Bifurcation Analysis of a Modified May–Holling–Tanner Predator–Prey Model with Allee Effect

In this paper, we investigate the dynamical behavior of a modified May–Holling–Tanner predator–prey model by considering the Allee effect in the prey and alternative food sources for the predator. The model is analyzed theoretically as well as numerically to determine all local codimension one and two bifurcations. Using the local parametrization method and Hopf bifurcation theorem, we analyze the Hopf bifurcation and compute its corresponding normal form coefficient to reveal its criticality. We specially derive normal form of the system near Bogdanov–Takens and generalized Hopf bifurcations, to determine possible bifurcation scenarios near each bifurcation. For Bogdanov–Takens, we further determine a set of parameters for which curves of saddle-node, Hopf and Homoclinic bifurcations can be observed. By numerical continuation technique, we compute several curves of equilibria and bifurcations, and detect different bifurcation points on these curves. For the computed bifurcation points, we numerically compute their corresponding normal form coefficients. We specially compute a family of limit cycles and limit point cycle bifurcation on this family.