Delay-induced Bogdanov–Takens bifurcation in a Leslie–Gower predator–prey model with nonmonotonic functional response

• The effects of the hunting delay on the dynamics of the predator–prey model are investigated.
• The procedure for the computation of normal forms of type-II Bogdanov–Takens bifurcation is addressed.
• The dynamical classification near the Bogdanov–Takens bifurcation point is determined.
• The delay-induced homoclinic orbit, heteroclinic orbit and subcritical Hopf bifurcation are obtained.

This work is concerned with the dynamics of a Leslie–Gower predator–prey model with nonmonotonic functional response near the Bogdanov–Takens bifurcation point. By analyzing the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the delay inducing the Bogdanov–Takens bifurcation is obtained. In this case, the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. The bifurcation diagram near the Bogdanov–Takens bifurcation point is drawn according to the obtained normal form. We show that the change of delay can result in heteroclinic orbit, homoclinic orbit and unstable limit cycle.