Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications

In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhaes to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie-Gower predator-prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.