Hopf and Bogdanov–Takens Bifurcations of a Delayed Bazykin Model

In this work, the Hopf and Bogdanov–Takens bifurcations of a delayed Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response are studied. Sufficient conditions for the existence of Hopf bifurcation are established. In the Bogdanov–Takens bifurcation, the dynamics near the nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. Some numerical simulations, such as the distribution of eigenvalues, the bifurcation diagrams of Hopf and Bogdanov–Takens bifurcations and phase portraits, are given to illustrate the theoretical criteria. The theoretical and numerical simulation results illustrate that there is supercritical Hopf bifurcation and subcritical Bogdanov–Takens bifurcation in this model. We show that, the dynamics of prey and predator are very sensitive to parameters and delay perturbations which can play a great role in controlling and regulating the number of biological populations.