Dynamics of a delayed predator-prey model with Allee effect and Holling type II functional response

In this paper, a delayed with Holling type II functional response (Beddington-DeAngelis) and Allee effect predator-prey model is considered. The growth of the prey is affected by the parameter $M$, which defines the Allee effect. In addition, the delay $\tau$ also influences the logistic growth of the prey, which can be interpreted as the maturity time or the gestation period. In the study of the characteristic equation, we observe that the delay $\tau$ also depends on the parameter $M$, which affects the dynamics in the prey population. Considering the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. On the other hand, we find that the system can also suffer a Hopf bifurcation in the positive equilibrium when the delay passes through a sequence of critical values. In particular, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, an explicit algorithm is provided applying the normal form theory and center manifold reduction for the functional differential equations. Finally, numerical simulations that support the theoretical analysis are included.