator rivalry and prey social behavior. The predator rivalry affects hugely the behavior of predator and the prey, so we are interested in analyzing the complex dynamics generated by the presence of predator harvesting and its influence on the behavior of the solution. For the mathematical purpose, we will concentrate on analyzing the system in the absence of diffusion where the existence of Hopf bifurcation is investigated; also, we proved that the presence of the time lags generates interesting patterns, which it can lead to instability even Hopf bifurcation. The stability of the homogeneous and nonhomogeneous periodic solution generated by the presence of Hopf bifurcation is established using the normal form. The mathematical obtained results are checked numerically using graphical representations.