Center manifolds theorem for parameterized delay differential equations with applications to zero singularities

The goal of this paper is to develop a center manifold theory for delay differential equations with parameters. As applications, we use the center manifold theorem to establish fold and Bogdanov–Takens bifurcations. In particular, we obtain the versal unfoldings of delayed predator–prey systems with predator harvesting at the Bogdanov–Takens singularity.