Zero singularities of codimension two in a delayed predator-prey diffusion system

In this paper, a delayed predator-prey diffusion system with Neumann boundary condition is addressed. By complex computation and rigorous analysis, it is obtained that the unique positive equilibrium of the system under consideration is a Bogdanov-Takens (B-T) singularity under certain conditions, whereas this is not true for the corresponding system without diffusion. By using the normal form theory and the center manifold reduction theorem for partial functional differential equations (PFDEs), the normal form of the B-T singularity is provided. Finally, numerical simulations are carried out to illustrate the theoretical results presented.