Diffusion effect and stability analysis of a predator–prey system described by a delayed reaction–diffusion equations

In this paper, we consider a delayed reaction–diffusion equations which describes a two-species predator–prey system with diffusion terms and stage structure. By using the linearization method and the method of upper and lower solutions, we study the local and global stability of the constant equilibria, respectively. The results show that the free diffusion of the delayed reaction–diffusion equations has no effect on the populations when the diffusion is too slow; otherwise, the free diffusion has a certain influence on the populations, however, the influence can be eliminated by improving the parameters to satisfy some suitable conditions.