To investigate the influence of Allee effect on population dynamics, the center theorem and bifurcation theory were used to study the bifurcation of a discrete predator-prey system with Allee effect on prey. The results indicate that an appropriate increase in the growth rate of the prey population will lead to the emergence of predators, the natural enemies of the prey by the transcritical bifurcation; Under the appropriate Allee effect, the system is stable, but if the Allee effect exceeds the critical value the system will undergo a double period flip bifurcation. The bifurcation parameter diagram also shows that as the Allee effect further strengthens, this flip bifurcation will lead to chaos in the system. From a biological perspective, in a certain ecological environment, a species that reproduces to a certain extent will encounter its natural enemies through "natural selection". An appropriate Allee effect is beneficial for the stability of the predator-prey system, but if the Allee effect of the prey is too strong, it will cause a sharp decrease in predators and an effective restriction of no natural enemies. The prey will undergo a brief surge, exceed the environmental capacity, and then die out. To reduce the adverse effects of the bifurcations on the system, the state feedback is used to control the transcritical bifurcation and facilitate hybrid control to control the flip bifurcation. The numerical simulation results show complete consistency with the theoretical analyses.